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Curvature of ellipsoids and other surfaces
Author(s) -
Harris W. F.
Publication year - 2006
Publication title -
ophthalmic and physiological optics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.147
H-Index - 66
eISSN - 1475-1313
pISSN - 0275-5408
DOI - 10.1111/j.1475-1313.2006.00382.x
Subject(s) - ellipsoid , curvature , principal curvature , mathematics , principal axis theorem , mathematical analysis , spheroid , point (geometry) , surface of revolution , sagittal plane , transverse plane , geometry , quadratic equation , surface (topology) , plane (geometry) , physics , mean curvature , medicine , biochemistry , chemistry , structural engineering , astronomy , engineering , in vitro , radiology
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.