I was searching for a standard equation of an egg like curve that I can tweak and transform to a ellipse and a circle. I found this link with example $x^2+\left(1.4^x 1.6y\right)^2=1$. A generic version looks like the following $$ax^{2}+(\gamma^{x} by)^{2} = c$$ If I put $\gamma=1$ I get a general equation of ellipse. If I plot the example equation given in that site I get an egg shaped curve as shown below. So my first question is, is this equation known to be a generalisation of ellipse and circle ? But this equation doesn't look like to produce convex shapes for all values, there is a relation between $a, b, c$ and $\gamma$ that produces convex shape. That is why I feel like this is even more generalisation of an ellipse.
Now coming to the second part of my question. Given a Convex Polygon I want to find out the tightest ellipse that bounds the polygon. However in sometimes The input polygon may be somewhat like a trapezium, in that case I want the tightest Oval as shown above that fits the polygon inside. I assume if I can fit an oval to a polygon that oval will look like an ellipse if my input polygon is a rectangle. If oval is defined as the equation above how can I find out the smallest such oval that fits a given convex polygon.