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Implementation
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dip[ins_][g_]:=$DisplayFunction[Insert[g,ins,{1,1}]]
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Monkey Saddle
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Here is the standard parametrization of the monkey saddle:
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monkeysaddle[u,v]
:[font = output; output; inactive; preserveAspect; endGroup; ]
{u, v, u^3 - 3*u*v^2}
;[o]
3 2
{u, v, u - 3 u v }
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Let us calculate its mean curvature
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mcurvature[monkeysaddle][u,v]
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(27*u*(-u^4 + 2*u^2*v^2 + 3*v^4))/
(1 + 9*u^4 + 18*u^2*v^2 + 9*v^4)^(3/2)
;[o]
4 2 2 4
27 u (-u + 2 u v + 3 v )
-------------------------------
4 2 2 4 3/2
(1 + 9 u + 18 u v + 9 v )
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We can also plot the mean curvature.
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Plot3D[mcurvature[monkeysaddle][u,v]//Evaluate,
{u,-1.5,1.5},{v,-1.5,1.5}];
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% End of Graphics
MathPictureEnd
:[font = text; inactive; preserveAspect; ]
Finally we plot the monkey saddle and use the mean curvature
to color it.
:[font = input; preserveAspect; startGroup; ]
ParametricPlot3D[
Append[monkeysaddle[r Cos[theta],r Sin[theta]],
Hue[(mcurvature[
monkeysaddle][r Cos[theta],r Sin[theta]]+0.88)/
1.76]]//Evaluate,
{r,0,1},{theta,0,2Pi},
Boxed->False,Axes->None,
Lighting->False,
PlotPoints->{20,40},
PlotRange->All];
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 266; pictureHeight = 287; endGroup; endGroup; endGroup; ]
%!
%%Creator: Mathematica
%%AspectRatio: 1.07565
MathPictureStart
%% Graphics3D
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% End of Graphics
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