Re: problems plotting curvature

[Heike Gramberg]

It looks like the dummy variables x, y in X1, X2, LL, MM, and NN are interfering with
x, y in the definition of your ColorFunction in the third plot. I'm not entirely sure why the
second plot didn't cause any problems, though. The easy way around this would be to
replace x and y in your ColorFunction definition to s and t, say. A safer option would be
to use a Block structure for your derivatives, e.g.

X1[u_, v_] := Block[{x}, D[vf[x, v], x] /. {x -> u}];

etc.

Heike.

On 21 Apr 2011, at 08:09, CoolGenie wrote:

> Hello,
> I'm trying to write a program for computing and plotting Gaussian
> curvature. Basically what I want is to set different colors on the
> surface depending on the curvature. My code is as follows
>
> Clear[vf, u, v, x, y, X1, X2, EE, FF, GG, n, nn, LL, MM, NN, KG];
> vf[u_, v_] :=
> vf[u, v] = {(2 + Cos[v]) Cos[u], (2 + Cos[v]) Sin[u], Sin[v]};
> (* wektory styczne *)
> X1[u_, v_] := D[vf[x, y], x] /. {x -> u, y -> v};
> X2[u_, v_] := D[vf[x, y], y] /. {x -> u, y -> v};
> (* wspolczynniki pierwszej formy podstawowej *)
> (* symbol E jest zajety wiec uzywamy EE, FF, GG *)
> EE[u_, v_] := Dot[X1[u, v], X1[u, v]];
> FF[u_, v_] := Dot[X1[u, v], X2[u, v]];
> GG[u_, v_] := Dot[X2[u, v], X2[u, v]];
> (* wektor normalny do powierzchni *)
> nn[u_, v_] := Cross[X1[u, v], X2[u, v]];
> n[u_, v_] := Normalize[nn[u, v]];
> (* wspolczynniki drugiej formy podstawowej *)
> LL[u_, v_] := Dot[D[vf[x, y], x, x], n[x, y]] /. {x -> u, y -> v};
> MM[u_, v_] := Dot[D[vf[x, y], x, y], n[x, y]] /. {x -> u, y -> v};
> NN[u_, v_] := Dot[D[vf[x, y], y, y], n[x, y]] /. {x -> u, y -> v};
> (* krzywizna Gaussa *)
> KG[u_, v_] :=
> KG[u, v] = (LL[u, v]*NN[u, v] -
> MM[u, v]*MM[u, v])/(EE[u, v]*GG[u, v] - FF[u, v]*FF[u, v] +
> 10^-10);
> (*min =First[FindMinimum[{KG[u,v], 0<=u<=2Pi,0<=v<=2Pi},u,v]]
> max=First[FindMaximum[{KG[u,v], 0<=u<=2Pi,0<=v<=2Pi},u,v]]*)
>
> ParametricPlot3D[vf[u, v], {u, 0, 2 Pi}, {v, 0, 2 Pi},
> ColorFunction -> Function[{x, y, z, u, v}, Hue[u]],
> ColorFunctionScaling -> False]
>
> ParametricPlot3D[{u, v, KG[u, v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi},
> ColorFunction ->
> Function[{x, y, z, u, v}, Hue[(KG[u, v] - min)/(max - min)]],
> ColorFunctionScaling -> False]
>
> ParametricPlot3D[vf[u, v], {u, 0, 2 Pi}, {v, 0, 2 Pi},
> ColorFunction -> Function[{x, y, z, u, v}, Hue[KG[u, v]]],
> ColorFunctionScaling -> False]
>
> This produces 3 plots: the first one gives out a torus, colored
> according to u and the picture is ok. The second one gives Gaussian
> curvature plotted as a surface above the axes of u and v, colored
> according to the value of this curvature. Again the plot is ok. The
> problem is with the third plot where I try to color the torus with KG.
> When calling this function, I get the errors:
>
> General::ivar: 2.3258915732471794` is not a valid variable. >>
>
> and so on and the torus is not colored in full: some areas are white.
> I can upload a picture or you could run your code and see that it's
> not ok.
> Does anyone have a solution for this problem?
> Regards,
> Przemek
>

[CoolGenie]

This helped, thank you!

P.