The cycloid is the curve one sees when looking an a moving bycicle with one part of one wheel illuminated. It is combined by two movements---the circular movement x=sin(t), y=cos(t) of the light around the wheel, and the straight movement x=t, y=1 of the bicycle. The superposition of both movement yields the parametric equations x=t-sin(t), y=1-cos(t)
The slope of the trajectory is (dy/dt)/(dx/dt) = sin(t)/(1-cos(t)) .
The length of the trajectory has first been found by Roberval and Wren. It is . The expression under the radical simplifies to 1-2cos(t). This is ... Therefore the length of one "bridge" is 8.
The area between the trajectory and the x-axis has independently been found by Torricelli, Fermat and Descartes. It is . We get a value of 3 P...