On curves with constant curvatures
AbstractOne of the fields of research of Computer Aided Geometric Design is approximating complex curves by simpler curves. Curves with constant curvatures are useful tools for these purposes. However, parametrizations of such curves are not always easily given. In this paper we will derive several necessary and sufficient geometric conditions for a curve to have constant curvatures, both in Euclidean geometry and in affine geometry.
Wilhelm Blaschke and Kurt Reidemeister. Vorlesungen
über Differentialgeometrie und Geometrische
Grundlagen von Einsteins Relativitätstheorie:
II Affine Differentialgeometrie. Springer
Berlin Heidelberg, 1923.
Manfredo P. do Carmo. Differential Geometry of
Curves and Surfaces. Prentice-Hall, Upper Saddle
River, New Jersey, 1976.
H. Glück. Higher Curvatures in Euclidean Spaces.
The American Mathematical Monthly, 73:699–704,
Heinrich W. Guggenheimer. Differential Geometry.
McGraw-Hill Book Company, Inc., New York,
Felix Klein. Vergleichende Betrachtungen uber
neuere geometrische Forschungen. 1872.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted under the conditions of the Creative Commons Attribution-Share Alike (CC BY-SA) license and that copies bear this notice and the full citation on the first page.