I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D curve on an elliptical surface described nicely in this post. Could someone kindly help me how this surface bounded by $AB$, $AC$, and $BC$ can be put into an equation? Thanks in advance.
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We can try to approach this as an interpolation problem. Sample the arcs in a few middle points, you'll get a dataset $(x_i, y_i, z_i)$, albeit only at the boundaries. Such highly irregular scattered data interpolation could be dealt with by using radial basis functions, with the thin plate spline as a natural kernel for stress analysis. The RBF interpolant will be a sum of as many terms as there were points, but it is a function you can evaluate on your FEM mesh inside the patch.
with a parametrized solution you might have a parameter for a point M between A and B and another for the point on the ellipsoid between C and M