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# What is the result of a rotation followed by a translation? What is the rotocenter of the resulting rotation?

I have something that tells me that a rotation followed by a translation is always a rotation...but what is the rotocenter for the resulting rotation?

### 1 Answer

- BenLv 610 years agoFavorite Answer
Notice that rotations always have exactly one fixed point, the center of rotation. So we just need to answer "What point gets fixed when you rotate and then translate?"

This might be a bit difficult to answer without a picture, but here's an attempt. Consider the translation vector and the center of the given rotation. They define a triangle with two vertices at the endpoints of the vector, and the last vertex at the rotocenter. Move the vector about the plane (keeping its direction of course) until the triangle formed is isoceles and the angle formed at the orthocenter is the angle of rotation. (There are actually two such positions; we want the one that makes the vector point opposite to the direction of rotation.) The terminal point of the vector is the rotocenter for the composite rotation: rotating by the given rotation puts us at the tail of the vector, then translating gets us back.

(Once you know that that point is fixed, it follows that the composition is really a rotation. Since rotations and translations are the only isometries that are orientation-preserving, the composition is one of these two types. And translations don't fix any points.)