# Parameterizing triangular meshes with complex geometric constraints

OData: XML JSON What's this?Given a 3D triangle mesh, our task is to compute a parameterization, or in other words, to flatten it to the plane. There are many applications in computer geometry and graphics, which benefit from a mesh parameterization being available for a mesh, such as (i) texturing, (ii) surface fitting, or (iii) mesh repair and restructuring. Parameterization is a topic of intense research; the task is quite challenging, besides geometric optimization, computational efficiency is also a major concern.

It is a fundamental result of differential geometry, that a surface with nonzero (Gaussian) curvature cannot be mapped to the plane without distortion. Thus, parameterization methods aim at minimizing certain distortion measures, such as the preservation of (i) angles, (ii) areas, or (iii) congruency of mesh triangles.

Most practical applications require that a parameterization should satisfy certain constraints. Previous work has focused primarily on two major kinds of constraints: for texturing it is often necessary to constrain discrete positions, while for quadrilateral remeshing it is crucial that sharp edges map to constant coordinate lines in the parameter plane. This thesis concerns higher level, feature-based geometric constraints. Examples of such constraints are that (i) a sequence of edges is to be mapped to a straight line, (ii) a closed sequence of edges is to be mapped to a circle, (iii) a planar curve, or (iv) a developable region is to preserve its shape during the parameterization -- while also minimizing some sort of distortion measure, at the same time.

Parameterization with geometric constraints is a novel research topic. First, we survey and evaluate previous literature, then we examine how to to enforce geometric constraints by extending known parameterization algorithms. We present a novel method for constrained mesh parameterization, which can be considered a variant of the widely used As-Rigid-As-Possible and vector-field based parameterization algorithms, and which is capable of enforcing any and all geometric constraints under consideration. We also examine methods for enforcing geometric constraints via the iterative deformation of a previously computed parameterization. We implement the mathematical algorithms, and demonstrate their capabilities using several examples. As a practical application of our algorithms, we compute optimal parameterizations for approximating data points by free-form surfaces.