Let S be a C⁴ surface and x(u) be a curve embedded in S. Our goal is to devise a curve x:u -> S that minimizes some energy functional E (e.g. geodesic distance).

Let F be a quadratic function from d-dimensional parameter space to R. We can find the global minimum of F in d-space. But we want the minimum of F (with respect to a metric tensor) on some manifold in d-space. This minimum is the point on closest (again wrt metric tensor) to the minimum of F in d-space.

We can use a predictor-corrector method to move from an arbitrary surface point to the minimum on the surface. The predictor is just a small step into space (moving off the surface ) toward the global minimum in d-space. The corrector snaps this new position back to the surface .

Curvature of computed by flattening the "cone" created by the target global minimum point p and the individual stepping points along the surface. Need to pick a step size based on this curvature. They estimate the curvature (and the distance to the goal point p) can be computed after a single iteration.

Thus we proceed through the space of curves x(u) on the surface S until we converge on the one that minimizes the appropriate energy functional.