Apparent Contour of Torus

This model is the apparent projection of a torus onto the plane. Consider a translucent torus positioned over a plane with light shining through it. The apparent contour is the shadow of the projection. Notice that some points on the plane have four points of the torus between them and the light source, others have only two points between them, while other points have none. The apparent contour separates the plane into regions with the above property.

The equation of the apparent contour model is very complicated. We start with a geometric torus in 3-space whose "hole" is in the z-direction. We then rotate the torus azimuthally by theta radians. We then do a (Grobner basis) computation that gives us the set of (x,y) values such that there exists some z-value so that (x,y,z) is on the torus and there exists a vertical tangent vector to the torus at (x,y,z).

The model has three parameters: small_radius is the height of of the torus (the "thickness" of an bagel); big_radius is the major radius of the torus (the distance from the "hole" of a bagel to the middle of the doughy portion); and theta is the angle of rotation from the horizontal (-Pi < theta < Pi).

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