Prove that the composition of a translation and a rotation i

Solution

Let R be a rotation around O by an angle and let T be a translation. We need to investigate T R.

Let l be the line parallel to the direction of the translation T. Let l₁ be the line through O perpendicular to l and let l₂ be the line through O such that R = r₁ r₂, where r_i is reflection with respect to l_i (clearly, l₂ makes the angle /2 with l₁). Then the translation T may be seen as a composition of r₁ and some other reflection r₃ (with respect to the line l₃ parallel to l₂ and orthogonal to l): T = r₃ r₁.

So, we have

T R = (r₃ r₁) (r₁ r₂) = r₃ r₂.

As l₃ is parallel to l₁, they make the same angle with l₂, so the rotation r₃ r₂ is exactly by the same angle as r₁ r₂. The center of the rotation is the intersection of the lines l₂ and l₃.