Consider the unit circle: Center O (0,0), Right P (1,0), Up Q (0,1). The standard parameterization is . The same formula works in your application: Find O,P,Q and use
Find a parameterisation of the circle of radius one formed by intersecting the plane and the sphere .
(Hint: Start by finding two orthogonal unit vectors in the plane).
Starting with the hint, my first vector is and my second vector is
Where do I go from here?