Let A be a point which moves with constant angular speed on the circle with centre O and radius 1, and B a point wich moves with an angular speed which is q times (q > 1) that of A on the circle with centre O and radius r (r > 1). We study the envelope of the straight lines AB. Both limit cases r = 1 and r = +∞ (with q constant) are epicycloids; the relation between the shapes of these epicycloids does not depend on q.