A Variational Framework for Curve Shortening in Various Geometric Domains

Geodesics measure the shortest distance (either locally or globally) between two points on a curved surface and serve as a fundamental tool in digital geometry processing. Suppose that we have a parameterized path (gamma)(t) = x(u(t),v(t)) on a surface x = x(u,v) with gamma(0)=p and gamma(1)=q . We formulate the two-point geodesic problem into a minimization problem integral H-1(0)(?xu(u ')(t)+xv(v ')(t)?)dt , where H(s) satisfies H(0)=0,H '(s) > 0 and H ''(s) >= 0 for s > 0 . In our implementation, we choose H(s)=(e)s(2)-1 and show that it has several unique advantages over other choices such as H (s) = s(2) and H(s) = s. It is also a minimizer of the traditional geodesic length variational and able to guarantee the uniqueness and regularity in terms of curve parameterization. In the discrete setting, we construct the initial path by a sequence of moveable points {xi}(n)(i=1) and minimize n-expressionry sumexpressiontion H-n(i=1)(?x(i)-x(i+1)?) . The resulting points are evenly spaced along the path. It's obvious that our algorithm can deal with parametric surfaces. Considering that meshes, point clouds and implicit surfaces can be transformed into a signed distance function (SDF), we also discuss its implementation on a general SDF. Finally, we show that our method can be extended to solve a general least-cost path problem. We validate the proposed algorithm in terms of accuracy, performance and scalability, and demonstrate the advantages by extensive comparisons.

Automated summary

Geodesics play a crucial role in digital geometry processing, providing a way to measure the shortest distance between points on a curved surface. We address the two-point geodesic problem by formulating it as a minimization problem and introducing a specific function H(s). Our method not only guarantees the uniqueness and regularity of curve parameterization, but also demonstrates advantages in handling parameterized surfaces, meshes, point clouds, implicit surfaces, and solving the general least-cost path problem.