Homogeneous Coordinates Part 1

Homogeneous coordinates provide a powerful way to approach the projective plane analytically. By describing each point in the plane as the ratio of three numbers, homogeneous coordinates allow one to refer to points on the `line at infinity' on the projective plane, and hence describe the projective plane in its entirety. Important notions like those of collineations and correlations see clearer under homogeneous coordinates and this leads us to a better understanding of the true nature of conic sections. In this, first part, we examine the basic definitions and ideas behind modeling the plane via homogeneous coordinates.

Details about the polarities refered to can be seen in my video:
`An Introduction To Counter Space'

[grahamparkhouse]

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.

[KarlLew]

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.

[ashishjain]

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.

[tusharnakini]

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?

[eugeneskokowski]

Great explanation! It's clear that you really love math and that's so inspiring! Thank you for this work.

[callmedeno]

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.

[myName-dgqm]

Great stuff Rich! Im not prepared to dig in just yet, but maybe by the end of the month. Thanks again!

[hedronsciences]

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.

[Андреич-сн]

The title reads Homogeneous Coordinates Part 1. Is Part 2 available?

[malharjajoo]

at 35:30 - he meant y and z ( not y and x )

[williamwesner]

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.