You know the equation for a unit circle in the xy plane (z-axis normal) is:

Moving the center and scaling the radius is the easy part.

The main difficulty is changing plane of the circle. Rotating from a z-axis normal to one with normal vector (a,b,c).

So we want to find the rotation transformation that takes the vector (0,0,1) to (a,b,c).

There are an infinite number of different rotations.

The most intuitive one would be to rotate about the axis

by

.

There is a well known formula to find the rotation matrix given axis and angle:

Rotation matrix - Wikipedia, the free encyclopedia
Once you find your rotation matrix

. You new parameteric equations are

given by: