International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 26, Pages 1633-1644
doi:10.1155/S016117120320819X
On some new properties of the spherical curvature of
stereographically projected analytic curves
Department of Mathematics, The Pennsylvania State University, Mont Alto 17237-9799, PA, USA
Received 2 August 2002
Copyright © 2003 Stephen M. Zemyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We discover new information about the spherical curvature of
stereographically projected analytic curves. To do so, we first
state formulas for the spherical curvature and spherical torsion
of the curves on S2 which result after stereographically
projecting the image curves of analytic, univalent functions
belonging to the class 𝒮. We then derive results
concerning the location of the critical points of the spherical
curvature, considered both as a function of one and two
variables. Further analysis leads to a maximum principle for the
spherical curvature functions.