Discrete Conformal Deformation: Algorithm and Experiments (1412.6892v1)

Abstract: In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle the surface with any topology with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss-Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.

• The paper introduces a novel method that prescribes curvature using vertex scaling and cocircular diagonal switches on triangulated meshes.
• It employs a variational approach with convex energy minimization and Newton’s method, ensuring stability and accurate curvature deformation.
• Numerical experiments validate the algorithm’s robustness and convergence across surfaces with arbitrary topology, enhancing its applicability in computational geometry.

Overview of Discrete Conformal Deformation for Triangulated Surfaces

The paper introduces a novel definition of discrete conformality specifically tailored for triangulated surfaces endowed with flat cone metrics. This work is centered on developing an algorithm to address the problem of prescribing curvature on such surfaces. It details a strategy to deform the curvature by constructing a discrete conformal map between an initial triangulated mesh and its deformed counterpart. The approach can manage surfaces with arbitrary topology, irrespective of the presence of boundaries, and accommodates any prescribed curvature satisfying the Gauss-Bonnet theorem. The authors support their theoretical contributions with numerical experiments demonstrating both convergence and algorithmic performance.

Key Concepts and Methodology

The heart of this research lies in the discrete conformal deformation, which utilizes two primary operations: vertex scaling and cocircular diagonal switch. The operations work on a triangulation $T$ formed by a set of vertices $V$, edges $E$, and triangles $F$. Edge lengths are modified in a controlled manner through vertex scaling by a function $w$ known as the discrete conformal factor. This scaling ensures the preservation of a key geometric invariant known as the length cross ratio, and the condition that the adjusted edge lengths satisfy triangle inequalities.

To resolve potential non-existence issues of discrete conformal factors when prescribing curvature, the authors introduce the diagonal switch mechanism. This operation is employed preemptively before a potential degeneracy occurs. Specifically, a cocircular diagonal switch is executed when the sum of angles opposite a given edge in adjacent triangles exceeds $\pi$. This stabilization approach maintains a valid and robust deformation process across various triangulations.

Analytical and Computational Framework

The problem is tackled through a variational perspective, where the discrete conformal factors form the domain of a convex energy function whose minimization yields the desired curvature deformation. The algorithm employs Newton's method, facilitated by estimating gradients and Hessians of this energy efficiently. The authors also discuss the intricacies of energy decomposition, taking into account the potential non-convexity introduced by combinatorial changes in triangulation.

The concept of a discrete conformal map is rigorously defined, extending the work from Euclidean to more general polygonal metrics. This map is constructed across shared refinements between initial and deformed triangulations and maintains certain projective properties crucial to the geometry of the triangulation.

Numerical Examples and Results

The paper includes numerical trials that underscore the robustness and efficiency of the proposed algorithm. Convergence is verified across various edge cases, demonstrating the method's applicability to both simple and complex mesh configurations. The performance metrics, highlighted through comparisons with existing conformal mapping techniques, solidify the proposed method’s place among contemporary geometric processing tools.

Implications and Future Directions

The advancements presented hold practical implications for fields reliant on computational geometry, such as computer graphics, geometric modeling, and image processing. The ability to prescribe curvature precisely while accommodating complex topological features opens new avenues for advancements in these areas.

Future research may extend the framework to consider hyperbolic metrics and possibly spherical configurations. The authors’ reference to hyperbolic analogs suggests a broader applicability of discrete conformal principles to diverse geometric contexts. Additionally, exploring potential synergies with inversive distance circle packing might address outstanding rigidity questions, enhancing practical algorithms for surface flattening and mapping tasks in computer-aided design and engineering.

In conclusion, this work effectively sets a foundation for further inquiries into discrete geometry and its computational applications, while providing a robust toolkit for practical curvature manipulation on triangulated surfaces.