what is the formula to find the angle of rotation of a shape?

Rotation Formula

A formula which transforms a given coordinate system by rotating it through a counterclockwise angle  about an axis . Referring to the above figure (Goldstein 1980), the equation for the "fixed" vector in the transformed coordinate system (i.e., the above figure corresponds to an alias transformation), is

(1) (2) (3)

(Goldstein 1980; Varshalovich et al. 1988, p. 24). The angle  and unit normal  may also be expressed as Euler angles. In terms of the Euler parameters,

(4)

The  rotation matrix can be calculated in the Wolfram Language as follows:

With[{n = {nx, ny, nz}}, Cos[phi] IdentityMatrix[3] + (1 - Cos[p]) Outer[Times, n, n] + Sin[p] {{0, n[[3]], -n[[2]]}, {-n[[3]], 0, n[[1]]}, {n[[2]], -n[[1]], 0}}