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

Question

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!

Addendum, more equations!

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

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 $b=\frac{7a}{10}.$

Answer 1

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.

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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

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