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.
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...'
A general affine coordinate change in 3d is Ax + b, where A is a 3-by-3 matrix and b is a 3d vector. If you add a fourth coordinate that you set to 1, you can write it as a 4-by-4 matrix. That's homogeneous coordinates.
Makes an affine space linear. It was a natural extension to provide direct transformations that encapsulate symmetry transformations, introduced by Mobius in his barycentric calculus.
The problem with math is that there's more you need to know than there's class time to teach it to. Projective space is a good example. It comes up in several different ways, but you're probably not going to see them all.
The motivation is better than most, trying to treat infinity as just another case instead of an exceptional one, but, to me at least, it still feels like a "hack" to put in a third dimension and then insist on dividing by it to reproject whenever you want to do anything useful.
I'm a little used to the trick but it'd be nice to give some deeper motivation of why the trick works in the first place and why this particular representation.
Lemme try. Have you ever seen a railroad vanishing in the horizon? You know the rails never meet and yet, they definitely meet in infinity before your very eyes. That's because you see a 2D projection of a 3D space. In 3D, you'll never find the intersection. It gives one of the best intuitions on how to work with points in infinity.
If you want to study how a polynomial P(X,Y) behaves at infinity, you can't set eg X,Y=infinity, but you can take P(X/W,Y/W) when W goes to zero. Now clear denominators by multiplying by an appropriate power of W and you get a homogeneous polynomial in X,Y,W. You can now set W=0 directly. Scaling a solution cX,cY,cW doesn't change the original solution since (cX)/(cW) = X/W, etc.
I guess the important part is clearing denominators turns problems with division into linear problems.
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.
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...'
A general affine coordinate change in 3d is Ax + b, where A is a 3-by-3 matrix and b is a 3d vector. If you add a fourth coordinate that you set to 1, you can write it as a 4-by-4 matrix. That's homogeneous coordinates.
Makes an affine space linear. It was a natural extension to provide direct transformations that encapsulate symmetry transformations, introduced by Mobius in his barycentric calculus.
The problem with math is that there's more you need to know than there's class time to teach it to. Projective space is a good example. It comes up in several different ways, but you're probably not going to see them all.
The motivation is better than most, trying to treat infinity as just another case instead of an exceptional one, but, to me at least, it still feels like a "hack" to put in a third dimension and then insist on dividing by it to reproject whenever you want to do anything useful.
I'm a little used to the trick but it'd be nice to give some deeper motivation of why the trick works in the first place and why this particular representation.
Lemme try. Have you ever seen a railroad vanishing in the horizon? You know the rails never meet and yet, they definitely meet in infinity before your very eyes. That's because you see a 2D projection of a 3D space. In 3D, you'll never find the intersection. It gives one of the best intuitions on how to work with points in infinity.
If you want to study how a polynomial P(X,Y) behaves at infinity, you can't set eg X,Y=infinity, but you can take P(X/W,Y/W) when W goes to zero. Now clear denominators by multiplying by an appropriate power of W and you get a homogeneous polynomial in X,Y,W. You can now set W=0 directly. Scaling a solution cX,cY,cW doesn't change the original solution since (cX)/(cW) = X/W, etc.
I guess the important part is clearing denominators turns problems with division into linear problems.