planerotation

I am working on a geometry problem that requires finding the intersection of two parabolic arcs in any rotation. I was able to intesect a line and a parabolic arc by rotating the plane to align the arc with an axis, but two parabolas cannot both align with an axis. I am working on deriving the formulas, but I would like to know if there is a resource already available for this. I'd first define the equation for the parabolic arc in 2D without rotations: x(t) = ax² + bx + c y(t) = t; You can now apply the rotation by building a rotation matrix: s = sin(angle) c = cos(angle) matrix = | c -s | |

The issue we are trying to solve the issue of locating a point in two different representations of a plane. The first plane we have is rotated to create perspective; the second is a 2d view of that same plane. We have 4 points on each of the plans that we know to be equivalent. The question is if we have an arbitrary point in plane 1, how do we find the corresponding point in plane 2? It is best probably to illustrate the use case in order to best clarify the question. We have an image illustrated on the left. Projective plane 2D layout diagram of space So the givens that we have are the red