mathematics - Affine transformation: rotate a plane parallel to another plane

I am writing a function that extrudes a 2D shape along a 3D spline, as found in 3D modeling software.

I need a way of translating a set of points P so that they all lie in a new plane L (preserving distances between points).

• P is all points where Z = 0;


• L is defined by (0,0,0), with the normal vector N.

I am using Processing, it has common matrix and vector functionality (but no quaternions) A conceptual or pseudo-code answer is OK.

Answer

I need a way of translating a set of points P

I suppose you mean rotating here?

Let Z = (0,0,1). If cross(N,Z) has length 0, it means that all your points already lie in the desired plane. Otherwise we can build a basis of the target plane:

• U = normalize(cross(N,Z))


• V = cross(N,U)

Now to transform a point P = (x,y,0) so that it lies in your target plane, simply do this:

P2 = x * U + y * V