Ellipse in intrinsic coordinates

[chronon]

Does anyone know an equation for an ellipse (or other conics) in intrinsic coordinates, that is direction and curve length. It's the sort of thing that looks like it should be simple, but I've a feeling it may in fact be somewhat messy.

 

[traianus]

See this link and hopefully there is what you need

http://mathworld.wolfram.com/Ellipse.html

See also

http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html