Does anyone know how to formulate a set of parametric equations that generate an egg-shape?
NOTE. I've asked several people, including my calculus professor, and in every case, what I got were variations on the standard, symmetrical ellipse. What I want can be defined in a few ways; a form whose curvature is a combination of two or more different ellipses. Or, an egg-shaped section from a 3-d hyperbolic form, i.e., revolve:
$$y=\frac{1}{x}$$
then cut it at an angle yielding an egg-form.
Thank you for your time!
NOTE. See fig.
For My Example
$$y=\frac{1}{x}$$
$$y=x-2.02$$
Both on the same plane.
Intersecting the surface $z=1/\sqrt{x^2+y^2}$ with a plane $z=kx+b$ gives a curve in 3-space. Rotating this curve by $\arctan k$ about the $y$-axis gives a curve in a plane parallel to $(xy)$-plane, having parametric equations (for $b^2\ge4k$): $$ \begin{align} &x={1\over2\sqrt{1+k^2}} \left({k^2-1\over k}b+{k^2+1\over k}\sqrt{b^2-4k\cos t}\right)\\ \\ &y={2\sin t\over b+\sqrt{b^2-4k\cos t}}\\ \end{align} $$ This can give an egg shape for suitable values of $b$ and $k$. Here's the result for $k=1$ and $b=2.02$: