I applied to the Projective Geometry course in Master Math in my hometown University ( I wont mention it again, I am totally ashamed) .
It is a total disaster regarding quality and intensity as Euler might have rightly suspected. Unfortunately, I can not go and study elsewhere since i work and live where I work and live. Anyway , I will go to these lectures since I need some structure behind what I try to learn. But:
Is there anywhere on the Internet decent distance learning possibilities of projective geometry, but absolutely with video lectures? Without video I guess I wont be able to study systematically. Books /written courses are not enough- I must see and hear.
I would be very grateful for such information. Of course I will do my own web search, but in case anyone knows. Introductory/bachelor level in a decent university would be enough.
Thanks.
QUOTE (Ivars+Sep 1 2008, 03:29 PM)
It is a total disaster regarding quality and intensity as Euler might have rightly suspected.
Well at least you learned something.
Well at least you learned something.
QUOTE (Ivars+Sep 1 2008, 03:29 PM)
Is there anywhere on the Internet decent distance learning possibilities of projective geometry, but absolutely with video lectures?
You mean you want to do as little work as possible and have someone hand you information on a plate. If you genuinely want to learn about projective geometry, you will have to do some reading (like everyone else). There are some notes here, but I strongly suspect you lack the necessary prerequisites.
You mean you want to do as little work as possible and have someone hand you information on a plate. If you genuinely want to learn about projective geometry, you will have to do some reading (like everyone else). There are some notes here, but I strongly suspect you lack the necessary prerequisites.
Euler, good link, thank you.
I strongly suspect I don't have the requisites, either, but some of my members do!
I strongly suspect I don't have the requisites, either, but some of my members do!
QUOTE (Euler+Sep 1 2008, 05:32 PM)
You mean you want to do as little work as possible and have someone hand you information on a plate. If you genuinely want to learn about projective geometry, you will have to do some reading (like everyone else). There are some notes here, but I strongly suspect you lack the necessary prerequisites.
Hi Euler,
Thanks for link, it starts easy , but require to much algebraic prerequisites from me. I am looking for something more geometrical, introductionary ala descriptive/synthetic geometry- especially F.Klein style, late 19th century style, I like all his texts on geometry.
I know myself well enough, I need lectures not because I do not want to read but to have external structure superimposed on me. I read too much of different authors ( I have perhaps 5 proj. geomery books (Coxeter, Edwards, collections of articles, Cayley, Clifford) and some 20 downloaded books-Coxeter, Veblen, Coolidge, Yaglom, Bolayi, Lobachevski, Gauss, Reyes, more modern etc. ) and have difficulties to decide which and when to follow, in the end after e.g. 1-2-3 months there is no real progress.
Knowing myself I guess that is what lectures is for. Structure and accountability. That is the reason I will continue to go to the ones I criticised because they give structure to my learning.
Hi Euler,
Thanks for link, it starts easy , but require to much algebraic prerequisites from me. I am looking for something more geometrical, introductionary ala descriptive/synthetic geometry- especially F.Klein style, late 19th century style, I like all his texts on geometry.
I know myself well enough, I need lectures not because I do not want to read but to have external structure superimposed on me. I read too much of different authors ( I have perhaps 5 proj. geomery books (Coxeter, Edwards, collections of articles, Cayley, Clifford) and some 20 downloaded books-Coxeter, Veblen, Coolidge, Yaglom, Bolayi, Lobachevski, Gauss, Reyes, more modern etc. ) and have difficulties to decide which and when to follow, in the end after e.g. 1-2-3 months there is no real progress.
Knowing myself I guess that is what lectures is for. Structure and accountability. That is the reason I will continue to go to the ones I criticised because they give structure to my learning.
If you don't want to do the work, don't be surprised that you wont learn anything about projective geometry. This is all a bit predictable...
I do not know what is Your problem with lectures. I found MIT physics video lectures on web, they are just fabulous! 1 hour and so many things immedeately clear and well remembered. Now I see what we miss here in Latvia. Decent universities. You may be happy to have one (or few) in UK.
Yes, your problem is you're Latvian, not that you're a lazy ignorant fool who doesn't know how to put in any effort to learning. 
Ok,
I learned first piece of understanding in geometry, hopefully correct.
Affine coordinates (and transformations) in e.g. plane are only a subcase of more general homogeneous coordinates ( and projective transformations). Affine coordinate systems are obtained from triangle ones when one of the lines forming triangle coordinates is sent to infinity.
Similarly , in case of 3D, affine coordinate systems are special case of tetrahedral coordinate system, with one plane forming tetrahedron sent to infinity.
Affine coordinates imply that 4th coordinate of position in 3D space is measured relative to plane at infinity.
Affine coordinates are only good for limited portions of space, or differential geometry, as are affine transformations.
In space in its totality, the only valid geometry is complex projective (descriptive) one in its totality. (perhaps infinite dimensional a well to allow for continuity, expansion in power series etc.?). As I have understood , complex projective space lies in the foundations of quantum mechanics.
So for understanding of possible movements and transformations of space in its totality, triangular ( tetrahedral) coordinates are always to be preferred over affine. Triangular because triangle is the simplest figure in plane formed from straight lines. (as is tetrahedron in space).
I just found the above is not entirely correct, as I have missed out projective coordinates....
I learned first piece of understanding in geometry, hopefully correct.
Affine coordinates (and transformations) in e.g. plane are only a subcase of more general homogeneous coordinates ( and projective transformations). Affine coordinate systems are obtained from triangle ones when one of the lines forming triangle coordinates is sent to infinity.
Similarly , in case of 3D, affine coordinate systems are special case of tetrahedral coordinate system, with one plane forming tetrahedron sent to infinity.
Affine coordinates imply that 4th coordinate of position in 3D space is measured relative to plane at infinity.
Affine coordinates are only good for limited portions of space, or differential geometry, as are affine transformations.
In space in its totality, the only valid geometry is complex projective (descriptive) one in its totality. (perhaps infinite dimensional a well to allow for continuity, expansion in power series etc.?). As I have understood , complex projective space lies in the foundations of quantum mechanics.
So for understanding of possible movements and transformations of space in its totality, triangular ( tetrahedral) coordinates are always to be preferred over affine. Triangular because triangle is the simplest figure in plane formed from straight lines. (as is tetrahedron in space).
I just found the above is not entirely correct, as I have missed out projective coordinates....
Well I found three interesting lectures, finally.
They are not elementary, but at least are not boring. The talk is about osculating curves as solutions of differential equations, which is very very interesting.
Osculating curves video lecture 1
Though it is a Russian site, the lectures is French-Eugene Ghys and speaks in English.
They are not elementary, but at least are not boring. The talk is about osculating curves as solutions of differential equations, which is very very interesting.
Osculating curves video lecture 1
Though it is a Russian site, the lectures is French-Eugene Ghys and speaks in English.
A/N, Euler,
Ivars asked to be referred to some distance learning courses - and you two jump all over him. You´re joint behaviour on this thread is a bloody disgrace and reveals you as nothing but the intolerant, close-minded bullies that we have all come to know. All normal, balanced human beings will recognise your aggressive and arrogant behaviour not as that of people who speak from a position of strength but more that of weak and inadequate cowards.
You have a limited knowledge of some physics and you feel entitled to look down on those who know a fraction less - ever hear the expression "tuppence halfpenny looking down on tuppence"?
Ivars asked to be referred to some distance learning courses - and you two jump all over him. You´re joint behaviour on this thread is a bloody disgrace and reveals you as nothing but the intolerant, close-minded bullies that we have all come to know. All normal, balanced human beings will recognise your aggressive and arrogant behaviour not as that of people who speak from a position of strength but more that of weak and inadequate cowards.
You have a limited knowledge of some physics and you feel entitled to look down on those who know a fraction less - ever hear the expression "tuppence halfpenny looking down on tuppence"?
QUOTE (Edward 3+Apr 6 2009, 07:22 PM)
You´re joint behaviour on this thread...
Seriously, learn the difference between "you're" and "your".
Seriously, learn the difference between "you're" and "your".
QUOTE (Edward 3+Apr 6 2009, 08:22 PM)
Ivars asked to be referred to some distance learning courses - and you two jump all over him.
Past experience says he isn't interested in learning, only giving the impression he wants to learn. It's a common crank behaviour.
Past experience says he isn't interested in learning, only giving the impression he wants to learn. It's a common crank behaviour.
QUOTE (Edward 3+Apr 6 2009, 08:22 PM)
You´re joint behaviour on this thread is a bloody disgrace and reveals you as nothing but the intolerant, close-minded bullies that we have all come to know
You'll find that Euler and I are happy to help people who want to learn, there's plenty of examples of that in our posting histories. But its clear Ivars isn't interested in learning. That's the behaviour of Ivars we've all come to know.
QUOTE (Edward 3+Apr 6 2009, 08:22 PM)
All normal, balanced human beings will recognise your aggressive and arrogant behaviour not as that of people who speak from a position of strength but more that of weak and inadequate cowards.
It's a little silly you attempt to berate us for 'aggressive and arrogant behaviour' after you make claims about knowing some mathematical things better than Fields Medal winner. Or after you attack people for grammar yet, as Euler points out, you struggle with the difference between you're and your.
Furthermore, Euler and I have proven many times that we know a lot more maths and physics than the average person. And me saying that isn't arrogance, it's a statement of fact. The list of people on these forums who have any anything relating to projective geometry is a very short one. Euler and I would both be on that list and as such know a bit of what is required, in terms of reading material and levels of effort, needed to get your name on that list. Ivars isn't even close.
Furthermore, Euler and I have proven many times that we know a lot more maths and physics than the average person. And me saying that isn't arrogance, it's a statement of fact. The list of people on these forums who have any anything relating to projective geometry is a very short one. Euler and I would both be on that list and as such know a bit of what is required, in terms of reading material and levels of effort, needed to get your name on that list. Ivars isn't even close.
QUOTE (Edward 3+Apr 6 2009, 08:22 PM)
You have a limited knowledge of some physics and you feel entitled to look down on those who know a fraction less
I would hardly call the difference between our knowledge and Ivars as 'a fraction'. That implies he's even close to us. He's not. And I'm willing to put that to the test by posting a few homework questions from course I teach to see if Ivars can do them. We all know he won't.
I don't pretend that Euler or I have perfect knowledge of everything. There's a huge gulf between my knowledge/ability and the big names in physics. But that doesn't mean there isn't always a gulf between my knowledge in some areas of physics and what the average person on the street knows. Spend 7 years doing something and its likely you'll know more about it than someone who doesn't do it at all. Or is that too arrogant a statement for you?
I don't look down on Ivars because he knows less physics than I do. I have no problem with people who don't know much physics. My girlfriend failed her GCSE maths exam. Infact, I've never gone out with anyone who does a science based topic! She isn't an 'academic person' but then she holds no delusions to that effect either, she's hard working, honest and a great person. I judge her on those things. I do, however, call a lying, delusion idiot a lying delusional idiot. Ivars knows nothing of projective geometry and isn't interested in learning. If he were rational he'd realise he needs to know a lot of prerequisite topic before hitting projective geometry.
Is it arrogant to make the statement "I've done what you're trying to do and here's my advice...." ? Yet when we suggest Ivars doesn't bite off more than he can chew, we're arrogant. When we say "I can do this" because we've done it before, sat and passed exams on it and in some things even published papers on it, is it arrogant? When you call us arrogant yet think you know more about mathematical analysis than any and all mathematicians ever, despite having never done any maths beyond high school, how come you aren't seeing you're a hypocrite?
I don't pretend that Euler or I have perfect knowledge of everything. There's a huge gulf between my knowledge/ability and the big names in physics. But that doesn't mean there isn't always a gulf between my knowledge in some areas of physics and what the average person on the street knows. Spend 7 years doing something and its likely you'll know more about it than someone who doesn't do it at all. Or is that too arrogant a statement for you?
I don't look down on Ivars because he knows less physics than I do. I have no problem with people who don't know much physics. My girlfriend failed her GCSE maths exam. Infact, I've never gone out with anyone who does a science based topic! She isn't an 'academic person' but then she holds no delusions to that effect either, she's hard working, honest and a great person. I judge her on those things. I do, however, call a lying, delusion idiot a lying delusional idiot. Ivars knows nothing of projective geometry and isn't interested in learning. If he were rational he'd realise he needs to know a lot of prerequisite topic before hitting projective geometry.
Is it arrogant to make the statement "I've done what you're trying to do and here's my advice...." ? Yet when we suggest Ivars doesn't bite off more than he can chew, we're arrogant. When we say "I can do this" because we've done it before, sat and passed exams on it and in some things even published papers on it, is it arrogant? When you call us arrogant yet think you know more about mathematical analysis than any and all mathematicians ever, despite having never done any maths beyond high school, how come you aren't seeing you're a hypocrite?
It was your adolescent behaviour I was referring to
QUOTE (Edward 3+Apr 7 2009, 10:03 AM)
It was your adolescent behaviour I was referring to
Oh come now Eddy.
Oh come now Eddy.
So true old chap - one should know better!!
What is interesting for anyone is that the guy says that so called consequent Taylor polynomials for a plane curve ( which You get as consequent polynomial approximations of any curve (e.g differentiable function y(x) which is differentiable near the point of interest) are disjoint and never cross in general case.
That should be taught in school as well.
The interesting thing about this and about disjoint osculating circles and higher degree osculating curves is - and this is specially for bukh:
The dimension - thickness of such curve- constitutes a point size, 0-dimension, in mathematics.
What Guys says it , in my interpretation, that if You move from one osculating circle to the next, since they are disjoint, You are moving One 0 - dimensional point at a time.
Guys also says that in case of conics, there is more subtle ordering then this- that between 2 lines that are disjoint there is also something that is less then disjoint but not continuous in 1 dimension as well- so some kind of fractional ordering.
If we think of continuous curve, the degree of disjoint it then 0
If we move across osculating conics, the degree of disjoint is 1-they are disjoint
However, there are degrees of discreteness/continuity in between at least for conics - Ghys could not say what is the geometric meaning of such a fact.
There could be such degree of disjointness also for higher degree osculating differential equations. Or there may be not, since , for degrees of equations higher than 2 the number of locally disjoint osculating curves is smaller than the number of crossings of 2 such curves, so they cross somewhere, in case of cubics in 1 point, in case of sextics in 10 additional points, but they do not EVER cross locally. Funny.
I have to watch the video once more, it was pretty subtle.
The other 2 videos by Ghys are excellent as well, I will comment on them later.
That should be taught in school as well.
The interesting thing about this and about disjoint osculating circles and higher degree osculating curves is - and this is specially for bukh:
The dimension - thickness of such curve- constitutes a point size, 0-dimension, in mathematics.
What Guys says it , in my interpretation, that if You move from one osculating circle to the next, since they are disjoint, You are moving One 0 - dimensional point at a time.
Guys also says that in case of conics, there is more subtle ordering then this- that between 2 lines that are disjoint there is also something that is less then disjoint but not continuous in 1 dimension as well- so some kind of fractional ordering.
If we think of continuous curve, the degree of disjoint it then 0
If we move across osculating conics, the degree of disjoint is 1-they are disjoint
However, there are degrees of discreteness/continuity in between at least for conics - Ghys could not say what is the geometric meaning of such a fact.
There could be such degree of disjointness also for higher degree osculating differential equations. Or there may be not, since , for degrees of equations higher than 2 the number of locally disjoint osculating curves is smaller than the number of crossings of 2 such curves, so they cross somewhere, in case of cubics in 1 point, in case of sextics in 10 additional points, but they do not EVER cross locally. Funny.
I have to watch the video once more, it was pretty subtle.
The other 2 videos by Ghys are excellent as well, I will comment on them later.
QUOTE (Ivars+Apr 7 2009, 03:01 PM)
What is interesting for anyone is that the guy says that so called consequent Taylor polynomials for a plane curve ( which You get as consequent polynomial approximations of any curve (e.g differentiable function y(x) which is differentiable near the point of interest) are disjoint and never cross in general case.
This is clearly false, and serves as an immediate indication that you don't understand the first thing about those lectures. Case in point: Taylor polynomials have the form:
f(a) + (x-a)f'(a) + ... + (1/n!) (x-a)^n f^(n)(b)
where b is in (a,x). It is clear that for each n, these polynomials are guaranteed to agree at x=a. The result is obvious anyway, from the construction of a Taylor polynomial. Anyone who doesn't see it immediately doesn't have the first clue about Taylor's theorem (i.e. Ivars).
This is clearly false, and serves as an immediate indication that you don't understand the first thing about those lectures. Case in point: Taylor polynomials have the form:
f(a) + (x-a)f'(a) + ... + (1/n!) (x-a)^n f^(n)(b)
where b is in (a,x). It is clear that for each n, these polynomials are guaranteed to agree at x=a. The result is obvious anyway, from the construction of a Taylor polynomial. Anyone who doesn't see it immediately doesn't have the first clue about Taylor's theorem (i.e. Ivars).
QUOTE (Ivars+Apr 7 2009, 03:01 PM)
That should be taught in school as well.
This is a good example of your behaviour. You clearly don't understand what you just attempted to talk about, but attempted to make out that you understood it so well, and that you found it so simple, it should be taught to school children.
Once again, you've shown us that you're an idiot who tries to make out he knows more than he does. Well done.
This is a good example of your behaviour. You clearly don't understand what you just attempted to talk about, but attempted to make out that you understood it so well, and that you found it so simple, it should be taught to school children.
Once again, you've shown us that you're an idiot who tries to make out he knows more than he does. Well done.
QUOTE (Edward 3+Apr 7 2009, 11:03 AM)
It was your adolescent behaviour I was referring to
Before you throw too many stones I suggest you move out of your glass house Mr Pot.
Before you throw too many stones I suggest you move out of your glass house Mr Pot.
Strongly recommended:
Dimensions Movie
You can watch the movie online or download it . Beautiful pictures and explanations, from simple to quite advanced.
Chapter 1, dimension two, is very elementary. Secondary school students should be able to appreciate it, but we think that, even if you know already what meridians and parallels are, you will enjoy the spectacle of the Earth rolling like a ball ! (Look here).
Chapter 2, dimension three, is still elementary, but requires a bit of imagination, and it has some philosophical elements... There are even some exercises to check that you have understood. For explanations, additional information and references, one can consult this page
Chaptres 3 and 4 get us into the fourth dimension. This is of course more difficult, and maybe it will make your head spin ! In order to understand everything, don't hesitate to push the pause button on your remote, to watch these chapters several times, and to consult this page where you will find references to additional information. But even if you do not feel like making the effort to understand it all, you can always sit back and enjoy the pictures !
Chaptres 5 and 6, complex numbers, contain an introduction to, well... complex numbers. In France, complex numbers are taught in the final year of secondary school. We don't see this as a replacement for a classic course, but we think that these chapters could accompany such a course in a pleasant way. If you learned about complex numbers a long time ago, and you forgot most of it..., this could refresh your memory. If you know nothing about complex numbers, you should push the pause button as often as you like, and try to understand using the references that we propose. These chapters are the most "school-like" of the film. To thank you for your efforts, chapter 6 ends with an amazing deep zoom scene.
Chaptres 7 and 8 give you an introduction to the Hopf fibration, which is not taught in secondary school, and not even in the first years at university. This is certainly not beginner's stuff ! On the other hand, it is quite pretty and deserves to be understood. Everything is explained in the film, but of course, things may go a bit fast. Here also, the references that we provide can be useful in case you have trouble understanding... Good luck, and enjoy the show !
Finally, chapter 9 is a special one. It shows the proof of a theorem of geometry. This proof uses nothing above the level of secondary school, and we could very well have put this chapter right after chapter 1. Without proofs for theorems mathematics would not exist, and we wanted to make this very clear at the end of a film that is essentially about mathematical objects.
Dimensions Movie
You can watch the movie online or download it . Beautiful pictures and explanations, from simple to quite advanced.
QUOTE
Nine chapters, two hours of maths, that take you gradually up to the fourth dimension. Mathematical vertigo guaranteed! Background information on every chapter: see "Details".
QUOTE (->
| QUOTE |
| Nine chapters, two hours of maths, that take you gradually up to the fourth dimension. Mathematical vertigo guaranteed! Background information on every chapter: see "Details". |
Chapter 1, dimension two, is very elementary. Secondary school students should be able to appreciate it, but we think that, even if you know already what meridians and parallels are, you will enjoy the spectacle of the Earth rolling like a ball ! (Look here).
Chapter 2, dimension three, is still elementary, but requires a bit of imagination, and it has some philosophical elements... There are even some exercises to check that you have understood. For explanations, additional information and references, one can consult this page
Chaptres 3 and 4 get us into the fourth dimension. This is of course more difficult, and maybe it will make your head spin ! In order to understand everything, don't hesitate to push the pause button on your remote, to watch these chapters several times, and to consult this page where you will find references to additional information. But even if you do not feel like making the effort to understand it all, you can always sit back and enjoy the pictures !
Chaptres 5 and 6, complex numbers, contain an introduction to, well... complex numbers. In France, complex numbers are taught in the final year of secondary school. We don't see this as a replacement for a classic course, but we think that these chapters could accompany such a course in a pleasant way. If you learned about complex numbers a long time ago, and you forgot most of it..., this could refresh your memory. If you know nothing about complex numbers, you should push the pause button as often as you like, and try to understand using the references that we propose. These chapters are the most "school-like" of the film. To thank you for your efforts, chapter 6 ends with an amazing deep zoom scene.
Chaptres 7 and 8 give you an introduction to the Hopf fibration, which is not taught in secondary school, and not even in the first years at university. This is certainly not beginner's stuff ! On the other hand, it is quite pretty and deserves to be understood. Everything is explained in the film, but of course, things may go a bit fast. Here also, the references that we provide can be useful in case you have trouble understanding... Good luck, and enjoy the show !
Finally, chapter 9 is a special one. It shows the proof of a theorem of geometry. This proof uses nothing above the level of secondary school, and we could very well have put this chapter right after chapter 1. Without proofs for theorems mathematics would not exist, and we wanted to make this very clear at the end of a film that is essentially about mathematical objects.
Quote AlphaNumeric
"Before you throw too many stones I suggest you move out of your glass house Mr Pot."
Relax A/N - all the stones are made of strings - wouldn´t hurt a fly. And no need for this Mr. Pot formality - just call me Crack !!
"Before you throw too many stones I suggest you move out of your glass house Mr Pot."
Relax A/N - all the stones are made of strings - wouldn´t hurt a fly. And no need for this Mr. Pot formality - just call me Crack !!
QUOTE (Edward 3+Apr 7 2009, 07:10 PM)
just call me Crack !!
I think it's spelt with a 'n', not a 'c'.
I think it's spelt with a 'n', not a 'c'.
Theorem ( Ghys) :
QUOTE
Proposition : If f (x) 2k+1 derivative does not vanish on [a,b], then the graphs of the polynomials T (f, x) of order 2 k are two by two disjoint.
QUOTE (Ivars+Apr 8 2009, 09:37 AM)
Proposition : If f (x) 2k+1 derivative does not vanish on [a,b], then the graphs of the polynomials T (f, x) of order 2 k are two by two disjoint.
Hey, I was just going to say that!
Hey, I was just going to say that!
Quote Kaeroll
"I think it's spelt with a 'n', not a 'c'."
Not so Kaeroll - A/N called me Mr.Pot - and I didn´t get all huffy like he did when I referred to him as the King of String - but I was just inviting him to dispense with the formalities and call me Crack - get it Crack Pot? Or, if you prefer, you can call me Pol - now, I don´t have to explain that to you as well, do I? Of course you´re free to call me Crank - that´s the title I picked up for asking too many questions !! In fact call me whatever you like - Trout/Fishy even calls me Kaneda - some guy who was a member here even before Fishy joined but he still hates the guy for some reason - actually I challenged Fishy on this topic but he funked the challenge.
all the best
??
"I think it's spelt with a 'n', not a 'c'."
Not so Kaeroll - A/N called me Mr.Pot - and I didn´t get all huffy like he did when I referred to him as the King of String - but I was just inviting him to dispense with the formalities and call me Crack - get it Crack Pot? Or, if you prefer, you can call me Pol - now, I don´t have to explain that to you as well, do I? Of course you´re free to call me Crank - that´s the title I picked up for asking too many questions !! In fact call me whatever you like - Trout/Fishy even calls me Kaneda - some guy who was a member here even before Fishy joined but he still hates the guy for some reason - actually I challenged Fishy on this topic but he funked the challenge.
all the best
??
QUOTE (Ivars+Apr 8 2009, 09:37 AM)
Proposition : If f (x) 2k+1 derivative does not vanish on [a,b], then the graphs of the polynomials T (f, x) of order 2 k are two by two disjoint.
This doesn't even make any sense. It's not even coherent. Please stop - you look pathetic.
This doesn't even make any sense. It's not even coherent. Please stop - you look pathetic.
What's the trident symbol mean then 
? I kinda fell over at that point on a previous set of notes - not quite got to the end of line one - but confession is good for the soul (hopefully) 
.
Edit .. looks like it ought to be a Greek letter but it isn't .. psi on it's side .. prongs pointing to the right ..
? I kinda fell over at that point on a previous set of notes - not quite got to the end of line one - but confession is good for the soul (hopefully) .Edit .. looks like it ought to be a Greek letter but it isn't .. psi on it's side .. prongs pointing to the right ..
Hi Ivars,
In the link given by Euler (thank you)
http://people.maths.ox.ac.uk/~hitchin/hitc...ve_geometry.pdf
Exercise:-
1. Let U1, U2 and U3 be the 2-dimensional vector subspaces of R3 defined by
x0 = 0, x0 + x1 + x2 = 0, 3x0 − 4x1 + 5x2 = 0 respectively. Find the vertices of the “triangle” in P2® whose sides are the projective lines P(U1), P(U2), P(U3).
Howzabout you and me try to answer this? There's going to be some humiliation involved .. but things can't really get much worse .. can they?
Starting now
:-
1/ x0 = 0
2/ x0 + x1 + x2 = 0,
3/ 3x0 − 4x1 + 5x2 = 0
To me 1/ looks like a 2D plane .. 2/ and 3/ look like lines .. I could be wrong .. your comments invited.
-C2.
Edit .. 2D plane .. ok .. this isn't easy .. let's just do our best.
In the link given by Euler (thank you)
http://people.maths.ox.ac.uk/~hitchin/hitc...ve_geometry.pdf
Exercise:-
1. Let U1, U2 and U3 be the 2-dimensional vector subspaces of R3 defined by
x0 = 0, x0 + x1 + x2 = 0, 3x0 − 4x1 + 5x2 = 0 respectively. Find the vertices of the “triangle” in P2® whose sides are the projective lines P(U1), P(U2), P(U3).
Howzabout you and me try to answer this? There's going to be some humiliation involved .. but things can't really get much worse .. can they?
Starting now
:-1/ x0 = 0
2/ x0 + x1 + x2 = 0,
3/ 3x0 − 4x1 + 5x2 = 0
To me 1/ looks like a 2D plane .. 2/ and 3/ look like lines .. I could be wrong .. your comments invited.
-C2.
Edit .. 2D plane .. ok .. this isn't easy .. let's just do our best.
Do you mean "∈" in "[v] ∈ P(V)". That's the "is element of" symbol from set theory which is a prerequisite.
And I found the problem trivial since P(U1)∩P(U2) = P(U1∩U2), and that I have a grasp of what a point in P(R^3) means.
And I found the problem trivial since P(U1)∩P(U2) = P(U1∩U2), and that I have a grasp of what a point in P(R^3) means.
Er..
If x0=0 then what remains runs straight up the x1,2 axes?
C'mon Ivars .. this is your thread..
C'mon Ivars .. this is your thread..
QUOTE (Confused2+Apr 10 2009, 02:52 PM)
Er..
If x0=0 then what remains runs straight up the x1,2 axes?
C'mon Ivars .. this is your thread..
And I agree!
Ivars, ar yuo dooing upseide dawn math?
The equation doesn't make sense, really:
I don't know what a "vanishing point" would be, but there is only one case that I know of that sort of, kind of, tangentially, like... uh, let me start again:
I consider 2k a whole thing --a complete entity in itself-- and not as a segment of any other equation.
Therefore, 2k+1 is redundant: I just know a single case where (2k) equals (k+1 times 2), and I did show it to you. I truly have no idea what you mean when you say (2k+1) because it literally squares the number of answers you can get that are right (or wrong)
You want to have 2k *plus one*?
FOINE.
You can't have them.
Not when both are content. If you want to have it, 2k has to be address, and that sweet little 1
- must necessarily be a *content* which may or may not be at the address... and if it is not there, then it's a certainty that it is visiting the neighbors(!!!) because of a moment...
I truly have no idea where this is going. I only know of one case of 2k = k+1, and it has nothing to do with anything you or anyone thinks it means, but they are redundant together. And I do know that the "2k" in one translation is already equal to the presence of a "2" in the logarithmic *root** of the other, and simultaneously equal to "plus one" in another because they are equival...
What is "vanishing point" supposed to mean in this case?
Soooooooo...
Iiiiiii mmmeeeaaaaaaannnnnnn...
Yooouuuu kknnnoooowwwww....
?????
Sorry, this makes no sense!
But methinks
1--RP got it right talking about "∈". Something can't both belong and not belong to an address.
2--Confused2 says:
I don't know what a "vanishing point" would be, but there is only one case that I know of that sort of, kind of, tangentially, like... uh, let me start again:
I consider 2k a whole thing --a complete entity in itself-- and not as a segment of any other equation.
Therefore, 2k+1 is redundant: I just know a single case where (2k) equals (k+1 times 2), and I did show it to you. I truly have no idea what you mean when you say (2k+1) because it literally squares the number of answers you can get that are right (or wrong)
You want to have 2k *plus one*?
FOINE.
You can't have them.
Not when both are content. If you want to have it, 2k has to be address, and that sweet little 1 - must necessarily be a *content* which may or may not be at the address... and if it is not there, then it's a certainty that it is visiting the neighbors(!!!) because of a moment...
I truly have no idea where this is going. I only know of one case of 2k = k+1, and it has nothing to do with anything you or anyone thinks it means, but they are redundant together. And I do know that the "2k" in one translation is already equal to the presence of a "2" in the logarithmic *root** of the other, and simultaneously equal to "plus one" in another because they are equival...
What is "vanishing point" supposed to mean in this case?
Soooooooo...
Iiiiiii mmmeeeaaaaaaannnnnnn...
Yooouuuu kknnnoooowwwww....
?????
Sorry, this makes no sense!
But methinks
1--RP got it right talking about "∈". Something can't both belong and not belong to an address.
2--Confused2 says:
1/ x0 = 0
2/ x0 + x1 + x2 = 0,
3/ 3x0 − 4x1 + 5x2 = 0
because he can't make a multiplication by zero and differentiate between the content and the address. The content has moved, it's just not where he thinks it is. The result of C2's equation is only zero if you don't know where the effort spent executing the equation -that is, the energy spent- went.
I don't know where it went, I just know it's there!
Sorry, folks. Now you know the reason I stopped commenting. I can't make head or tails of what is being said. C2's empty-space-filled-with-0's should be moving and jumping about, and... it's not.
Well said!
Hi rpenner,
This was most instructive . If only some of the other pro's here would sometimes really DO some math and use their finite lives to help people who ask questions.
I will try to replay Your calculation reasoning to see if I have understood them correctly and then ask a confusing ( for me) question:
So you replace real projective plane P²( R) with Real 3D space of projective coordinates R³ and the set of all points in P²( R) is then the set equivalent to all lines through origin in such space R³.
So you replace real projective plane P²( R) with Real 3D space of projective coordinates R³ and the set of all points in P²( R) is then the set equivalent to all lines through origin in such space R³.
Now we could just write this point as (p0, p1, p2) but the problem is that is the same line (and therefore the same point) as (−2p0, −2p1, −2p2). So we normalize, so that the leading non-zero term is 1. So if p0 ≠ 0, we write the point in P²( R) as [1, p1/p0, p2/p0] and if p0 = 0 and p1 ≠ 0, we write [0, 1, p2/p1] and if p0 = 0 and p1 = 0, we write [0, 0, 1].
So here You introduce homogeneous coordinates by "normalizing" the first projective coordinate to be =1 = p0/p0, and consequently , others p1/p0, p2/p0.
if p0=0 and p1≠0 we get p1/p0=oo or line with infinite slope in R³ in homogeneous coordinates . In non-normalized coordinates in , that would be any line through origin lying in a plane perpendicular to the relevant coordinate axis in R³ p0. With p1 taking all real values we get a set of such lines in R³ - a plane consisting of pencil of lines- which corresponds to set of points in P²( R) - a line in P²( R) , so called line at "infinity"?
If p0 =0 and p1 = 0, slope p1/p0=0/0 is indefinite and correspond to a single line through origin in R³ which is basically the coordinate axis p2 itself and lines paralel to it in R³ as p1, p0 = 0 .
This would be presented as a single point in P²( R) - point at infinity - where all lines parallel to p2 in R3 meet?
This is true as long as Real projective plane is considered as certain 2 dimensional combination of numbers from space R³.
This is true as long as Real projective plane is considered as certain 2 dimensional combination of numbers from space R³.
So the three points (as I suggested above) are:
P(U1∩U2)
x0 = 0 and x0 + x1 + x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = −λ
⇒ P(U1∩U2) = [0, 1, −1]
P(U1∩U3)
x0 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = (4/5)λ
⇒ P(U1∩U3) = [0, 1, 4/5]
P(U2∩U3)
x0 + x1 + x2 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = λ and x1 + x2 = −λ and −4x1 + 5x2 = −3λ
⇒ x0 = λ and −9x1 = 2λ and 9x2 = −7λ
⇒ P(U2∩U3) = [1, −2/9, −7/9]
Here You use the equations of Planes through origin in R³ corresponding to the given lines of "triangle" in P²( R) and find their meets using the coordinate information about lines forming the triangle.
My confusing question was: Klein, Cayley, Staudt ( I hope I am not wrong here) considered Real projective spaces as subcases of Complex projective spaces, not vice versa as construction with upping the dimensions by 1 may suggest.
If we would have a similar question with "triangle" in CP², would we than need 4 dimensional Real space to represent it and 6 dimensional Real space where to work the things out, both endowed with complex structure which probably has not much effect as long as there are no differential relations involved?
6 dimensional Real Space is also the space of line coordinates in Plucker sense ( x,y,z, X, Y, Z) which gives all lines in R³ as functions of their point (with fixed orientation of coordinate system, all points along line) and line (with fixed origin, all lines through the point) coordinates.
Line Geometry
Is there any relation between points in CP² and R^6 of Plucker line coordinates (even if all Plucker coordinates are kept real) ? In projective space P^5 with homogeneous coordinates, Plucker coordinates has 5 independent parameters, so in a sense, such Real Space is also the space of point/line coordinates in Plucker sense R^6 has a structure (constraint) as well.
The Plucker approach and served as one of foundations to the theory of screws - I think so at least, not sure. So could Complex structure on R^6 be helical as well and how is that represented ? Twist of I ( Penrose Road to Reality , Berry)?
Screw Theory
C'mon Ivars .. this is your thread..
And I agree!
Ivars, ar yuo dooing upseide dawn math?
The equation doesn't make sense, really:
QUOTE
Proposition : If f (x) 2k+1 derivative does not vanish on [a,b], then the graphs of the polynomials T (f, x) of order 2 k are two by two disjoint.
I don't know what a "vanishing point" would be, but there is only one case that I know of that sort of, kind of, tangentially, like... uh, let me start again:
I consider 2k a whole thing --a complete entity in itself-- and not as a segment of any other equation.
Therefore, 2k+1 is redundant: I just know a single case where (2k) equals (k+1 times 2), and I did show it to you. I truly have no idea what you mean when you say (2k+1) because it literally squares the number of answers you can get that are right (or wrong)
You want to have 2k *plus one*?
FOINE.
You can't have them.
Not when both are content. If you want to have it, 2k has to be address, and that sweet little 1
- must necessarily be a *content* which may or may not be at the address... and if it is not there, then it's a certainty that it is visiting the neighbors(!!!) because of a moment...I truly have no idea where this is going. I only know of one case of 2k = k+1, and it has nothing to do with anything you or anyone thinks it means, but they are redundant together. And I do know that the "2k" in one translation is already equal to the presence of a "2" in the logarithmic *root** of the other, and simultaneously equal to "plus one" in another because they are equival...
What is "vanishing point" supposed to mean in this case?
Soooooooo...
Iiiiiii mmmeeeaaaaaaannnnnnn...
Yooouuuu kknnnoooowwwww....
?????
Sorry, this makes no sense!
But methinks
1--RP got it right talking about "∈". Something can't both belong and not belong to an address.
2--Confused2 says:
QUOTE (->
| QUOTE |
| Proposition : If f (x) 2k+1 derivative does not vanish on [a,b], then the graphs of the polynomials T (f, x) of order 2 k are two by two disjoint. |
I don't know what a "vanishing point" would be, but there is only one case that I know of that sort of, kind of, tangentially, like... uh, let me start again:
I consider 2k a whole thing --a complete entity in itself-- and not as a segment of any other equation.
Therefore, 2k+1 is redundant: I just know a single case where (2k) equals (k+1 times 2), and I did show it to you. I truly have no idea what you mean when you say (2k+1) because it literally squares the number of answers you can get that are right (or wrong)
You want to have 2k *plus one*?
FOINE.
You can't have them.
Not when both are content. If you want to have it, 2k has to be address, and that sweet little 1
- must necessarily be a *content* which may or may not be at the address... and if it is not there, then it's a certainty that it is visiting the neighbors(!!!) because of a moment...I truly have no idea where this is going. I only know of one case of 2k = k+1, and it has nothing to do with anything you or anyone thinks it means, but they are redundant together. And I do know that the "2k" in one translation is already equal to the presence of a "2" in the logarithmic *root** of the other, and simultaneously equal to "plus one" in another because they are equival...
What is "vanishing point" supposed to mean in this case?
Soooooooo...
Iiiiiii mmmeeeaaaaaaannnnnnn...
Yooouuuu kknnnoooowwwww....
?????
Sorry, this makes no sense!
But methinks
1--RP got it right talking about "∈". Something can't both belong and not belong to an address.
2--Confused2 says:
1/ x0 = 0
2/ x0 + x1 + x2 = 0,
3/ 3x0 − 4x1 + 5x2 = 0
because he can't make a multiplication by zero and differentiate between the content and the address. The content has moved, it's just not where he thinks it is. The result of C2's equation is only zero if you don't know where the effort spent executing the equation -that is, the energy spent- went.
I don't know where it went, I just know it's there!
Sorry, folks. Now you know the reason I stopped commenting. I can't make head or tails of what is being said. C2's empty-space-filled-with-0's should be moving and jumping about, and... it's not.
Hi
I may be totally wrong, and perhaps also out of context. But perhaps Ivars tried to express how matter can form out from "time-flow", and therefore get the math opposite.
Or put differently, it is not possible to calculate math bottom-up, math is always upside down, whereas physics normally is bottom up.
Physics is the real McCoy, and math is a discipline useful to mimic physics, but only and always as approximations.
Physical world is based upon how smallest dimension-objects scale-wise arrange themselves in more and more complex shapes - and in a dynamic cinematographic fashion. This implicates that it is not possible to ever identify the smallest - never to know the origin of what we measure and perceive. Physics will always be with hidden variables, and physics will always be in need of renormalization in order to be calculable.
Mathematics on the other hand can never be calculated out from smallest - because smallest will always be hidden in a lower scale, will always be hidden as zero - with no information. And nothing can be calculated out from zero. On the other hand - it is always possible to calculate from upside down, and calculus is about how numbers are being made more and more accurate - but it is not possible to create any kind of starting conditions and then calculate anything meaningful up-wards so to speak.
I may be totally wrong, and perhaps also out of context. But perhaps Ivars tried to express how matter can form out from "time-flow", and therefore get the math opposite.
Or put differently, it is not possible to calculate math bottom-up, math is always upside down, whereas physics normally is bottom up.
Physics is the real McCoy, and math is a discipline useful to mimic physics, but only and always as approximations.
Physical world is based upon how smallest dimension-objects scale-wise arrange themselves in more and more complex shapes - and in a dynamic cinematographic fashion. This implicates that it is not possible to ever identify the smallest - never to know the origin of what we measure and perceive. Physics will always be with hidden variables, and physics will always be in need of renormalization in order to be calculable.
Mathematics on the other hand can never be calculated out from smallest - because smallest will always be hidden in a lower scale, will always be hidden as zero - with no information. And nothing can be calculated out from zero. On the other hand - it is always possible to calculate from upside down, and calculus is about how numbers are being made more and more accurate - but it is not possible to create any kind of starting conditions and then calculate anything meaningful up-wards so to speak.
Do you even have a clue as to anything you just said? Are you arguing only on a presumption of anothers disposition? Don't you ever shut up?
Bw/S
"Are you arguing only on a presumption of anothers disposition"
Yes - and ?
You are welcome to comment on the issue - !
"Don't you ever shut up?" !
(anothers being spelled an others - I think)
"Are you arguing only on a presumption of anothers disposition"
Yes - and ?
You are welcome to comment on the issue - !
"Don't you ever shut up?" !
(anothers being spelled an others - I think)
QUOTE (bukh+Apr 12 2009, 07:14 PM)
Bw/S
"Are you arguing only on a presumption of anothers disposition"
Yes - and ?
You are welcome to comment on the issue - !
"Don't you ever shut up?" !
(anothers being spelled an others - I think)
Who's disposition are you commenting on - Ivars, Euler, who? Maybe you don't exactly know what you wrote because you can't decide who you're replying too. So as long as you write nonsense, you're safe. Good show!
"Are you arguing only on a presumption of anothers disposition"
Yes - and ?
You are welcome to comment on the issue - !
"Don't you ever shut up?" !
(anothers being spelled an others - I think)
Who's disposition are you commenting on - Ivars, Euler, who? Maybe you don't exactly know what you wrote because you can't decide who you're replying too. So as long as you write nonsense, you're safe. Good show!
BwS
Hopefully it is commenting on both -
And BTW - I think it is quite common - yeah perhaps the most common - that one comment on an others disposition.
Of course with the exception of those persons who do not argue - but just spout some hateful words out in the blue.
Hopefully it is commenting on both -
And BTW - I think it is quite common - yeah perhaps the most common - that one comment on an others disposition.
Of course with the exception of those persons who do not argue - but just spout some hateful words out in the blue.
QUOTE (bukh+Apr 12 2009, 08:14 PM)
Hopefully it is commenting on both -
That's especially nice. I see you made a very special post that addresses the finer points in each.
That's especially nice. I see you made a very special post that addresses the finer points in each.
Well said!
BwS and Granouille
Thanks - very much - always nice to receive such positive response -
Thanks - very much - always nice to receive such positive response -
QUOTE (bukh+Apr 13 2009, 09:34 AM)
BwS and Granouille
Thanks - very much - always nice to receive such positive response -
Always nice to see total nonsense BS and then see the poster whine when it gets treated as such.
Thanks - very much - always nice to receive such positive response -
Always nice to see total nonsense BS and then see the poster whine when it gets treated as such.
QUOTE (Confused2+Apr 10 2009, 04:35 PM)
http://people.maths.ox.ac.uk/~hitchin/hitc...ve_geometry.pdf
Exercise:-
1. Let U1, U2 and U3 be the 2-dimensional vector subspaces of R3 defined by
x0 = 0, x0 + x1 + x2 = 0, 3x0 − 4x1 + 5x2 = 0 respectively. Find the vertices of the “triangle” in P²( R) whose sides are the projective lines P(U1), P(U2), P(U3).
Ok. This is very easy. A point in P²( R) is of the the form (x0, x1, x2) = (λp0, λp1, λp2) which is the equation of a line in R³ which pases through the orgin. Now we could just write this point as (p0, p1, p2) but the problem is that is the same line (and therefore the same point) as (−2p0, −2p1, −2p2). So we normalize, so that the leading non-zero term is 1. So if p0 ≠ 0, we write the point in P²( R) as [1, p1/p0, p2/p0] and if p0 = 0 and p1 ≠ 0, we write [0, 1, p2/p1] and if p0 = 0 and p1 = 0, we write [0, 0, 1].
Naturally (?) a plane through the origin in R³ corresponds to a line in P²( R) . And an intesection of planes through the origin is a line through the origin in R³, and so a point in P²( R).
So the three points (as I suggested above) are:
P(U1∩U2)
x0 = 0 and x0 + x1 + x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = −λ
⇒ P(U1∩U2) = [0, 1, −1]
P(U1∩U3)
x0 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = (4/5)λ
⇒ P(U1∩U3) = [0, 1, 4/5]
P(U2∩U3)
x0 + x1 + x2 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = λ and x1 + x2 = −λ and −4x1 + 5x2 = −3λ
⇒ x0 = λ and −9x1 = 2λ and 9x2 = −7λ
⇒ P(U2∩U3) = [1, −2/9, −7/9]
And if you understand the definitions, the first two points you can simply write down from calculations in your head.
Exercise:-
1. Let U1, U2 and U3 be the 2-dimensional vector subspaces of R3 defined by
x0 = 0, x0 + x1 + x2 = 0, 3x0 − 4x1 + 5x2 = 0 respectively. Find the vertices of the “triangle” in P²( R) whose sides are the projective lines P(U1), P(U2), P(U3).
Ok. This is very easy. A point in P²( R) is of the the form (x0, x1, x2) = (λp0, λp1, λp2) which is the equation of a line in R³ which pases through the orgin. Now we could just write this point as (p0, p1, p2) but the problem is that is the same line (and therefore the same point) as (−2p0, −2p1, −2p2). So we normalize, so that the leading non-zero term is 1. So if p0 ≠ 0, we write the point in P²( R) as [1, p1/p0, p2/p0] and if p0 = 0 and p1 ≠ 0, we write [0, 1, p2/p1] and if p0 = 0 and p1 = 0, we write [0, 0, 1].
Naturally (?) a plane through the origin in R³ corresponds to a line in P²( R) . And an intesection of planes through the origin is a line through the origin in R³, and so a point in P²( R).
So the three points (as I suggested above) are:
P(U1∩U2)
x0 = 0 and x0 + x1 + x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = −λ
⇒ P(U1∩U2) = [0, 1, −1]
P(U1∩U3)
x0 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = (4/5)λ
⇒ P(U1∩U3) = [0, 1, 4/5]
P(U2∩U3)
x0 + x1 + x2 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = λ and x1 + x2 = −λ and −4x1 + 5x2 = −3λ
⇒ x0 = λ and −9x1 = 2λ and 9x2 = −7λ
⇒ P(U2∩U3) = [1, −2/9, −7/9]
And if you understand the definitions, the first two points you can simply write down from calculations in your head.
Hi guys,
Excuse me I took a computer vacation but only for 5 days. Happy to see thread is busy!
Excuse me I took a computer vacation but only for 5 days. Happy to see thread is busy!
It's weird: given Ivars' talk of projective geometry, over the past 8 months, I fail to see any posts on projective geometry from him. In fact, the only posts he's given in this thread have indicated his complete misunderstanding of some basic results in elementary mathematics.
It's almost as if Ivars wants to make people believe he knows and understands more than he does. Surely only a complete idiot would do that?
It's almost as if Ivars wants to make people believe he knows and understands more than he does. Surely only a complete idiot would do that?
QUOTE (rpenner+Apr 13 2009, 08:17 PM)
Hi rpenner,
This was most instructive . If only some of the other pro's here would sometimes really DO some math and use their finite lives to help people who ask questions.
I will try to replay Your calculation reasoning to see if I have understood them correctly and then ask a confusing ( for me) question:
QUOTE
Ok. This is very easy. A point in P²( R) is of the the form (x0, x1, x2) = (λp0, λp1, λp2) which is the equation of a line in R³ which pases through the orgin
So you replace real projective plane P²( R) with Real 3D space of projective coordinates R³ and the set of all points in P²( R) is then the set equivalent to all lines through origin in such space R³.
QUOTE (->
| QUOTE |
| Ok. This is very easy. A point in P²( R) is of the the form (x0, x1, x2) = (λp0, λp1, λp2) which is the equation of a line in R³ which pases through the orgin |
So you replace real projective plane P²( R) with Real 3D space of projective coordinates R³ and the set of all points in P²( R) is then the set equivalent to all lines through origin in such space R³.
Now we could just write this point as (p0, p1, p2) but the problem is that is the same line (and therefore the same point) as (−2p0, −2p1, −2p2). So we normalize, so that the leading non-zero term is 1. So if p0 ≠ 0, we write the point in P²( R) as [1, p1/p0, p2/p0] and if p0 = 0 and p1 ≠ 0, we write [0, 1, p2/p1] and if p0 = 0 and p1 = 0, we write [0, 0, 1].
So here You introduce homogeneous coordinates by "normalizing" the first projective coordinate to be =1 = p0/p0, and consequently , others p1/p0, p2/p0.
if p0=0 and p1≠0 we get p1/p0=oo or line with infinite slope in R³ in homogeneous coordinates . In non-normalized coordinates in , that would be any line through origin lying in a plane perpendicular to the relevant coordinate axis in R³ p0. With p1 taking all real values we get a set of such lines in R³ - a plane consisting of pencil of lines- which corresponds to set of points in P²( R) - a line in P²( R) , so called line at "infinity"?
If p0 =0 and p1 = 0, slope p1/p0=0/0 is indefinite and correspond to a single line through origin in R³ which is basically the coordinate axis p2 itself and lines paralel to it in R³ as p1, p0 = 0 .
This would be presented as a single point in P²( R) - point at infinity - where all lines parallel to p2 in R3 meet?
QUOTE
Naturally (?) a plane through the origin in R³ corresponds to a line in P²( R) . And an intesection of planes through the origin is a line through the origin in R³, and so a point in P²( R).
This is true as long as Real projective plane is considered as certain 2 dimensional combination of numbers from space R³.
QUOTE (->
| QUOTE |
| Naturally (?) a plane through the origin in R³ corresponds to a line in P²( R) . And an intesection of planes through the origin is a line through the origin in R³, and so a point in P²( R). |
This is true as long as Real projective plane is considered as certain 2 dimensional combination of numbers from space R³.
So the three points (as I suggested above) are:
P(U1∩U2)
x0 = 0 and x0 + x1 + x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = −λ
⇒ P(U1∩U2) = [0, 1, −1]
P(U1∩U3)
x0 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = 0 and x1 = λ and x2 = (4/5)λ
⇒ P(U1∩U3) = [0, 1, 4/5]
P(U2∩U3)
x0 + x1 + x2 = 0 and 3x0 − 4x1 + 5x2 = 0
⇒ x0 = λ and x1 + x2 = −λ and −4x1 + 5x2 = −3λ
⇒ x0 = λ and −9x1 = 2λ and 9x2 = −7λ
⇒ P(U2∩U3) = [1, −2/9, −7/9]
Here You use the equations of Planes through origin in R³ corresponding to the given lines of "triangle" in P²( R) and find their meets using the coordinate information about lines forming the triangle.
My confusing question was: Klein, Cayley, Staudt ( I hope I am not wrong here) considered Real projective spaces as subcases of Complex projective spaces, not vice versa as construction with upping the dimensions by 1 may suggest.
If we would have a similar question with "triangle" in CP², would we than need 4 dimensional Real space to represent it and 6 dimensional Real space where to work the things out, both endowed with complex structure which probably has not much effect as long as there are no differential relations involved?
6 dimensional Real Space is also the space of line coordinates in Plucker sense ( x,y,z, X, Y, Z) which gives all lines in R³ as functions of their point (with fixed orientation of coordinate system, all points along line) and line (with fixed origin, all lines through the point) coordinates.
Line Geometry
Is there any relation between points in CP² and R^6 of Plucker line coordinates (even if all Plucker coordinates are kept real) ? In projective space P^5 with homogeneous coordinates, Plucker coordinates has 5 independent parameters, so in a sense, such Real Space is also the space of point/line coordinates in Plucker sense R^6 has a structure (constraint) as well.
The Plucker approach and served as one of foundations to the theory of screws - I think so at least, not sure. So could Complex structure on R^6 be helical as well and how is that represented ? Twist of I ( Penrose Road to Reality , Berry)?
Screw Theory
Since you are still dividing by zero, I have to believe that further discussion is pointless. Please read through the material and solve the next two problems.
Wow, nasty. I don't know anything on this subject but I did have to comment on something I read...
With the project I'm working on most of what I know is information I have read. However some of the best increases in my own understanding of what I have read came from the pictures that some of the text had. The reason being is that there are different ways people learn. To enlist I had to take an aptitude test and scored highest in visual/spacial. I learn the best by hands on and seeing picture/diagrams. When I do read something I need to know it enough already to be able to picture it in my head or else I may as well being reading another language. The problem is that if I know it enough to visualize it then reading doesn't really give me much more on the subject except maybe a bit more context.
It's not a matter of being lazy or wanting things spoon fed, it's just a matter of seeking out sources that best match ones learning ability. There is also nothing wrong with seeking out institutions, either they be online or not, to get taught the material by someone who is knowledgeable in the subject. Anyone here with a degree should be the last person to think that seeking out an institution or source material (in text or video format) is called being spoon fed.
Now I don't know the history of some of you and by some of the feed back I have read, I really don't want to know. What I do know is what I see here and what I see here is someone who really does want to know this subject. The only person here that actually showed maturity and respect by responding constructively is rpenner. Good on him.
QUOTE
You mean you want to do as little work as possible and have someone hand you information on a plate. If you genuinely want to learn about projective geometry, you will have to do some reading (like everyone else)
With the project I'm working on most of what I know is information I have read. However some of the best increases in my own understanding of what I have read came from the pictures that some of the text had. The reason being is that there are different ways people learn. To enlist I had to take an aptitude test and scored highest in visual/spacial. I learn the best by hands on and seeing picture/diagrams. When I do read something I need to know it enough already to be able to picture it in my head or else I may as well being reading another language. The problem is that if I know it enough to visualize it then reading doesn't really give me much more on the subject except maybe a bit more context.
It's not a matter of being lazy or wanting things spoon fed, it's just a matter of seeking out sources that best match ones learning ability. There is also nothing wrong with seeking out institutions, either they be online or not, to get taught the material by someone who is knowledgeable in the subject. Anyone here with a degree should be the last person to think that seeking out an institution or source material (in text or video format) is called being spoon fed.
Now I don't know the history of some of you and by some of the feed back I have read, I really don't want to know. What I do know is what I see here and what I see here is someone who really does want to know this subject. The only person here that actually showed maturity and respect by responding constructively is rpenner. Good on him.
QUOTE (H2O+Apr 16 2009, 01:17 AM)
However some of the best increases in my own understanding of what I have read came from the pictures that some of the text had. The reason being is that there are different ways people learn. To enlist I had to take an aptitude test and scored highest in visual/spacial. I learn the best by hands on and seeing picture/diagrams. When I do read something I need to know it enough already to be able to picture it in my head or else I may as well being reading another language. The problem is that if I know it enough to visualize it then reading doesn't really give me much more on the subject except maybe a bit more context.
It's not a matter of being lazy or wanting things spoon fed, it's just a matter of seeking out sources that best match ones learning ability. There is also nothing wrong with seeking out institutions, either they be online or not, to get taught the material by someone who is knowledgeable in the subject. Anyone here with a degree should be the last person to think that seeking out an institution or source material (in text or video format) is called being spoon fed.
Hi H20
You have opened a can of worms. Poor You. Wait to see what happens with your feedback.
But thanks anyway! It is nice to see a decent person who understand the differences in learning styles and corresponding needs.
.
Even Felix Klein had all the possible geometric constructions modeled in sculptural figures to get better understanding and for students.
By the way, so far NO ONE has given me even a hint where to look for VIDEO lectures.
I would like to add that I am also interested in Differential Geometry video lectures, Frenet moving frame, clasification of curves by curvature/torsion as functions of arclentgh, application of intrinsic curve parameters , especially Projective Differential geometry, differential and integral projective invariants of curves.
Things evolve....to keep head clear from I would love to get rid of all coordinate representations of curves/surfaces etc are they just complicate the visual picture.
After having got rid of coordinates, I would love to know how the curve defined by intrinsic parameters behave in fractional dimensional spaces, perhaps projective when dimensionality of space is made a parameter.
It's not a matter of being lazy or wanting things spoon fed, it's just a matter of seeking out sources that best match ones learning ability. There is also nothing wrong with seeking out institutions, either they be online or not, to get taught the material by someone who is knowledgeable in the subject. Anyone here with a degree should be the last person to think that seeking out an institution or source material (in text or video format) is called being spoon fed.
Hi H20
You have opened a can of worms. Poor You. Wait to see what happens with your feedback.
But thanks anyway! It is nice to see a decent person who understand the differences in learning styles and corresponding needs.
.Even Felix Klein had all the possible geometric constructions modeled in sculptural figures to get better understanding and for students.
By the way, so far NO ONE has given me even a hint where to look for VIDEO lectures.
I would like to add that I am also interested in Differential Geometry video lectures, Frenet moving frame, clasification of curves by curvature/torsion as functions of arclentgh, application of intrinsic curve parameters , especially Projective Differential geometry, differential and integral projective invariants of curves.
Things evolve....to keep head clear from I would love to get rid of all coordinate representations of curves/surfaces etc are they just complicate the visual picture.
After having got rid of coordinates, I would love to know how the curve defined by intrinsic parameters behave in fractional dimensional spaces, perhaps projective when dimensionality of space is made a parameter.
QUOTE (rpenner+Apr 15 2009, 03:08 PM)
Since you are still dividing by zero, I have to believe that further discussion is pointless. Please read through the material and solve the next two problems.
Hi rpenner
Symbolic division by zero in a context of extended Real Number Line ( or extended complex plane) is not a crime in itself. Perhaps we should not start this discussion here.
Division by zero Weinstein
Division by zero Wikipedia
Division by zero Wikipedia
Real projective line
The set R and oo is the real projective line, which is a one-point compactification of the real line. Here oo means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies -oo = oo which is necessary in this context. In this structure, a/0 = oo can be defined for nonzero a, and a/oo = 0. It is the natural way to view the range of the tangent and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either +\pi/2 or -pi/2 from either direction.
This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, oo + oo has no meaning in the projective line.
As to calculations, I am not so keen on particular results. Life is finite. I am more interested what happens if we move the origin in R^3 - what would happen with the corresponding PR^2? To me it seems that we will have to move the point at infinity which is actually not infinity in PR^2 since there is no distance in PR^2 ( the idea of point at infinity comes from affine representation of Projective space).
So it is better called ideal point, and the movement of this ideal point would then correspond to movement of origin of R^3 . When origin in R^3 traces out a curve in 3D space, the ideal point in RP^2 performs what movement? If origin traces a surface in R^3? If origin moves so that it fills volume in R^3?
If speed of movement of point particle is defined in R^3 as derivative of 2 vectors that are infinitesimally moved along trajectory (which means the space R^3 has to have affine connection) to perform vector difference operation, what is the projective RP^2 analogue of this operation of vector differentiation in R^3?
Hi rpenner
Symbolic division by zero in a context of extended Real Number Line ( or extended complex plane) is not a crime in itself. Perhaps we should not start this discussion here.
Division by zero Weinstein
QUOTE
There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^* not 0 in the extended complex plane C-* is defined to be a quantity known as complex infinity. This definition expresses the fact that, for z not , lim_(w->0) z/w=infty (i.e., complex infinity). However, even though the formal statement 1/0=infty is permitted in C-*, note that this does not mean that 1=0·infty. Zero does not have a multiplicative inverse under any circumstances.
Division by zero Wikipedia
QUOTE (->
| QUOTE |
| There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^* not 0 in the extended complex plane C-* is defined to be a quantity known as complex infinity. This definition expresses the fact that, for z not , lim_(w->0) z/w=infty (i.e., complex infinity). However, even though the formal statement 1/0=infty is permitted in C-*, note that this does not mean that 1=0·infty. Zero does not have a multiplicative inverse under any circumstances. |
Division by zero Wikipedia
Real projective line
The set R and oo is the real projective line, which is a one-point compactification of the real line. Here oo means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies -oo = oo which is necessary in this context. In this structure, a/0 = oo can be defined for nonzero a, and a/oo = 0. It is the natural way to view the range of the tangent and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either +\pi/2 or -pi/2 from either direction.
This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, oo + oo has no meaning in the projective line.
As to calculations, I am not so keen on particular results. Life is finite. I am more interested what happens if we move the origin in R^3 - what would happen with the corresponding PR^2? To me it seems that we will have to move the point at infinity which is actually not infinity in PR^2 since there is no distance in PR^2 ( the idea of point at infinity comes from affine representation of Projective space).
So it is better called ideal point, and the movement of this ideal point would then correspond to movement of origin of R^3 . When origin in R^3 traces out a curve in 3D space, the ideal point in RP^2 performs what movement? If origin traces a surface in R^3? If origin moves so that it fills volume in R^3?
If speed of movement of point particle is defined in R^3 as derivative of 2 vectors that are infinitesimally moved along trajectory (which means the space R^3 has to have affine connection) to perform vector difference operation, what is the projective RP^2 analogue of this operation of vector differentiation in R^3?
QUOTE (Ivars+Apr 16 2009, 05:47 AM)
Hi H20
You have opened a can of worms. Poor You. Wait to see what happens with your feedback.
But thanks anyway! It is nice to see a decent person who understand the differences in learning styles and corresponding needs..
Even Felix Klein had all the possible geometric constructions modeled in sculptural figures to get better understanding and for students.
By the way, so far NO ONE has given me even a hint where to look for VIDEO lectures.
I would like to add that I am also interested in Differential Geometry video lectures, Frenet moving frame, clasification of curves by curvature/torsion as functions of arclentgh, application of intrinsic curve parameters , especially Projective Differential geometry, differential and integral projective invariants of curves.
Things evolve....to keep head clear from I would love to get rid of all coordinate representations of curves/surfaces etc are they just complicate the visual picture.
After having got rid of coordinates, I would love to know how the curve defined by intrinsic parameters behave in fractional dimensional spaces, perhaps projective when dimensionality of space is made a parameter.
As you've aptly demonstrated in this thread, you're not in the least bit interested in learning. You are all talk. In addition, you're dishonest.
Over 8 months ago, you told us all you were going to learn about projective geometry and you were really interested in it. I said you wouldn't. I said you were all talk. And here we are, and you've demonstrated to us, once again, that you're all talk.
Let's see how you get on with your new quest: learning differential geometry. Come on, prove me wrong, actually learn some differential geometry.
You have opened a can of worms. Poor You. Wait to see what happens with your feedback.
But thanks anyway! It is nice to see a decent person who understand the differences in learning styles and corresponding needs.
.Even Felix Klein had all the possible geometric constructions modeled in sculptural figures to get better understanding and for students.
By the way, so far NO ONE has given me even a hint where to look for VIDEO lectures.
I would like to add that I am also interested in Differential Geometry video lectures, Frenet moving frame, clasification of curves by curvature/torsion as functions of arclentgh, application of intrinsic curve parameters , especially Projective Differential geometry, differential and integral projective invariants of curves.
Things evolve....to keep head clear from I would love to get rid of all coordinate representations of curves/surfaces etc are they just complicate the visual picture.
After having got rid of coordinates, I would love to know how the curve defined by intrinsic parameters behave in fractional dimensional spaces, perhaps projective when dimensionality of space is made a parameter.
As you've aptly demonstrated in this thread, you're not in the least bit interested in learning. You are all talk. In addition, you're dishonest.
Over 8 months ago, you told us all you were going to learn about projective geometry and you were really interested in it. I said you wouldn't. I said you were all talk. And here we are, and you've demonstrated to us, once again, that you're all talk.
Let's see how you get on with your new quest: learning differential geometry. Come on, prove me wrong, actually learn some differential geometry.
QUOTE (Euler+Apr 16 2009, 09:01 AM)
Let's see how you get on with your new quest: learning differential geometry. Come on, prove me wrong, actually learn some differential geometry.
Yes, probably results will again be dismal since I started with last works of Wilczynski.
But if anyone is interested, here are few links not so easy to find:
Differential properties of functions of complex variable which are invariant under linear transformations part 1
Part 2
Interpretation of the simplest projective integral invariant
Yes, probably results will again be dismal since I started with last works of Wilczynski.
But if anyone is interested, here are few links not so easy to find:
Differential properties of functions of complex variable which are invariant under linear transformations part 1
Part 2
Interpretation of the simplest projective integral invariant
QUOTE (Ivars+Apr 16 2009, 02:58 PM)
Yes, probably results will again be dismal since I started with last works of Wilczynski.
But if anyone is interested, here are few links not so easy to find:
Differential properties of functions of complex variable which are invariant under linear transformations part 1
Part 2
Interpretation of the simplest projective integral invariant
None of those links have anything to do with differential geometry.
For those who are actually interested in learning some differential geometry, here are some excellent notes:
http://www.dpmms.cam.ac.uk/~md384/snmeiwseis.pdf
But if anyone is interested, here are few links not so easy to find:
Differential properties of functions of complex variable which are invariant under linear transformations part 1
Part 2
Interpretation of the simplest projective integral invariant
None of those links have anything to do with differential geometry.
For those who are actually interested in learning some differential geometry, here are some excellent notes:
http://www.dpmms.cam.ac.uk/~md384/snmeiwseis.pdf
QUOTE (Euler+Apr 17 2009, 09:00 AM)
None of those links have anything to do with differential geometry.
For those who are actually interested in learning some differential geometry, here are some excellent notes:
http://www.dpmms.cam.ac.uk/~md384/snmeiwseis.pdf
No doubt about that, but
So books by Wilczynski, Lane called "Projective differential geometry..." and this article seems good enough to start with:
ON DIFFERENTIAL INVARIANTS OF PLANAR CURVES AND RECOGNIZING PARTIALLY OCCLUDED PLANAR SHAPES
At least You do not need all the terrible abstract notions of modern math at the beginning.
For those who are actually interested in learning some differential geometry, here are some excellent notes:
http://www.dpmms.cam.ac.uk/~md384/snmeiwseis.pdf
No doubt about that, but
QUOTE
clasification of curves by curvature/torsion as functions of arclentgh, application of intrinsic curve parameters , especially Projective Differential geometry, differential and integral projective invariants of curves.
So books by Wilczynski, Lane called "Projective differential geometry..." and this article seems good enough to start with:
ON DIFFERENTIAL INVARIANTS OF PLANAR CURVES AND RECOGNIZING PARTIALLY OCCLUDED PLANAR SHAPES
At least You do not need all the terrible abstract notions of modern math at the beginning.
Ivars, you are clearly just typing in words like 'differential geometry, curves, planes' into Google and then providing us with links to some of the longer titled ones. If you were serious about learning differnential geometry, you'd get ahold of some 'introduction to differential geometry' book (I think Nakahara is a great book but needs undergrad knowledge) and use that, or a set of lecture notes.
Trying to learn maths from published papers is just stupid, because when writing papers authors assume a lot of assumed knowledge, they aren't making it readable to layman, they are trying to transmit new ideas and information to people in their area in as effecient a way as possible.
Spending a month reading about the notation of modern maths (which differential geometry uses a ton of, it's got a lot of recent developments) saves a year in banging your head against intractable notation.
But that would only matter to you if you weren't lying about learning some maths.
Trying to learn maths from published papers is just stupid, because when writing papers authors assume a lot of assumed knowledge, they aren't making it readable to layman, they are trying to transmit new ideas and information to people in their area in as effecient a way as possible.
QUOTE
At least You do not need all the terrible abstract notions of modern math at the beginning.
This is just a pathetic cop out. If someone wanted to learn electromagnetism its a billion times easier to learn some basic mathematical methods and then use vector notations or differential forms than to just open Maxwell's original paper and read that, which doesn't have any of the very elegant and powerful (ie not a little abstract) notation used in modern maths. Spending a month reading about the notation of modern maths (which differential geometry uses a ton of, it's got a lot of recent developments) saves a year in banging your head against intractable notation.
But that would only matter to you if you weren't lying about learning some maths.
QUOTE (Ivars+Apr 17 2009, 04:54 PM)
ON DIFFERENTIAL INVARIANTS OF PLANAR CURVES AND RECOGNIZING PARTIALLY OCCLUDED PLANAR SHAPES
At least You do not need all the terrible abstract notions of modern math at the beginning.
Strike 2. That's a (unremarkable) paper on invariants of curves under rigid motions. This is not differential geometry. I've given you some notes - if you're serious about learning you'll go through them. Of course, we all know you wont, because as we've witnessed, you're all talk.
At least You do not need all the terrible abstract notions of modern math at the beginning.
Strike 2. That's a (unremarkable) paper on invariants of curves under rigid motions. This is not differential geometry. I've given you some notes - if you're serious about learning you'll go through them. Of course, we all know you wont, because as we've witnessed, you're all talk.
QUOTE (Ivars+Apr 7 2009, 02:01 PM)
Strongly recommended:
Dimensions Movie
You can watch the movie online or download it . Beautiful pictures and explanations, from simple to quite advanced.
That was an awesome movie. Great visuals and can be even enjoyed by people who are not math experts.
It is good at giving the non-expert an idea how they can come up with this stuff.
Dimensions Movie
You can watch the movie online or download it . Beautiful pictures and explanations, from simple to quite advanced.
That was an awesome movie. Great visuals and can be even enjoyed by people who are not math experts.
It is good at giving the non-expert an idea how they can come up with this stuff.
QUOTE (Cusa+Apr 19 2009, 06:22 PM)
Why do things shrink as they move into distance?
I believe this is a question of dimensionality and relativity.
This is like watching stupid develop right before your eyes.
I believe this is a question of dimensionality and relativity.
This is like watching stupid develop right before your eyes.
Hi AN, Euler, et al,
I know focusing on projective transformations is limiting, but focusing on metric is even more so. The reason I think projective differential and integral invariants are important is that we see, observe the world projectively- and that is why computer vision article is not so dumb- it has picked up where mathematicians had left the topic almost 100 years ago ( Halphen, Wilczynsky, Lane, Forsyth, Salmon, Cayley, Klein, Lie etc) .
Since physics work with observations, I bet heavily the theories that are true up to observation depend on projective properties of spaces - and that is well known ( CP2 in quantum mechanics) .
By the way, projective curvature of curves is invariant to projective transformations, and when it is constant, in plane, we end up with so called anharmonic curves or Lie_Klein W curves. These are curves that transform onto itself via projective transformations of space.
Thanks to all that has watched Dimensions movie. That should be revisited after some reading:)
Also to rpenner: the fundamental triangle of projective space is just a special case of cubic ( having 3 inflection points, in general) so TRIANGLE itself is just a special case of cubic with vertexes at inflection points. Fundamental tetrahedron in projective space is a special case of quartic, having in general 4 inflection points.
I find this type of generalization very useful since it takes away the special role polygons have in elementary Euclidean geometry. They are just sub cases of more general concepts. I find it easier to move from general to particular. [Moderator: There are a large number of words where the poster has failed to comprehend. It's really hard for me to understand what the poster hopes to gain, since I have not seen him complete the remaining problems on the self-contained PDF tutorial.]
According to Gauss-Bonnet theorem, there is no difference between Euler characteristic of a triangle and cubic, so topologically they are equivalent. In this sense, projective geometry plus Gauss Bonnet theorem seems to be a good tool to study topology via curves and surfaces not via abstract symbols....At least for dimensions one can visualize.
Since 4D objects may be recognized by there projection to 3D, it should be possible to develop some intuition in that direction as movies suggested. For example, what 4D curves are creating anharmonic curves as their projection on 3D? What 3D surfaces are creating 2D anharmonic curves via usual perspective projection we use in seing? What type of curves (functions) remain invariant via projection from infinite dimensional projective space to 2D?
Ivars
I know focusing on projective transformations is limiting, but focusing on metric is even more so. The reason I think projective differential and integral invariants are important is that we see, observe the world projectively- and that is why computer vision article is not so dumb- it has picked up where mathematicians had left the topic almost 100 years ago ( Halphen, Wilczynsky, Lane, Forsyth, Salmon, Cayley, Klein, Lie etc) .
Since physics work with observations, I bet heavily the theories that are true up to observation depend on projective properties of spaces - and that is well known ( CP2 in quantum mechanics) .
By the way, projective curvature of curves is invariant to projective transformations, and when it is constant, in plane, we end up with so called anharmonic curves or Lie_Klein W curves. These are curves that transform onto itself via projective transformations of space.
Thanks to all that has watched Dimensions movie. That should be revisited after some reading:)
Also to rpenner: the fundamental triangle of projective space is just a special case of cubic ( having 3 inflection points, in general) so TRIANGLE itself is just a special case of cubic with vertexes at inflection points. Fundamental tetrahedron in projective space is a special case of quartic, having in general 4 inflection points.
I find this type of generalization very useful since it takes away the special role polygons have in elementary Euclidean geometry. They are just sub cases of more general concepts. I find it easier to move from general to particular. [Moderator: There are a large number of words where the poster has failed to comprehend. It's really hard for me to understand what the poster hopes to gain, since I have not seen him complete the remaining problems on the self-contained PDF tutorial.]
According to Gauss-Bonnet theorem, there is no difference between Euler characteristic of a triangle and cubic, so topologically they are equivalent. In this sense, projective geometry plus Gauss Bonnet theorem seems to be a good tool to study topology via curves and surfaces not via abstract symbols....At least for dimensions one can visualize.
Since 4D objects may be recognized by there projection to 3D, it should be possible to develop some intuition in that direction as movies suggested. For example, what 4D curves are creating anharmonic curves as their projection on 3D? What 3D surfaces are creating 2D anharmonic curves via usual perspective projection we use in seing? What type of curves (functions) remain invariant via projection from infinite dimensional projective space to 2D?
Ivars
Classic Ivars post - many words, but absolutely no substance. Now how far have you got with those differential geometry notes? Same as all the others? Not done anything?
Come on, prove to us all there's more to you than just talk. Let's go through the first couple of pages of the notes I've given you. That's just two pages!
Come on, prove to us all there's more to you than just talk. Let's go through the first couple of pages of the notes I've given you. That's just two pages!
QUOTE (Ivars+Apr 20 2009, 08:32 AM)
know focusing on projective transformations is limiting, but focusing on metric is even more so.
Projective geometry is not 'limiting' and the properties of a metric space a huge, they cover all of Riemannian geometry. Entire libraries can be filled on just one of those topics! You really haven't a clue.
QUOTE (Ivars+Apr 20 2009, 08:32 AM)
The reason I think projective differential and integral invariants are important is that we see, observe the world projectively- and that is why computer vision article is not so dumb- it has picked up where mathematicians had left the topic almost 100 years ago ( Halphen, Wilczynsky, Lane, Forsyth, Salmon, Cayley, Klein, Lie etc) .
Oh look, Ivars Wiki'd for mathematicians whose names appear in an article on projective geometry.
Oh look, Ivars Wiki'd for mathematicians whose names appear in an article on projective geometry.
QUOTE (Ivars+Apr 20 2009, 08:32 AM)
Since physics work with observations, I bet heavily the theories that are true up to observation depend on projective properties of spaces - and that is well known ( CP2 in quantum mechanics) .
Given you know neither the maths or the physics and you can't do any mathematics of CP2, I don't think you're in a position to be making predictions.
QUOTE (Ivars+Apr 20 2009, 08:32 AM)
By the way, projective curvature of curves is invariant to projective transformations, and when it is constant, in plane, we end up with so called anharmonic curves or Lie_Klein W curves. These are curves that transform onto itself via projective transformations of space.
This is so obviously lifted from a website it's painful. You don't even bother with a half arsed segway, you just plonk in some soundbite on projective geometry your Googling got you.
QUOTE (Ivars+Apr 20 2009, 08:32 AM)
Also to rpenner: the fundamental triangle of projective space is just a special case of cubic ( having 3 inflection points, in general) so TRIANGLE itself is just a special case of cubic with vertexes at inflection points. Fundamental tetrahedron in projective space is a special case of quartic, having in general 4 inflection points.
I find this type of generalization very useful since it takes away the special role polygons have in elementary Euclidean geometry. They are just sub cases of more general concepts. I find it easier to move from general to particular.
According to Gauss-Bonnet theorem, there is no difference between Euler characteristic of a triangle and cubic, so topologically they are equivalent. In this sense, projective geometry plus Gauss Bonnet theorem seems to be a good tool to study topology via curves and surfaces not via abstract symbols....At least for dimensions one can visualize.
Since 4D objects may be recognized by there projection to 3D, it should be possible to develop some intuition in that direction as movies suggested. For example, what 4D curves are creating anharmonic curves as their projection on 3D? What 3D surfaces are creating 2D anharmonic curves via usual perspective projection we use in seing? What type of curves (functions) remain invariant via projection from infinite dimensional projective space to 2D?
I find this type of generalization very useful since it takes away the special role polygons have in elementary Euclidean geometry. They are just sub cases of more general concepts. I find it easier to move from general to particular.
According to Gauss-Bonnet theorem, there is no difference between Euler characteristic of a triangle and cubic, so topologically they are equivalent. In this sense, projective geometry plus Gauss Bonnet theorem seems to be a good tool to study topology via curves and surfaces not via abstract symbols....At least for dimensions one can visualize.
Since 4D objects may be recognized by there projection to 3D, it should be possible to develop some intuition in that direction as movies suggested. For example, what 4D curves are creating anharmonic curves as their projection on 3D? What 3D surfaces are creating 2D anharmonic curves via usual perspective projection we use in seing? What type of curves (functions) remain invariant via projection from infinite dimensional projective space to 2D?
And more vacuous copy and pasting. Are you that pathetic you have to lie to people you know will see through you. Euler and I both teach and mark undergraduates and it doesn't take long before you develop the ability to see whose copied and whose not. You're so obviously mindlessly parroting websites its pathetic.
QUOTE (AlphaNumeric+Apr 20 2009, 10:12 AM)
Oh look, Ivars Wiki'd for mathematicians whose names appear in an article on projective geometry.
Given you know neither the maths or the physics and you can't do any mathematics of CP2, I don't think you're in a position to be making predictions.
The only 2 I have not read yet in original is Halphen since I can not find his thesis and following works in English, and Lie since I am not ready yet. Of course, there is still much much more. Have You, by the way?
As to copy paste, I am glad You find my thoughts so well encapsulating the ideas that You take them for copy pastes.
As to CP2, I said projective space is used because we talk about observables,and mentioned the fact CP2 is behind quantum physics is because I read about it in R.Penroses Road to reality where he explicitly states that, I did not say that I know the details of its workings in physics.
As to Projective/metric restrictiveness, I was only mentioning the fact that metric geometry makes a 7 parameter subgroup of 15 parameter projective group and thus is more restrictive or less general ( in 3D case). That does not mean there can not be things in metric geometry that does not hold under projective transformations but are complex in itself, as You correctly state.
Given you know neither the maths or the physics and you can't do any mathematics of CP2, I don't think you're in a position to be making predictions.
The only 2 I have not read yet in original is Halphen since I can not find his thesis and following works in English, and Lie since I am not ready yet. Of course, there is still much much more. Have You, by the way?
As to copy paste, I am glad You find my thoughts so well encapsulating the ideas that You take them for copy pastes.
As to CP2, I said projective space is used because we talk about observables,and mentioned the fact CP2 is behind quantum physics is because I read about it in R.Penroses Road to reality where he explicitly states that, I did not say that I know the details of its workings in physics.
As to Projective/metric restrictiveness, I was only mentioning the fact that metric geometry makes a 7 parameter subgroup of 15 parameter projective group and thus is more restrictive or less general ( in 3D case). That does not mean there can not be things in metric geometry that does not hold under projective transformations but are complex in itself, as You correctly state.
QUOTE (Ivars+Apr 20 2009, 09:30 PM)
The only 2 I have not read yet in original is Halphen since I can not find his thesis and following works in English, and Lie since I am not ready yet. Of course, there is still much much more. Have You, by the way?
No, I haven't read the thesis of Halphen or anyone else you mention. Infact, I would say I've only ever read the thesis of two people, both of whom are in my area of work and who are less than 30 years of age. In the vast majority of cases reading a PhD thesis which is older than say 20 years is likely to be less illuminating than reading more recent publications or books. Why? Because in those 20 years more powerful mathematical tools will be developed to streamline methods and more work will have been done.
For instance, the original publication of Maxwell's which contain his electromagnetic equations are horrible to read. They didn't have any of the VERY useful notations like vector calculus or differential forms in his day, they did everything using stuff like quaternions or component by component. Reading Maxwell's original paper is only useful for historical perspective. Reading a book on vector calculus and differential forms is infinitely more useful.
I haven't read any work by Lie but I've read books on Lie algebras written by other people since Lie. Infact, I've got one in arms reach. I've also read several lecture courses on them. And, unlike you, I've actually done questions relating to Lie algebras and Lie groups. Unlike you, I make sure I can do the maths. Unlike you I have been tested by sitting exams on it and unlike you they didn't think blurting out BS copy and paste paragraphs is showing understanding.
You've read a number of peoples thesis but I bet you can't do a single question Euler or I could ask you on that area. You've shown in this thread you will avoid doing anything which requires a working understanding, now jumping from projective geometry to differential geometry. And your grasp of what differential geometry is is so poor you Googled for some papers and then linked us to the results, which were nothing to do with differential geometry! Because rather than get a book entitled "An Introduction to Differential Geometry" and work through it, you prefer to delude yourself into thinking what you're doing at the moment is of any use at all.
I've spent 2 hours this evening (and 10 yesterday) doing a bunch of differential geometry on projective spaces because it relates to my work. I don't use Penrose's 'Road to Reality' as a textbook, even though I own it, it's a 'oh, that's an interesting result' book. I don't read published papers from Gauss, Riemann, Lie or Cartan to learn how to describe Kahler moduli spaces, I use textbooks and about 100 A4 sheets of paper doing examples, working things through and calculating things until I've got a firm grasp of it. Neither Euler or myself or anyone reads a book on differential geometry and says "Yep, I can do that now" and can. Every single one of us goes through a ton of paper doing exercises, proving results, practising methods. I am confident that you have barely put pen to paper. In the last 8 months. I generally go through one of those 500 sheet packs every 5 to 10 days.
No, I haven't read the thesis of Halphen or anyone else you mention. Infact, I would say I've only ever read the thesis of two people, both of whom are in my area of work and who are less than 30 years of age. In the vast majority of cases reading a PhD thesis which is older than say 20 years is likely to be less illuminating than reading more recent publications or books. Why? Because in those 20 years more powerful mathematical tools will be developed to streamline methods and more work will have been done.
For instance, the original publication of Maxwell's which contain his electromagnetic equations are horrible to read. They didn't have any of the VERY useful notations like vector calculus or differential forms in his day, they did everything using stuff like quaternions or component by component. Reading Maxwell's original paper is only useful for historical perspective. Reading a book on vector calculus and differential forms is infinitely more useful.
I haven't read any work by Lie but I've read books on Lie algebras written by other people since Lie. Infact, I've got one in arms reach. I've also read several lecture courses on them. And, unlike you, I've actually done questions relating to Lie algebras and Lie groups. Unlike you, I make sure I can do the maths. Unlike you I have been tested by sitting exams on it and unlike you they didn't think blurting out BS copy and paste paragraphs is showing understanding.
You've read a number of peoples thesis but I bet you can't do a single question Euler or I could ask you on that area. You've shown in this thread you will avoid doing anything which requires a working understanding, now jumping from projective geometry to differential geometry. And your grasp of what differential geometry is is so poor you Googled for some papers and then linked us to the results, which were nothing to do with differential geometry! Because rather than get a book entitled "An Introduction to Differential Geometry" and work through it, you prefer to delude yourself into thinking what you're doing at the moment is of any use at all.
I've spent 2 hours this evening (and 10 yesterday) doing a bunch of differential geometry on projective spaces because it relates to my work. I don't use Penrose's 'Road to Reality' as a textbook, even though I own it, it's a 'oh, that's an interesting result' book. I don't read published papers from Gauss, Riemann, Lie or Cartan to learn how to describe Kahler moduli spaces, I use textbooks and about 100 A4 sheets of paper doing examples, working things through and calculating things until I've got a firm grasp of it. Neither Euler or myself or anyone reads a book on differential geometry and says "Yep, I can do that now" and can. Every single one of us goes through a ton of paper doing exercises, proving results, practising methods. I am confident that you have barely put pen to paper. In the last 8 months. I generally go through one of those 500 sheet packs every 5 to 10 days.
Talk about getting reamed!
http://en.wikipedia.org/wiki/Units_of_paper_quantity
http://www.merriam-webster.com/dictionary/ream[2] (Sense 4)
http://en.wikipedia.org/wiki/Units_of_paper_quantity
http://www.merriam-webster.com/dictionary/ream[2] (Sense 4)
QUOTE (AlphaNumeric+Apr 20 2009, 08:54 PM)
I generally go through one of those 500 sheet packs every 5 to 10 days.
Do you print 2 pages per side, front and back?
Do you print 2 pages per side, front and back?
Alphanumeric and Euler, do you use computers and computer graphics software a lot? Those old time mathematicians could not do that (not even pocket calculators poor souls ) Has the computer made things possible that never were before in the world of math?
Ivars, that movie of dimensions had a lot of visual content. Are there any others?
Do mathematicians and math inspired artists displaying their works online?
I admit, I lean towards the art side of things. Do people who are actual mathematicians interested in how things look? Is it a driving desire to try to see these other dimensions and other visual results? When I was in school years ago back in the 1970's, math was rather dull because there wasn't much to see.
) Has the computer made things possible that never were before in the world of math?Ivars, that movie of dimensions had a lot of visual content. Are there any others?
Do mathematicians and math inspired artists displaying their works online?
I admit, I lean towards the art side of things. Do people who are actual mathematicians interested in how things look? Is it a driving desire to try to see these other dimensions and other visual results? When I was in school years ago back in the 1970's, math was rather dull because there wasn't much to see.
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