Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.
Marden, Albert School of Mathematics, University of Minnesota, Minneapolis, Minnesota.
- Additional Readings
A surface obtained by rotating a circle about a line that lies in the plane of the circle but that has no points in common with the circle (see illustration). The equations x = u cos v, y = u sin v, z = [r2 − (u − b)2], b > r > 0, represent the upper half of the torus obtained by rotating about the z axis a circle of radius r whose center is the point (b, 0, 0). The parameter u represents the distance of a point P of the torus from the z axis, and v is the angle of rotation. According to whether b < u ≤ b + a or b − a ≤ u < b or u = b, the corresponding point P is elliptic, hyperbolic, or parabolic, respectively, and the Gauss curvature of the surface at P is positive, negative, or zero. See also: Differential geometry
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