Jacobian. This oversampling is done via rejection sampling as illustrated by
Diaconis, Holmes, and Shahshahani (2013).</p></li>
<li><p>The more analytic technique is to invert the Jacobian symbolically in order
to define an interior-preserving parameterization \(q:S\to f(M)\), as
illustrated for 2-manifolds by Arvo (2001). Sampling \(P\) uniformly on
\(S\) then produces a uniform sample \(q(P)\) on \(f(M)\). The
interior-preserving map also enables stratified sampling on the manifold via
stratification of t