Polynomial curve expressed in homogeneous coordinates (blue)
Quite often small facts or thoughts have big impacts on me.
The other day in my Electrical Engineering 209AS course, my Professor Ankur Mehta briefly and sort of nonchalantly pointed out that any translation of an object in any spatial dimension can simply be represented as a rotation about an axis positioned at infinity.
While this was sort of a side note to his overall point regarding homologous coordinate systems and corresponding matrix transformations, I felt my mind slip into wonder at that satisfying thought. Such subtle intricacies of mathematics hold so much gravity about them when understood in the manor with which they deserve. Concepts of infinity and many-dimensional matrix transformations can often be difficult to fully comprehend and visualize innately. I believe it's through the small, somewhat quiet moments which one feels a connection to the universe, with the clicking of a small concept being employed. I swear that feeling is like the locking pins in a ratcheting mechanism, having just rotated past a new set of sprocket teeth. In the same way, it prevents one from rotating in the opposing direction - somehow as a means to prevent backward motion in understanding.
The following is taken from wiki:Homogeneous Coordinates; read if you'd like, it's quite beautiful.
The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, (x, y, 1) is such a system of homogeneous coordinates for the point (x, y). For example, the Cartesian point (1, 2) can be represented in homogeneous coordinates as (1, 2, 1) or (2, 4, 2). The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates.