I'm new to projective geometry. As of my understanding, when we want to map 3d scene to a plane, we can use homogeneous coordinates to imitate the center of projection as eyes and a projective plane that is 1 unit far away from the eyes. Therefore a point in the projective plane corresponds to a line in 3d space and a line corresponds to a plane and all the rays pass through the center of projection.
But when I come across degenerate conics, all the rays converge at one point on a projective plane and it confused me a lot.
My question is: Do all rays need to converge at the origin of the homogeneous coordinate? Appreciate if anyone can help me with it!
If you want to treat the lines in 3d space as equivalent to the points of a projective plane, then they all have to pass through a single point. If you do anything with coordinates, treating that point as the origin makes sense so that you can use linear transformations to keep that one point fixed.
If you do the above, and use a right circular cone with its vertex at the center, then you intersect a plane orthogonal to the cone's axis in a circle. A plane elsewhere but not through the vertex would get intersected in an ellipse, parabola or hyperbola. So your single right circular cone would correspond to a number of different conic sections.
But this would be only a very limited view. You would only get circles centered around the origin of your plane. The position of a conic section in the plane would be tied to its shape. And indeed, degenerate conics consisting of one or two lines would be completely out of the question. So this doesn't do justice to the rich family of possible conic sections.
So instead let's start with a less projective view. You have a cone, anywhere, and you intersect it with a plane. You get a conic section on the plane. Now we switch to a projective setup: connect all the points of the conic section with the origin, and fill any gaps at infinity (e.g. using a limit process). Now in the typical (non-degenerate) case you get some elliptic cone or similar. Which you can view as a linear transformation of a right circular cone in 3d, and therefore as a protective transformation of a circle around the origin in your projective plane. Any two non-degenerate conic sections with real points (as opposed to only complex points on them) can be transformed to any other using a projective transformation.
But in some cases you get something else. In the case of a pair of lines in the plane, you get a corresponding pair of planes through the origin in 3d. This is a qualitatively different configuration. You can't get there from a linear transformation of a right cone. Or speaking in the projective plane, no projective transformation can turn a non-degenerate conic section into a degenerate one (although you can get arbitrarily close).
So the picture of a cone in an arbitrary position is useful to imagine the richness of situations to consider. The picture of lines through the origin is useful to visualize the topology of the projective plane (and the equivalence classes of homogeneous coordinates). By keeping them distinct and not forcing both pictures to be the same, you can get the best of both worlds.