Consider the following two equations:
1. ax + by + cz = 0.
2. x^2 + y^2 + z^2 = r^2
The curve of intersection is a circle centered along the z axis. Projection onto the xy-plane is an ellipse, and the circle itself is in a rotated plane.
But...does anybody here have a solution for deriving the parametric equations which describe this curve of intersection?
Furthermore, I'm also interested in deriving parametric equations for the surface which lies on this sphere, but has a boundary which is the curve of intersection first described. From what I can so far determine, it is a rotated hemisphere about the xy-plane with center about the z-axis.
I guess a more general question would be how to find new parametric equations for surfaces and curves rotated about the xy-plane?