Special Affine Curvature

Author

Eric W. Weisstein
August 13, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/SpecialAffineCurvature.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SpecialAffineCurvature.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

Examples

curves={
{"circle",a{Cos[t],Sin[t]}},
{"ellipse",{a Cos[t],b Sin[t]}},
{"hyperbola",{a Sec[t],b Tan[t]}},
{"parabola",a{2t,t^2}}
};

Grid[{#[[1]],FullSimplify[SpecialAffineCurvature[#[[2]],t],0<b<a&&-Pi<t<Pi]}&/@curves,Dividers->All,Alignment->{{Left,Center},Automatic}]//TraditionalForm

Special Cases

Hyperbola

FullSimplify[-(((Sec[t])^2)/(((-a b ((Sec[t])^3)))^(2/3))),a>0&&b>0&&-Pi/2<t<Pi/2]