numerical approximation of Jacobian from distance function
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lowrank commented
Instead of giving an accurate Jacobian analytically, it would be easier to get the Jacobian using finite difference on distance function. It involves the second derivative of the distance function, and should be still quite sensitive to rounding errors.
lowrank commented
it is done for ellipse, with numerical computation of Laplacian for distance function. It should be accurate if the high-precision library is available.
Currently, an error of