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Spherical curves and quadratic relationships for special functions

Published online by Cambridge University Press:  17 February 2009

Even Mehlum  and
Jet Wimp
Even Mehlum
Affiliation:
Central Institute for Industrial Research, Oslo, Norway.
Jet Wimp
Affiliation:
Department of Mathematical Sciences, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A.
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Abstract

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We show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

Type
Research Article
Information
The ANZIAM Journal , Volume 27 , Issue 1 , July 1985 , pp. 111 - 124
Copyright
Copyright © Australian Mathematical Society 1985

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