This thesis introduces discrete conformal variational problems as a versatile toolkit for the construction and manipulation of smooth surfaces in three-dimensions. 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, so we instead ask for minimizers of the Willmore (squared mean curvature) energy subject to a conformality constraint. Conformal transformations are desirable in applications because they preserve angles, and consequentially also mesh quality and the fidelity of geometric data. The conformal structure of a surface can be specified in terms of finitely many geometric parameters, and therefore provides a suitable interface for the free form design of surfaces. We term these surfaces conformal surface splines. Until now, however, there has been no systematic study of discrete conformal variational problems.

The main contribution of this thesis is analysis and numerical computation of discrete constrained Willmore surfaces. We present an efficient algorithm for computing discrete (conformally) constrained Willmore surfaces using triangle meshes of arbitrary topology. We also introduce free boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. We demonstrate the applicability of our framework to geometric modeling and mathematical visualization.

To understand the Möbius invariant discretization of the Willmore energy underlying conformal surface splines, we describe a new quaternionic description of the conformal three-sphere, along with realizations of the spaces of circles, spheres, and point pairs in Euclidean three-space. We give an interpretation of the Willmore energy as the curvature of a quaternionic connection that has a clear geometric interpretation in terms of mean curvature spheres rolling over the surface. Building on this interpretation, we prove that the Möbius invariant discretization of the Willmore energy is equal to the curvature of a discrete connection defined by rolling the edge circumspheres. Conservation laws for discrete Willmore surfaces are also derived, finding applications in the prescription of tangent planes at point constraints.