# 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. Oct 31, 2019 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 Oct 31, 2019 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. Oct 31, 2019 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