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Locus on the surface of a cube

Source: Czech and Slovak Olympiad 1972, National Round, Problem 2

July 10, 2024
geometry3D geometrygeometric transformationrotationLocus

Problem Statement

Let ABCDA′B′C′D′ABCDA'B'C'D' be a cube (where ABCDABCD is a square and AA′∥BB′∥CC′∥DD′AA'\parallel BB'\parallel CC'\parallel DD'). Furthermore, let R\mathcal R be a rotation (with respect some line) that maps vertex AA to B.B. Find the set of all images X=R(C)X=\mathcal R(C) such that XX lies on the surface of the cube for some rotation R(A)=B.\mathcal R(A)=B.
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