I touched on homogeneous coordinates at school 57 years ago and today plucked my old textbook by E A Maxwell off the bookshelf, blew off the dust, and read some of it before looking on the internet to get a more up-to-date perspective, upon which I found your video. Excellent tutorial. Thank you. I'm thinking of using homogeneous coordinates to express rigid body displacements and forces, because with the line at infinity, every 2D displacement becomes a rotation, and I think the same will be true in 3D as well. I expect this has been investigated by others, but my experience of engineers is that they are so in the thrall of computer software that their incentive to delve down into fundamentals is rather limited.
Very nice idea ! I'm not sure if it has been done. It seems you can get a lot of mileage out of the idea. You should be able to re-express many laws of physics purely in terms of rotations. I think there are lots of new ideas that could be obtained by thinking about these elements at infinity, because usage of Cartesian coordinates usually stops people thinking about them. I also guess that new perceptions of physics can be obtained by considering the imaginary points and lines of projective geometry, but that is another story th-cam.com/video/Tb7_He_UIfY/w-d-xo.html
Thank you, Richard. Just read your encouraging response. I love the link you give at the end and I must watch you more. There's much to learn! I'm getting into homogeneous co-ordinates in 3D space, and am okay with points and planes, which are duals of each other. But I'm struggling with how to manage lines. Can you recommend where I should look for guidance? My first aim is to model my engineering shapes by surface triangles. Until I'm confident in the basics I'm not competent to delve into 3D homogeneous mechanics.
@@grahamparkhouse1577 Sure. I would recommend H.S.M Coexter's book `Projective Geometry'. You could just dive straight in at chapter 12, but that just covers homogeneous coordinates in 2D. I'm not sure I've come across much literature focused on homogeneous coordinates in 3D, but I'm sure you could find it with google. One of my favorite books on the projective geometry, which focuses a lot on 3D stuff, is Lawrence Edwards book `Projective Geometry'. This book is a masterpiece. It uses (almost) no algebra at all, and goes really deeply into projective geometry using pure english. I spend about a year studying it. However, this book does not use any coordinates, and so you might find its approach a bit long winded and non-computational, and the book may not help you much with 3D homogeneous coordinates specifically. I am curious about what engineering shapes you seek to model.
Richard, thank you for this reply - only just read it. I really need to go 3D. Lawrence Edward's book sounds fascinating. Your description reminds me of 'Geometry and the Imagination' by Hilbert and Cohn-Vossen, an amazing book which is almost entirely free of anything numeric, yet the mathematical insights are mind blowing. I expect you know it. Many years ago I read Geometrical Methods of Mathematical Physics with a theoretical physicist who helped me through it. So my appetite for adventure is well whetted. But I'm busy at the moment. I have just read the amazon reviews of Lawrence Edward's book, or rather the one review that had anything to say, which I found encouraging. I love the circular points at infinity, and I know that they are in the domain I should be looking at. I am going to buy it, because I think understanding is what I need at the moment. My experience is that the numbers fall into place very easily once the concepts are grasped. Thank you for the recommendation.
wow! Thank you so much for this simply beautiful explanation of the expressive power of homogenous coordinates. They allow us to fly through geometry without the encumbrance of a plethora of axioms and afford us expansive views of relationships never before seen. Your simple and clear exposition dispenses with the modern clutter of music and graphics that would have impeded clear understanding of the topic. Thank you.
I really appreciate Richard taking the time to create this series. Overall series is unique and allows non-university and university students both to learn about this interesting and important area. However: 1. Regarding production: a) Would have appreciated the lighting to be better and the board to be bigger? Shadow and reflections make it harder to follow; lighting needs improvement. 2. Regarding content: a) suggest starting with intuitive explanation first and then ramping up to abstract. Intuitive interpretation which folks are familiar with such as cartesian coordinate, how they tie up with homogeneous coordinates really helps students visualize, follow and also remember (generally abstract is harder to remember). I suppose mathematicians think more abstractly so it is also a difference in thinking and style.
thanks very much. I was looking at these coordinate systems for projection in games. Saw some other videos and I understood the 'procedure' but you've helped me understand the concept.
Ok cool, my guess is your really gonna dig this, its based on coxeter's chapter. The material here, plus parts 2 & 3 (coming) provide the fastest route to understanding what projections & conics are that I've found
is it true that in your example: if 3x+2y-2=0 and x is positive then y is negative e.v.v. that we can say that if ax + by - c = 0 then x and y oppose each other in sign, if c= not zero? that`s a nice line of thinking to write (muy + nuz) = (muy + z) and that then it equals to or tends to (y) if y approaches or is equal to infinity (or zero). that i like. is it correct to state that at 43:30 the fourth point D can be written in the terms of the previous point A, B and C if and only if D is collinear with any pair of A, B and C? very nice video. thanks.
I highly recommend the dynamic geometry graphing software Cinderella2. It blows Geogebra out of the water in terms of computational speed and robustness, and even has physics simulation options. It has Euclidean, spherical, and hyperbolic modes, as well as polar display modes which are all viewable side by side. cinderella.de/tiki-index.php?page=Download+Cinderella.2&bl
I touched on homogeneous coordinates at school 57 years ago and today plucked my old textbook by E A Maxwell off the bookshelf, blew off the dust, and read some of it before looking on the internet to get a more up-to-date perspective, upon which I found your video. Excellent tutorial. Thank you. I'm thinking of using homogeneous coordinates to express rigid body displacements and forces, because with the line at infinity, every 2D displacement becomes a rotation, and I think the same will be true in 3D as well. I expect this has been investigated by others, but my experience of engineers is that they are so in the thrall of computer software that their incentive to delve down into fundamentals is rather limited.
Very nice idea ! I'm not sure if it has been done. It seems you can get a lot of mileage out of the idea. You should be able to re-express many laws of physics purely in terms of rotations. I think there are lots of new ideas that could be obtained by thinking about these elements at infinity, because usage of Cartesian coordinates usually stops people thinking about them. I also guess that new perceptions of physics can be obtained by considering the imaginary points and lines of projective geometry, but that is another story th-cam.com/video/Tb7_He_UIfY/w-d-xo.html
Thank you, Richard. Just read your encouraging response. I love the link you give at the end and I must watch you more. There's much to learn! I'm getting into homogeneous co-ordinates in 3D space, and am okay with points and planes, which are duals of each other. But I'm struggling with how to manage lines. Can you recommend where I should look for guidance? My first aim is to model my engineering shapes by surface triangles. Until I'm confident in the basics I'm not competent to delve into 3D homogeneous mechanics.
@@grahamparkhouse1577 Sure. I would recommend H.S.M Coexter's book `Projective Geometry'. You could just dive straight in at chapter 12, but that just covers homogeneous coordinates in 2D. I'm not sure I've come across much literature focused on homogeneous coordinates in 3D, but I'm sure you could find it with google. One of my favorite books on the projective geometry, which focuses a lot on 3D stuff, is Lawrence Edwards book `Projective Geometry'. This book is a masterpiece. It uses (almost) no algebra at all, and goes really deeply into projective geometry using pure english. I spend about a year studying it. However, this book does not use any coordinates, and so you might find its approach a bit long winded and non-computational, and the book may not help you much with 3D homogeneous coordinates specifically. I am curious about what engineering shapes you seek to model.
Richard, thank you for this reply - only just read it. I really need to go 3D. Lawrence Edward's book sounds fascinating. Your description reminds me of 'Geometry and the Imagination' by Hilbert and Cohn-Vossen, an amazing book which is almost entirely free of anything numeric, yet the mathematical insights are mind blowing. I expect you know it. Many years ago I read Geometrical Methods of Mathematical Physics with a theoretical physicist who helped me through it. So my appetite for adventure is well whetted. But I'm busy at the moment. I have just read the amazon reviews of Lawrence Edward's book, or rather the one review that had anything to say, which I found encouraging. I love the circular points at infinity, and I know that they are in the domain I should be looking at. I am going to buy it, because I think understanding is what I need at the moment. My experience is that the numbers fall into place very easily once the concepts are grasped. Thank you for the recommendation.
wow! Thank you so much for this simply beautiful explanation of the expressive power of homogenous coordinates. They allow us to fly through geometry without the encumbrance of a plethora of axioms and afford us expansive views of relationships never before seen. Your simple and clear exposition dispenses with the modern clutter of music and graphics that would have impeded clear understanding of the topic. Thank you.
I really appreciate Richard taking the time to create this series. Overall series is unique and allows non-university and university students both to learn about this interesting and important area. However:
1. Regarding production: a) Would have appreciated the lighting to be better and the board to be bigger? Shadow and reflections make it harder to follow; lighting needs improvement.
2. Regarding content: a) suggest starting with intuitive explanation first and then ramping up to abstract. Intuitive interpretation which folks are familiar with such as cartesian coordinate, how they tie up with homogeneous coordinates really helps students visualize, follow and also remember (generally abstract is harder to remember). I suppose mathematicians think more abstractly so it is also a difference in thinking and style.
Great explanation! It's clear that you really love math and that's so inspiring! Thank you for this work.
I guess, there is a typo in the line joining the points (1,0,0) and (0,1,0) at 26:52. It should be [0,0,1] instead of [1,0,0]. No?
Yes, you're right.
was quite confused for a minute or two
yep, i spotted that too ! In simple words the line should have a "1" where the 2 points ( that it passes through ) do not have a "1".
Thank you, I appreciate your work
thanks very much. I was looking at these coordinate systems for projection in games. Saw some other videos and I understood the 'procedure' but you've helped me understand the concept.
Great stuff Rich! Im not prepared to dig in just yet, but maybe by the end of the month. Thanks again!
Ok cool, my guess is your really gonna dig this, its based on coxeter's chapter. The material here, plus parts 2 & 3 (coming) provide the fastest route to understanding what projections & conics are that I've found
Absolutely! :-D
The title reads Homogeneous Coordinates Part 1. Is Part 2 available?
th-cam.com/video/a3YNx2sC-5Y/w-d-xo.html
is it true that in your example: if 3x+2y-2=0 and x is positive then y is negative e.v.v. that we can say that if ax + by - c = 0 then x and y oppose each other in sign, if c= not zero?
that`s a nice line of thinking to write (muy + nuz) = (muy + z) and that then it equals to or tends to (y) if y approaches or is equal to infinity (or zero). that i like.
is it correct to state that at 43:30 the fourth point D can be written in the terms of the previous point A, B and C if and only if D is collinear with any pair of A, B and C?
very nice video. thanks.
North, East, South, Well, Done
at 35:30 - he meant y and z ( not y and x )
I highly recommend the dynamic geometry graphing software Cinderella2. It blows Geogebra out of the water in terms of computational speed and robustness, and even has physics simulation options. It has Euclidean, spherical, and hyperbolic modes, as well as polar display modes which are all viewable side by side.
cinderella.de/tiki-index.php?page=Download+Cinderella.2&bl
Very inspiring
Anyone who did not understand the point on line test , look at slide 5 - www.cs.cornell.edu/courses/cs664/2008sp/handouts/cs664-9-camera-geometry.pdf