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Curvature of ellipsoids and other surfaces.

Harris WF

Optometric Science Research Group, Department of Optometry, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa. wfh@na.rau.ac.za

From differential geometry one obtains an expression for the curvature in any direction at a point on a surface. The general theory is outlined. The theory is then specialised for surfaces that are represented parametrically as height over a transverse plane. The general ellipsoid is treated in detail as a special case. A quadratic equation gives the principal directions at the point and, hence, the principal curvatures associated with them. Equations are obtained for ellipsoids in general that are generalisations of Bennett's equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

Published 21 August 2006 in Ophthalmic Physiol Opt, 26(5): 497-501.
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