# How to predict the shape of the circle from any point of view

by OFF_Smog
Tags: circle, point, predict, shape, view
 P: 582 consider the simpler problem of a circle on a hinge along its horizontal diameter. The circle is oriented so that initially its plane is vertical. Let phi be the angle the plane of the circle makes with the vertical axis as we rotate it around the hinge. now suppose we have a light source (at infinity) and look at the shadow cast by the circle on a nearby vertical plane. Initially (phi=0) the shadow is a circle of the same radius as the circle. As we rotate the circle on its hinge, the shadow changes to an ellipse in the same manner as the camera in your diagram viewing the circle from different angles. The horizontal axis of the shadow corresponds to a in your diagram. It is constant and always equal to the radius of the circle. The height of the shadow (b) changes with phi and a simple diagram will show you that its value is $$b=a\cos\phi$$ I will leave you to relate phi back to the angles in your diagram.