Conformal Surface Splines

TL;DR

This paper presents a new framework for controlling conformal surface splines through boundary conditions and point constraints, enabling precise manipulation of surface geometry for applications in modeling and visualization.

Contribution

It introduces novel boundary conditions and constraints for conformal immersions, enhancing controllability and discretization compatibility in surface spline modeling.

Findings

Enables manipulation of surface geometry via point constraints.

Provides a discretization compatible with discrete conformal equivalence.

Demonstrates applications in geometric modeling and visualization.

Abstract

We introduce a family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. Our free boundary conditions fix the metric on the boundary, up to a global scale, and admit a discretization compatible with discrete conformal equivalence. We also introduce constraints on the conformal scale factor, enforcing rigidity of the geometry in regions of interest, and describe how in the presence of point constraints the conformal class encodes knot points of the spline that can be directly manipulated. To control the tangent planes, we introduce flux constraints balancing the internal material stresses. The collection of these point constraints provide intuitive controls for exploring a subspace of conformal immersions interpolating a fixed set of points in space. We demonstrate the applicability of our framework to geometric modeling, mathematical visualization, and form finding.