In Homogeneous Coordinates, the point at infinity is represented by a vector pointing to "the horizon".
Is there an equivalent representation of the Point at infinity in geometric algebra (or in clifford algebra)?
I'm thinking that if a blade A has lower dimension than B, then B is infinitely larger than A (like a line has infinite points, and an area has infinite lines), then in some sense B is a kind of infinite compared to A. Homogeneous coordinates just add an extra dimension, to be able to represent the point at infinity, so all of it hints a connection, between infinity and extra dimensions, and I would like to know it more precisely.
Yes. There is quite a lot of recent literature on the subject, and the terms you need to look up are "Conformal Geometric Algebra" and "Projective Geometric Algebra". David Hestenes introduces the former here and here, and Pablo Colapinto goes into further detail in his master's thesis here. Some particularly good resources on the latter are by Eric Lengyel and Charles Gunn.
In projective geometric algebra, we add an extra dimension with a null basis vector, in the same manner as for homogeneous coordinates. The blades through the origin correspond to points, lines, planes, etc. of the projective space. Blades where the extra dimension is zero correspond to points, lines, etc. at infinity.
In conformal geometric algebra we add two extra dimensions, either of opposite signature or both null. These correspond to the choice of origin and the point at infinity. Conformal geometric algebra completely unifies lines and planes with circles and spheres - a flat plane is simply a sphere passing through the point at infinity.