Graduate Student Seminar, 4:45pm December 8, 2003


Ion Dinca


Geometry of Quadrics


The locus of points in R^3 equidistant form a fixed point and a plane (sphere) is a paraboloid of revolution (respectively elipsoid of revolution if the point is inside the sphere or a a sheet of a hyperboloid of revolution with 2 sheets if the point is outside the sphere).

If we reflect the quadric in its tangent plane at each point we get a 2 dimensional family of quadrics isometric to the given one, all whose foci pass through the given plane (sphere) and if we put these pictures together we get the rolling (without slipping) of a quadric (the reflected one) on an isometric surface ( the given quadric), such that the 2 quadrics always come in contact at points which correspond under the isometry). In a flurry of activity at the end of the 19^th and the begining of the 20^th century, the theory of (isometric) deformation of quadrics was mainly developed by L. Bianchi, P. Calapso, G. Darboux, C. Guichard and J. Weingarten (L. Bianchi and C. Guichard got the prize of the French Academy for this work, Weingarten got his earlier mainly for isometric deformations of surfaces of revolution), and states that for the particular quadrics of revolution considered above the same picture is true: If we roll a paraboloid of revolution on an isometric surface its focus will describe a minimal surface, and if we roll an ellipsoid (a sheet of a hyperboloid with 2 sheets) of revolution on an isometric surface, its foci will describe a constant mean curvature (CMC) surface. Conversely, all minimal and CMC surfaces can be realized in this way in a 2 dimensional fashion.

To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert