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Re: ParametricPlot3D from 5.2 to 6.0

• To: mathgroup at smc.vnet.net
• Subject: [mg80566] Re: ParametricPlot3D from 5.2 to 6.0
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Sun, 26 Aug 2007 03:05:51 -0400 (EDT)
• References: <fagtvh$91l$1@smc.vnet.net>

Francois LE COAT wrote:

>Hi,
>
>With the following notebook <http://eureka.atari.org/vrml/etoile.nb>
>I had the following rendering <http://eureka.atari.org/vrml/etoilm.gif>
>using Mathematica v. <= 5.2. With Mathematica 6.0 the interpolation
>method order seems to have changed. I didn't managed to obtain a similar
>rendering with the "star" 3D surface.
>
>Can someone help to find how to describe the "star" the same way again ?
>
>Thanks by advanced.
>
>Regards,
>
>--
>Fran=E7ois LE COAT
>Author of Eureka 2.12 (2D Graph Describer, 3D Modeller)
><http://eureka.atari.org/>
><http://fon.gs/eureka/>
>
>  
>
I like the effect the Max has of speeding up the output.
This code works in 5.x Mathematica here: ( I don't have 6.0: I doubt iy 
will run on my older operating system )
xmin := 0
xmax := 2*Pi
ymin := 0
ymax := Pi
ro[t_, p_] := Max[(1 - 3*Sin[n*p]*Cos[5*t]), 0]
xxx[t_, p_] := ro[t, p]*Sin[p]*Cos[t]
yyy[t_, p_] := ro[t, p]*Sin[p]*Sin[t]
zzz[t_, p_] := ro[t, p]*Cos[p]
Table[ParametricPlot3D[{yyy[t, p], -xxx[t, p], zzz[t, p]}, {t, xmin, 
xmax}, {
    p, ymin, ymax}, PlotPoints -> {
    100, 50}, Axes -> False, Boxed -> False, PlotRange -> All], {n, 1, 5}]
FullSimplify[ExpandAll[xxx[t, p]^2 + yyy[t, p]^2 + zzz[t, p]^2]]

Coming from the FullSimplify:
Cos[5*t]=Sin[5*t+Pi/2]

Your "star" like output is a "fake"
as these are "rose" like parametrics.

Code for a "true" star-like parametric from :
http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/barthdecic/index.html

Clear[x, y, z, p, a]
a = 1;
p = (1 + Sqrt[5])/2;
x = r*Sin[p0]*Cos[t];
y = r*Sin[p0]*Sin[t];
z = r*Cos[p0];
implicit = ExpandAll[8 (x^2 - p^4 y^2) (y^2 - p^4 z^2) (z^2 - p^4 x^2) 
(x^4 +
y^4 + z^4 - 2 x^2 y^2 - 2 x^2 z^2 - 2 y^2 z^2) +
a (3 + 5 p) (x^2 + y^2 + z^2 - a)^2*(x^2 + y^2 + z^2 - (2 - p) a)^2];
Clear[p0, t];
(* flat radial version*)
Table[Plot3D[-Re[r] /. Solve[implicit == 0 , r][[n]], {p0, 0, Pi}, {t, 0, 2*
              Pi}], {n, 1, 10}];
(* projection onto sphere as radial*)
Clear[ro, xxx, yyy, zzx]
xmin := 0
xmax := 2*Pi
ymin := 0
ymax := Pi
ro[t_, p0_] := -Re[r] /. Solve[implicit == 0 , r][[n]]
xxx[t_, p0_] := ro[t, p0]*Sin[p0]*Cos[t]
yyy[t_, p0_] := ro[t,
    p0]*Sin[p0]*Sin[t]
zzz[t_, p0_] := ro[t, p0]*Cos[p0]
Table[ParametricPlot3D[{yyy[t, p0], xxx[t, p0], zzz[t, p0]}, {t, xmin, 
xmax},
  {p0, ymin, ymax}, PlotPoints -> {40, 20},
  PlotRange -> All], {n, 1, 10}]

It is kind of "clunky" and doesn't work very well,
but it is a better "real" star than yours apparently.
Roger Bagula