Parametric equation for Taubin heart surface

We know the implicit equation for building a Taubin's heart surface: $$\\\left(x^2+\frac{9y^2}{4}+z^2-1\right)^3-x^2 z^3-\frac{9y^2 z^3}{80}=0$$ Can we convert to parametric equation equivalent?

What tools can we use to solve this?

If the answer is positive, i am interested about parametric equation for this surface

• You can start by building a local ON coordinate system aligned with tangent plane and normal vector somehow. Then "integrate" your way forward on this surface and letting your ON system follow the trajectory. – mathreadler Oct 31 '19 at 14:14
• Thanks for your answer! But i am not a mathematician. Please, describe your answer in more detail, may be with computer algebra system – PavelDev Oct 31 '19 at 16:38
• I am also not a mathematician, but an engineer. :o) If I can find a good explanation then I will write an answer. Right now I am not so sure my answer will make sense, but if I come up with it then I will come back and write. – mathreadler Oct 31 '19 at 16:44

I am not sure, if it's too late for helping, but you can try below an alternative equation to plot a parametric heart surface.

Julia's parametric heart surface equation:

x=Sin[v](15 Sin[u]-4 Sin[3u])

y=8 Cos[v]

z=Sin[v](15 Cos[u]-5 Cos[2u]-2 Cos[3u]-Cos[2u])

In Wolfram Mathematica Code:

ParametricPlot3D[{Sin[v](15Sin[u]-4 Sin[3 u]),8Cos[v],Sin[v](15Cos[u]-5 Cos[2u]-2 Cos[3 u]-Cos[2 u])},{u,0,2Pi},{v,0,Pi},Axes->True,AxesLabel-> {"x","y","z"},PlotStyle->Red,PlotLabel->Style["Julia's Parametric Heart Surface",14,Bold],ViewPoint->Front]


Julia's parametric heart surface