how to calculate new 3D point location after shifting

Question

There are three 3D points on a rigid body P1(x1,y1,z1), P2(x2,y2,z2) and P3(x3,y3,z3). After tilting the body new locations of P1 and P2 are known. How to calculate the new location P3.

Answer 1

Let $Q$ be the orthogonal projection of $P_3$ on the straight line $P_1 P_2$. Let $t\in \mathbb{R}$ such that $Q = (1-t)P_1 + t P_2$. Let $r$ be the distance $P_3 Q$, and let $P_1^\prime,P_2^\prime,P_3^\prime$ be the images of the 3 points by the transformation. Let $Q^\prime = (1-t)P_1^\prime + t P_2^\prime$, then $P_3^\prime$ can be any point on the circle $C(Q^\prime, r)$ in the plane orthogonal to the line $(P_1^\prime, P_2^\prime)$.