Given 2 points on a rigid body, how to mathematically obtain the new position when one of the point is moved in space?

Question

A rigid body in space is defined by a point Xm (xm,ym,zm) and an orientation specified by vector N (na,nb,nc). The body has, on it, another point X1(x1,y1,z1) rigidly constrained to Xm. If the Body is moved to a new position and orientation X’m (x’m,y’m,z’m) and N’(n’a,n’b,n’c), find X’1(x’1, y’1,z’1), the new position of X1 assuming rigidity of the constraint is maintained?

Answer 1

(Sketch) So let $\bf N$ represent the vector defined by the three Euler angles. Assume you know how to calculate the rotation matrix $R_{\bf N}$ associated to $\bf N$ (which can be quite involved!). At that point the solution is simple: $$ \bf X_1'=\bf X_m'+R_{\bf N'}R_{\bf N}^\top\cdot (\bf X_1-\bf X_m), $$ where $(\cdot)^\top $ is the transposition.