This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fréchet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. Moreover, to implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc.). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing.

We present a data structure that makes it easy to run a large class of algorithms from computational geometry and scientific computing on extremely poor-quality surface meshes. Rather than changing the geometry, as in traditional remeshing, we consider intrinsic triangulations which connect vertices by straight paths along the exact geometry of the input mesh. Our key insight is that such a triangulation can be encoded implicitly by storing the direction and distance to neighboring vertices. The resulting signpost data structure then allows geometric and topological queries to be made on-demand by tracing paths across the surface. Existing algorithms can be easily translated into the intrinsic setting, since this data structure supports the same basic operations as an ordinary triangle mesh (vertex insertions, edge splits, etc. ). The output of intrinsic algorithms can then be stored on an ordinary mesh for subsequent use; unlike previous data structures, we use a constant amount of memory and do not need to explicitly construct an overlay mesh unless it is specifically requested. Working in the intrinsic setting incurs little computational overhead, yet we can run algorithms on extremely degenerate inputs, including all manifold meshes from the Thingi10k data set. To evaluate our data structure we implement several fundamental geometric algorithms including intrinsic versions of Delaunay refinement and optimal Delaunay triangulation, approximation of Steiner trees, adaptive mesh refinement for PDEs, and computation of Poisson equations, geodesic distance, and flip-free tangent vector fields.

Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than a fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles ( e.g. , positive, or within a given range), and sophisticated boundary conditions ( e.g. , convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.

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Smooth curves and surfaces can be characterized as minimizers of squared curvature bending energies subject to constraints. In the univariate case with an isometry (length) constraint this leads to classic non-linear splines. For surfaces, isometry is too rigid a constraint and instead one asks for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. We present an ecient algorithm for (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology with or without boundary. Our conformal class constraint is based on the discrete notion of conformal equivalence of triangle meshes. The resulting non-linear constrained optimization problem can be solved eciently using the competitive gradient descent method together with appropriate Sobolev metrics. The surfaces can be represented either through point positions or dierential coordinates. The latter enable the realization of abstract metric surfaces without an initial immersion. A versatile toolkit for extrinsic conformal geometry processing, suitable for the construction and manipulation of smooth surfaces, results through the inclusion of additional point, area, and volume constraints.