Equation of a cylinder with a profile / ellipsoid I have a geometry of a cylinder that is curved along both the lengths where there is generally a height of a cylinder. I am aware of the equation of a cylinder. I wanted to know what could the equation of a curved cylinder be as a function of the x,y, and z coordinates.? Also, could this curved cylinder be called an ellipsoid?
I have attached a geometry herewith along with the dimensions.
2D

3D


That would probably be called a "barrel shape". The equation depends on the type of curve you choose for the barrel. If it's a circular curve, if the $y$ axis is along the axis of symmetry of the barrel and we count the centre of the barrel as the origin. then the equation for the barrel would be something like
$$x^2+z^2=r^2$$
for the $x$ and $z$ coordinates, and
$$(r+h)^2+y^2=R^2$$
Here $R$ is the radius of curvature of the barrel side and $h<R$ is a constant which expresses an offset from the origin of the centre of the radius of curvature.
So we end up like
$$(\sqrt{x^2+z^2}+h)^2+y^2=R^2.$$
In the case of your diagram, it looks like you have $h=80 \text{mm}$ and $R=90 \text{mm}$.
I couldn't persuade Wolfram Alpha to do a 3D plot of this, but here is a 2D version.

In practice you would do this parametrically.
