# Creating a 3D ellipsoid from a 2D ellipse (rotated around a symmetric axis)

Hello fellow mathematicians,

I am trying to generate the equation for a 3D tapered spheroid, so that I may obtain its contour plot. I am using Mathematica and/or Wolfram Alpha.

The tapered ellipse $(x^2 + y^2)^2 = 1.2 x^3 + {0.36}xy^2$ is to be rotated around the x-axis to form an egg-shape.

The image below is generally what it should look like, but I do not have the code that produced this image. The difference between my spheroid and the one in the image is that mine has a major axis of $1.2$ whereas the picture denotes a major axis of $1$.

Picture

An equation/code to produce this figure in WA or Mathematica would be most appreciated!

Thank you all!

Generalized tapered ellipse (the above is the special case $a=1.2$): $$(x^2+y^2)^2 = ax^3 + \frac{3a}{10}xy^2$$

Parametrics for the generalized tapered ellipse:

\begin{align} x &=\left(\frac{a}{2}-\frac{b}{2}\sin^2\left(\frac{t}{2}\right)\right)\left(1+\cos t\right) \nonumber \\ y &=\left(\frac{a}{2}-\frac{b}{2}\sin^2\left(\frac{t}{2}\right)\right)\sin t \label{eq:2} \end{align}

where $\forall ((a,b)\in\mathbb{R})\ |\ (a,b)>0$, $a$ and $b$ are the lengths of the major and minor axes of the ellipse. The minor axis is fixed in the general ellipse, but is appropriately represented in the parametrics. Note that the general equation is simply the Cartesian form of the parametrics where minor axis $b=\frac{7a}{10}.$

• Ellipse and ellipsoide are wrong terms here and might be confusing. Your shape is egg-shapes or oval, but not an ellipse. Feb 22, 2018 at 19:52
• Mm, how about tapered prolate spheroid? Feb 22, 2018 at 19:56
• If this is the technical term ;) Feb 22, 2018 at 19:57
• It is, I just thought ellipse/oid was slightly less of a mouthful :) Feb 22, 2018 at 19:58
• I agree that stating this term over and over sounds a bit bloated :D Feb 22, 2018 at 20:06

If you are okay with an implicitely given surface, then just replace all $y^2$ by $y^2+z^2$ to generate the associated solid of revolution around the $x$-axis. This gives you

$$(x^2+y^2+z^2)^2=1.2 x^3+0.36x(y^2+z^2).$$

You can look at the result here.

Finding an explicit parametrization of this surface is exactly as hard as finding one for the 2D-egg-shape. If you have a parametrization $t\mapsto(x(t),y(t))$ of your curve, then the surface can be parametrized by

$$(t,\theta)\;\mapsto\; \begin{pmatrix}x(t)\\ \cos(\theta)y(t)\\ \sin(\theta) y(t)\end{pmatrix}$$

where $\theta\in[0,2\pi)$. In your case this should give the "beautiful" formula

$$(t,\theta)\;\mapsto\;\frac12\left(a-b\sin^2\left(\frac t2\right)\right)\begin{pmatrix} 1+\cos(t)\\ \sin(t)\cos(\theta)\\ \sin(t)\sin(\theta) \end{pmatrix}.$$

• Ahh, very much appreciated. Thank you! imaginary upvote Feb 22, 2018 at 20:08
• Actually, if you are curious, I could give you the parametrization for the 2D curves. In hindsight, I probably should have included that in the OP, but no worries, your equation is quite sufficient. Feb 22, 2018 at 20:14
• @Equinox Sure, include the parametrization in your post, then I can include the one for the surface or write a note on how to find it. Feb 22, 2018 at 20:15
• @Equinox You are welcome. Even if you lack a tiny amount of reputation to upvote, feel free to accept the answer or wait if other useful answers are incoming. Either way, one day it would be nice to accept one of the answers to this question. :) Feb 22, 2018 at 20:16
• Yep I'll probably wait another few hours before the accept, nothing personal! Feb 22, 2018 at 20:17