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...'
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.
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.
And what this means geometrically is that the translation becomes a shear in the 4d space, which is a linear transform. It’s easier to visualize if you think in 2D; the matrix
(1 b)
(0 1)
maps a point (x, 1) to (x+b, 1) which is exactly what we want, while vectors (which we model as points (x, 0)) remain at (x, 0) as they should - translation shouldn’t act on vectors!
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.
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.
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.
It's a nice extension of the plane to get some results. For example https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem (Gemini recommends me this video, but I didn't see it https://www.youtube.com/watch?v=5OYLnMyesX4&t ). The idea is that in the complex protective plane, two lines intersect in exactly a point, and two circles intersect in exactly 4 points (yes 4). It's very easy to count intersections there, so after you counted them there you can try to count how many of them are in the usual plane and how many are "fake" intersections in the extension.
There are many similar results, but this is one of the most easy to understand.
There are 3 different methods to extend geometric spaces (i.e. affine spaces or vector spaces) with a set of points at infinity.
The projective method extends an N-dimensional space with a set of points at infinity that forms an (N-1)-dimensional space.
That means that a projective straight line has 1 point at infinity, a projective plane has 1 straight line at infinity, a 3-dimensional projective space has a plane at infinity and so on.
The other 2 methods are exemplified by the extended real number line and the extended complex plane.
The extended real number line has 2 points at infinity, not 1 point like a projective line. This method is seldom used in higher dimensions, but nonetheless it works in any dimension, i.e. a plane can be completed with a circle at infinity, a 3-dimensional space can be completed with a sphere at infinity, and so on.
The extended complex plane has only 1 point at infinity, not a straight line like a projective plane. This method can also be used for spaces with any number of dimensions, which are always completed with a single point at infinity, regardless of the number of dimensions. This method is used in the so-called inversive geometry (which studies invariants to transformations that preserve angles in spaces with 3 or more dimensions).
While the other 2 methods of handling infinities are also useful in certain applications, the projective method of handling infinities is by far the most beautiful in my opinion, because it allows the unification of many concepts, simplifying a lot their handling. When I first learned projective geometry when young, it was quite a revelation. I consider only very few parts of mathematics and of sciences in general as beautiful as projective geometry.
When thinking about a problem, there is absolutely no need to think about the hack of homogeneous coordinates. That hack corresponds with a particular model of a projective space, where an N-dimensional projective space is mapped to a hypersphere embedded in an N+1 dimensional linear space, e.g. a projective plane is mapped to a sphere (with double cover).
When you make computations, homogeneous coordinates are the easiest, but when you think qualitatively it is easy to think directly in a projective plane or in a 3-dimensional projective space, without using any extra dimension. E.g. in the latter you have an extra plane at infinity. Each point in that plane corresponds with a straight line in the 3D space, and also with any other straight line parallel to it. Each straight line at infinity corresponds with a plane in the 3D space, and also with all planes parallel to it. Each circle at infinity corresponds with a conical surface in the 3D space, and also with all other conical surfaces that have the same vertex angle and the axes parallel between themselves (and also with all hyperboloids with 1 or 2 sheets that approach asymptotically those conic surfaces towards infinity). A polygon at infinity corresponds with a pyramidal surface and with all other pyramidal surfaces with parallel axes and equal angles. And so on, any geometric figure at infinity will correspond with some surfaces or curves in the 3D space, which end at infinity into that geometric figure. If you understand which is the figure at infinity in which a surface of 3D space ends, you can understand in what kind of surface the original surface can be transformed, when the figure from infinity is brought to a finite distance by a projective transformation, and another section of the surface goes to infinity.
For a 3D space, it is slightly more difficult to visualize a flat plane at infinity, instead of visualizing a single point at infinity or an enclosing sphere at infinity, as in the other 2 methods of handling infinities, but the rewards are greater, so the effort of understanding the projective method is worthwhile.
Consider a simple embedding of words into a 'vector' of dimension d.
One can, however, homogenize it by adding an extra dimension such that the first d dimension catches only the sense of the word and that extra dimension indexes through its synonyms. The sense being the common projective shadow of those words.
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.
My mother's master's thesis was on the projective plane! She took (plane) geometric theorems in the regular space, transformed to the projective plane, did another transform that is valid there (you can swap points and lines, or something like that - I don't recall exactly), then transformed back to non-projective space, and had a new theorem, which was true because the original theorem was true.
That’s cool! Probably the second transformation was to dualize. Points in the projective plane correspond to lines in the dual projective plane and vice versa.
The "directions space", which is embedded in an (N+1)-dimensional space is equivalent with an N-dimensional projective space, but it is not the projective space.
There is a bijection between these 2 spaces, so you can compute anything about a projective space using the "directions space" with an extra dimension, but nonetheless they are different things.
The geometric properties of projective transformations are different in the projective space itself and in the "directions space" corresponding to it. E.g. in the former the projective transformations preserve straight lines and certain ratios of the distances measured along those lines, so they transform geometric figures in a certain way.
An example that makes obvious that the 2 spaces are different is that if you make a map of the Earth using a gnomonic projection, the surface of the Earth corresponds to the "directions space" (where the direction is specified by longitude and latitude), while the flat map, which is distorted, corresponds to the projective space. (While the gnomonic projection is related to projective geometry, the orthographic projection is related to affine geometry and the stereographic projection is related to the conformal and inversive geometries).
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...'
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.
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.
And what this means geometrically is that the translation becomes a shear in the 4d space, which is a linear transform. It’s easier to visualize if you think in 2D; the matrix
maps a point (x, 1) to (x+b, 1) which is exactly what we want, while vectors (which we model as points (x, 0)) remain at (x, 0) as they should - translation shouldn’t act on vectors!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.
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.
"Scaling a solution cX,cY,cW..." you are in fact stating the definition of homogeneity here.
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.
It's a nice extension of the plane to get some results. For example https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem (Gemini recommends me this video, but I didn't see it https://www.youtube.com/watch?v=5OYLnMyesX4&t ). The idea is that in the complex protective plane, two lines intersect in exactly a point, and two circles intersect in exactly 4 points (yes 4). It's very easy to count intersections there, so after you counted them there you can try to count how many of them are in the usual plane and how many are "fake" intersections in the extension.
There are many similar results, but this is one of the most easy to understand.
There are 3 different methods to extend geometric spaces (i.e. affine spaces or vector spaces) with a set of points at infinity.
The projective method extends an N-dimensional space with a set of points at infinity that forms an (N-1)-dimensional space.
That means that a projective straight line has 1 point at infinity, a projective plane has 1 straight line at infinity, a 3-dimensional projective space has a plane at infinity and so on.
The other 2 methods are exemplified by the extended real number line and the extended complex plane.
The extended real number line has 2 points at infinity, not 1 point like a projective line. This method is seldom used in higher dimensions, but nonetheless it works in any dimension, i.e. a plane can be completed with a circle at infinity, a 3-dimensional space can be completed with a sphere at infinity, and so on.
The extended complex plane has only 1 point at infinity, not a straight line like a projective plane. This method can also be used for spaces with any number of dimensions, which are always completed with a single point at infinity, regardless of the number of dimensions. This method is used in the so-called inversive geometry (which studies invariants to transformations that preserve angles in spaces with 3 or more dimensions).
While the other 2 methods of handling infinities are also useful in certain applications, the projective method of handling infinities is by far the most beautiful in my opinion, because it allows the unification of many concepts, simplifying a lot their handling. When I first learned projective geometry when young, it was quite a revelation. I consider only very few parts of mathematics and of sciences in general as beautiful as projective geometry.
When thinking about a problem, there is absolutely no need to think about the hack of homogeneous coordinates. That hack corresponds with a particular model of a projective space, where an N-dimensional projective space is mapped to a hypersphere embedded in an N+1 dimensional linear space, e.g. a projective plane is mapped to a sphere (with double cover).
When you make computations, homogeneous coordinates are the easiest, but when you think qualitatively it is easy to think directly in a projective plane or in a 3-dimensional projective space, without using any extra dimension. E.g. in the latter you have an extra plane at infinity. Each point in that plane corresponds with a straight line in the 3D space, and also with any other straight line parallel to it. Each straight line at infinity corresponds with a plane in the 3D space, and also with all planes parallel to it. Each circle at infinity corresponds with a conical surface in the 3D space, and also with all other conical surfaces that have the same vertex angle and the axes parallel between themselves (and also with all hyperboloids with 1 or 2 sheets that approach asymptotically those conic surfaces towards infinity). A polygon at infinity corresponds with a pyramidal surface and with all other pyramidal surfaces with parallel axes and equal angles. And so on, any geometric figure at infinity will correspond with some surfaces or curves in the 3D space, which end at infinity into that geometric figure. If you understand which is the figure at infinity in which a surface of 3D space ends, you can understand in what kind of surface the original surface can be transformed, when the figure from infinity is brought to a finite distance by a projective transformation, and another section of the surface goes to infinity.
For a 3D space, it is slightly more difficult to visualize a flat plane at infinity, instead of visualizing a single point at infinity or an enclosing sphere at infinity, as in the other 2 methods of handling infinities, but the rewards are greater, so the effort of understanding the projective method is worthwhile.
Consider a simple embedding of words into a 'vector' of dimension d.
One can, however, homogenize it by adding an extra dimension such that the first d dimension catches only the sense of the word and that extra dimension indexes through its synonyms. The sense being the common projective shadow of those words.
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.
My mother's master's thesis was on the projective plane! She took (plane) geometric theorems in the regular space, transformed to the projective plane, did another transform that is valid there (you can swap points and lines, or something like that - I don't recall exactly), then transformed back to non-projective space, and had a new theorem, which was true because the original theorem was true.
That’s cool! Probably the second transformation was to dualize. Points in the projective plane correspond to lines in the dual projective plane and vice versa.
Yes, dualize. Her title was "Metric Specializations of Projection Plane Duals".
Always wondered why they don't just call it "directions space" or "magnitudeless coordinates" instead.
The "directions space", which is embedded in an (N+1)-dimensional space is equivalent with an N-dimensional projective space, but it is not the projective space.
There is a bijection between these 2 spaces, so you can compute anything about a projective space using the "directions space" with an extra dimension, but nonetheless they are different things.
The geometric properties of projective transformations are different in the projective space itself and in the "directions space" corresponding to it. E.g. in the former the projective transformations preserve straight lines and certain ratios of the distances measured along those lines, so they transform geometric figures in a certain way.
An example that makes obvious that the 2 spaces are different is that if you make a map of the Earth using a gnomonic projection, the surface of the Earth corresponds to the "directions space" (where the direction is specified by longitude and latitude), while the flat map, which is distorted, corresponds to the projective space. (While the gnomonic projection is related to projective geometry, the orthographic projection is related to affine geometry and the stereographic projection is related to the conformal and inversive geometries).
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