Why the Projective Plane Is Almost Inevitable
Making Peace with the Projective Plane
John D. Cook reflects on how projective space and homogeneous coordinates once felt like arbitrary abstractions, taught top-down with no motivation. He argues they're actually natural: adding a third coordinate to represent points at infinity eliminates special cases like vertical lines, and normalizing by that coordinate yields homogeneous coordinates. Programmers, especially in elliptic curve cryptography, rediscover this same trick concretely when forced to handle the mysterious point at infinity.
Exceptional cases and hand-wavy references to objects "at infinity" go away when moving to projective space.
- lopotubez
Since it can be shown that there is a bijection with lines through the origin and points on the projective plane, this means that if you trace families of parallel lines on the projective plane, that will correspond to the horizontal line through the origin or "point at infinity". It also shares the same direction with the parallel lines.
I imagine this when laying down thinking my arm is the 1-dim subspace and the ceiling is the projective plane, if I trace families of paralell lines, they all end up being the same line (I know this sounds stupid but it is an easy way for me to visualize it).
This also makes it clear where the general equation of lines come from. Imagine the same line through the origin [a:b:c] and another line presumedly perpendicular to it [x:y:z], then by Pythagoras it can be shown that ax + by + cz = 0, which is precisely the plane perpendicular to [a:b:c]. So if you imagine that plane also horizontal then that corresponds to the "horizon at infinity". The general equation of the line aX + bY + c = 0 is just the intersection with the projective plane with homogeneous coords (X,Y,1), for X = x/z and Y = y/z, z != 0. Points in the projective plane correspond to lines through the origin, and lines in the projective plane correspond to planes through the origin.
I always wonder why the projective plane is not the "default" plane that is taught in school when reality is seemingly projective.
- nh23423fefe
The only time i've understood applications of projective geometry was seeing rotations and translations unified using exponentials of bivectors. I still dont get homogenous coords. Like i get how f(lambda*x)=lambda^n*f(x) and that means f is like a homomorphism but i've never solved any problem sets and understood why one would do this in the first place. I get what stereographic projection is, but can't apply it to problems.
whereas i constantly see applications of other math objects i've learned about. i never hear myself think, 'oooh yeah thats projective and so...'
- xeonmc
Always wondered why they don't just call it "directions space" or "magnitudeless coordinates" instead.