rigid-body transformation

A rigid-body transformation is a way of moving an object through space that never bends, stretches, or warps it. Imagine picking up a solid wooden block and carrying it to a new spot, perhaps turning it as you go. The block ends up somewhere new and maybe facing a new way, but its shape is untouched: every corner is still the same distance from every other corner. That distance-preserving quality is exactly what makes the motion "rigid."

Any such motion is built from just two ingredients combined: a rotation (a turn about some axis) and a translation (a straight slide from one place to another). Together they can take an object from any starting pose to any ending pose. Crucially, no rigid transformation can ever change the object's size, squash its angles, or flip it into a mirror image; if a motion did any of those, it would no longer count as rigid. This is why it is the natural language for describing how real, solid robot parts move.

Their real power shows up when you chain them. The transformation from a robot's shoulder to its elbow, followed by the one from elbow to wrist, followed by wrist to fingertip, can all be composed into a single transformation from shoulder straight to fingertip. Doing them one after another, and folding the steps into one, is how a robot computes where its hand is from the angles of all its joints.