Combined Curvature Map Analysis

• Combined curvature maps are composite descriptors that integrate principal, mean, and Gaussian curvature to analyze manifold geometry.
• They are employed in both theoretical frameworks (e.g., optimal transport and curvature prescription) and practical tasks such as point cloud surface reconstruction.
• Estimation methods fuse local PCA or VCM with Weingarten map fitting to achieve robust curvature extraction even under noise and irregular sampling.

A combined curvature map is a composite geometric descriptor that encodes multiple intrinsic and extrinsic curvature quantities—typically principal, mean, and Gaussian curvature—across a manifold or a sampled point cloud. Such maps arise in both the theoretical context of geometric analysis (e.g., Gauss curvature prescription and optimal transport) and in algorithmic geometry processing (e.g., surface reconstruction, noise-robust curvature estimation). While the exact formulation or visualization of a “combined curvature map” is often application-dependent and implementation-specific, the unifying theme is the pointwise construction and interpretation of curvature-informatics as a single scalar field or as a colormap, integrating several curvature estimators for downstream analysis or visualization.

1. Theoretical Foundations: Curvature Prescription and the Combined Curvature–Map Problem

In geometric analysis, a prototypical combined curvature-map problem involves prescribing both the Gauss curvature and the image of the Gauss map for hypersurfaces, such as the graph of a convex function $u: \Omega\subset\mathbb{R}^n\to\mathbb{R}$. For the hypersurface $$\Sigma_u = \{(x,u(x))\in\mathbb{R}^{n+1}: x\in\Omega\}$$, the Gauss curvature at $(x,u(x))$ is given by

$$G(x) = \frac{\det D(N \circ \Phi_u)(x)}{\det D\Phi_u(x)},$$

where $N(x)$ is the outer unit normal and $\Phi_u(x) = (x, u(x))$. This leads to the Monge–Ampère equation,

$\det D^2 u(x) = K(x)\,(1+|Du(x)|^2)^{(n+2)/2},$

where $K(x)$ is the prescribed curvature. The “combined” aspect arises when one imposes an additional second boundary value: that the image of the Gauss map $x \mapsto N(x)\in S^n_-$ fills a prescribed subset $\Lambda\subset S^n_-$, translating into a system for both curvature and directional image (Guillen et al., 2020).

This system is equivalently recast via optimal transport as a map from source measure $\mu = K(x)\,dx$ on $\Omega$ to target measure $\nu = (-y_{n+1})\,d\mathrm{Vol}_{S^n}|_\Lambda$ on $\Lambda$, with cost $c(x,y) = x\cdot y/y_{n+1}$ and the optimal map coinciding with the Gauss map. This optimal transport structure encodes the combined curvature and map prescription.

2. Algorithmic Curvature Estimation and the Combined Curvature Map on Point Clouds

In practical geometry processing, the combined curvature map typically refers to a fused scalar field constructed from principal, mean, and Gaussian curvature estimators across discrete data. Algorithms for point cloud data sampled from a manifold $M\subset\mathbb{R}^N$ follow a multi-stage process (Spang, 2023, Cao et al., 2019):

1. Neighborhood selection: $k$-nearest neighbors or fixed-radius $\epsilon$-balls.
2. Tangent/normal basis estimation: via local PCA or Voronoi Covariance Measure (VCM).
3. Weingarten map estimation: Using local normal differences, solve a least-squares system to fit the local shape operator $S_p$.
4. Principal, mean, and Gaussian curvature extraction: Eigenvalues $\kappa_1, \kappa_2$ of $S_p$ yield $H=(\kappa_1+\kappa_2)/2$, $K=\kappa_1\kappa_2$.
5. Fusion into combined scalar map: Weighted sum $C = \alpha|H| + \beta|K|$, or derived invariants such as the shape index or curvedness.

The summary below tabulates common approaches to combined curvature mapping on point clouds:

Step	Method	Scalar(s) Produced
Neighborhood	k-NN, $\epsilon$-ball	Local point set
Basis Estimation	PCA, VCM	Tangent/normal at each point
Weingarten Map	Least-squares fit (WME/VWME)	Principal curvatures $\kappa_{1,2}$
Scalar Curvature Fusion	Linear, index-based, or custom	Combined map $C(x)$

Editor's term: “VWME” refers to Voronoi-augmented Weingarten Map Estimator.

3. Visualization and Normalization Strategies

The typical practice is to normalize the individual scalar curvature fields (mean $H$, Gaussian $K$, $|\kappa_1|$, $|\kappa_2|$) globally to $[0,1]$, and then encode them within a colormap for visualization. For example, $H$ may control hue, $|K|$ the intensity or saturation in an HSV or RGB mapping. Such a map is overlaid on the discrete mesh or point cloud (Spang, 2023).

Fusion strategies include:

• Linear fusion: $C = \alpha |H| + \beta |K|$ to emphasize different geometric features.
• Shape index: $$S = (2/\pi) \arctan((\kappa_2+\kappa_1)/(\kappa_2-\kappa_1)) \in [-1,1]$$.
• Bending energy: $\sqrt{H^2-K}$ for elastic energy applications.

4. Statistical and Numerical Properties

Combined curvature map estimation inherits statistical properties from its constituent estimators. For the Weingarten map via least squares, the mean-squared error converges as $\mathrm{MSE}=O(h^4) + O((nh^m)^{-1})$ for bandwidth $h$ and $m$-dimensional manifold, with optimal $h \asymp n^{-1/(m+4)}$ yielding $\mathrm{MSE} = O(n^{-2/3})$ for surfaces ($m=2$) (Cao et al., 2019).

Noise robustness is significantly enhanced by fusing VCM-estimated normals with Weingarten estimation (VWME): VCM exhibits $$\|\mathrm{VCM}_K - \mathrm{VCM}_{K^\epsilon}\|_\infty \le C d_H(K,K^\epsilon)^{1/2}$$, so small perturbations induce small normal errors (Spang, 2023). Under heavy noise or adversarial sampling, VCM-based combined curvature maps preserve higher cosine similarity of normals and lower mean-squared error for curvatures compared to PCA-only approaches.

5. Generalizations: Moment Maps and Coupled Curvature Equations

Beyond Euclidean immersions, combined curvature–map constructs arise in complex geometry and gauge theory via the notion of moment maps for coupled equations. For a product Kähler manifold $(X,\omega_0)$ with additional Kähler forms $\omega_1, \dots, \omega_k$, the combined moment map

$\mu = (\mu_0, \dots, \mu_k)$

encodes both scalar curvature and various coupled Ricci or generalized p-degree curvature conditions (Lee, 2020).

The vanishing of this combined moment map yields simultaneous solutions to a family of “coupled” geometric PDEs, including the cscK system, coupled Kähler–Yang–Mills, and deformed Hermitian Yang–Mills equations. The associated generalized Mabuchi functional is geodesically convex and its critical points correspond to solutions of the full coupled system.

6. Applications and Practical Implementations

Combined curvature maps underpin tasks in geometric analysis (e.g., boundary regularity in Gauss curvature prescription (Guillen et al., 2020)), differential geometry, and data-driven surface analysis. In computational geometry, they facilitate feature detection, segmentation, and denoising of point-sampled surfaces. Robust estimation methods—particularly those using VCM and VWME pipelines—are favored for their stability and applicability in arbitrary ambient dimension, especially under significant noise and for nonuniformly sampled data (Spang, 2023, Cao et al., 2019).

Implementation details such as neighborhood selection (best chosen as $k \sim O(\log n)$ for stability), kernel selection, and normalization are critical for robust, interpretable combined curvature visualizations.

7. Limitations and Outlook

Current approaches to combined curvature mapping depend critically on the reliability of local manifold structure estimation and are sensitive to sampling density, outliers, and noise. Although methods such as VWME attain high robustness, there remain open challenges for very high-dimensional embedding, extremely irregular point spacing, and surfaces with high curvature concentration. A plausible implication is that further advances in combined curvature mapping will require improved adaptive graph construction, anisotropic localization, or data-driven regularization for extreme regimes.

The combined curvature map paradigm continues to generalize through the intersection of optimal transport, moment map techniques, and robust discrete geometric analysis, informing advances in both theoretical and applied geometry.