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Mathematics Recreation Visual Arts Geometry Graphics and Visualization Wolfram Language

Name of this surface?

Bryan Lettner

Posted 10 years ago

I've been throwing some Sines and Cosines around with ParametricPlot3D, and I've found some interesting surfaces. Does this one have a name? ParametricPlot3D[ { Sin[u] Sin[v] Sin[v + u], Cos[u] Cos[v] Cos[v + u], Cos[u] Sin[v] Sin[v + u] }, {u, 0, Pi}, {v, 0, Pi}] It has some interesting features. Viewed from different angles, it can appear as a: -deltoid (tricuspoid) -parabola -cardioid -Cool bowl shape resembling spherical harmonics Attachments:

POSTED BY: Bryan Lettner

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Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 10 years ago

I do not know the answer. But possibly having the implicit form might help for looking it up. tpolys = TrigExpand[{Sin[u]Sin[v]Sin[v + u], Cos[u]Cos[v]Cos[v + u], Cos[u]Sin[v]Sin[v + u]}]; tp2 = tpolys /. {Cos[u] -> cu, Cos[v] -> cv, Sin[u] -> su, Sin[v] -> sv}; gb = GroebnerBasis[ Join[{x, y, z} - tp2, {cu^2 + su^2 - 1, cv^2 + sv^2 - 1}], {x, y, z}, {cu, cv, su, sv}]; implicit = First[gb] (* Out[154]= x^2 y + x^2 z - z^2 + y z^2 + z^3 *)

I do not know the answer. But possibly having the implicit form might help for looking it up.

  tpolys = TrigExpand[{Sin[u]*Sin[v]*Sin[v + u], Cos[u]*Cos[v]*Cos[v + u], Cos[u]*Sin[v]*Sin[v + u]}];
    tp2 = tpolys /. {Cos[u] -> cu, Cos[v] -> cv, Sin[u] -> su, Sin[v] -> sv};
    gb = GroebnerBasis[
       Join[{x, y, z} - tp2, {cu^2 + su^2 - 1, cv^2 + sv^2 - 1}], {x, y, z}, {cu, cv, su, sv}];
    implicit = First[gb]

    (* Out[154]= x^2 y + x^2 z - z^2 + y z^2 + z^3 *)

POSTED BY: Daniel Lichtblau

Bryan Lettner

Posted 10 years ago

Thank you Vitaliy. Unfortunately I'm using Mathematica 9 and it doesn't have this feature. I will do some more digging around and post here if I find it.