I have three points that represent a rigid body. The rigid body undergoes a planar transformation in $\Bbb R^3$ due to rotation and translation. I am working with angular velocity with nonzero $\vec i$, $\vec j$, and $\vec k$ components, and what I am trying to accomplish cannot be done easily in $\Bbb R^3$. How can I map my points to $\Bbb R^2$, perform operations (rotations), and then map back to $\Bbb R^3$?
Robjohn provided a solution here, 3D to 2D rotation matrix , but I'd like to understand the math behind his solution. Thank you
What robjohn does is this:
He selects three points in the plane. The vector $S$ is a unitary vector contined in the plane. The vector $T$ is another unitary vector contained in the plane and perpendicular to $S$, if you do a drawing you will see it clearly.
What he has now is a system of orthonormal vectors that generate the plane, so we have a basis for that plane. Now for every point in the plane, he has the three point coordinates $\lbrace U_k\rbrace$, multiplying (dot product) with $S$ and $T$ will project those coordinates over those vectors, and substracting $P$ before will make $P$ be the origin of the plane., the again, if you make a drawing I'm sure you will see it.
Noe for every point in the plane you will have two coordinates $x,y$, you cand do what ever you want with those, and then transform back.
Drawings really help a lot. Is there anything in particular that you don't understand?