help with ellipse equation
I've got an ellipse defined by a central (steep) radius of curvature (R0) and a p-value (shape factor) that is p=1-e^2
I've derived most of the formulas I need for calculating the local radius of curvature at given points. One problem I don't seem to be able to solve is the definition of the ellipse via the flat radius of curvature.
The p-values I've got define the radius of the ellipse at its steep apex and from there it becomes flatter.
I would want to define the ellipse via its flat radius and become steeper from there.
I know that
a = R0 * p
b = SQRT(R0^2*p)
Rflat = a^2/b
so Rflat = (R0 * p)^2 / SQRT R0*p)
can someone derive the expression for Rsteep as a function of Rflat and p?