A rotating circle in perspective

We are looking at a circle of radius $1@ placed in the plane $z=-2@, projected through perspective onto the plane $z=-1@. We rotate it around the axis $z=-2@, $y=0@ to see what effects are visible. Notice how the illusion of three dimensioanlity is created by a small trick. Even with the thickened disk, but without the cross-bar, the illusion is missing.

Rotate by grabbing the end node, translate by grabbing the middle one. On the left in each case is the view in perspective, on the right a side view.

There are a number of things to think about in these pictures. When the circle is on the axis, at first the rotation looks like a translation. If the circle is off the $z@ axis, there are two locations where you get a circular image. One is parallel to the perspective plane, of course. The perspective image seems in general to be an ellipse, although how to calculate its axis of symmetry is not obvious.

Bill Casselman
Mathematics Department
U. B. C.