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

#### Speaker:

Ion Dinca
#### Title:

Geometry of Quadrics
#### Abstract:

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