Miter Surface

Author

Eric W. Weisstein
June 3, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/MiterSurface.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

Mitre:=4x^2(x^2+y^2+z^2)-y^2(1-y^2-z^2)

Equation

Expand[Mitre]

AlgebraicDegree[Mitre,{x,y,z}]

Graphic

With[{r=1},ContourPlot3D[Mitre==0,{x,-r,r},{y,-r,r},{z,-r,r},BoxRatios->{1,1,1}]]

Volume

Integrate[Boole[Mitre<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

Integrate[Boole[Mitre<0],{z,-∞,∞},{y,-∞,∞},{x,-∞,∞}]//Timing

POVRay

<<Utilities`POVray`

POVCoefficients[Mitre,{x,y,z}]

Properties

FullSimplify[GaussianCurvature[Mitre,{x,y,z}]]