Name of this surface?

[bryanlettner]

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

[Vitaliyk (Vitaliy Kaurov)]

Wolfram Language has data on surfaces. You could dig through those.

SurfaceData["SampleEntities"]

SurfaceData["Properties"]

[bryanlettner]

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.

[danl (Daniel Lichtblau)]

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 *)