Analysis of the Geometric Heat Flow Equation: Computing Geodesics in Real-Time with Convergence Guarantees

TL;DR

This work tackles real-time computation of geodesics on Riemannian manifolds by analyzing the geometric heat flow (GHF) PDE and its Jacobi-flow variant for perturbations $J=\partial_\tau c$. It proves that the Jacobi heat flow is exponentially stable in $L_2$ when the curvature bound $\langle J, R(J, \partial_s c) \partial_s c\rangle_g < 4\langle J, J\rangle_g$ holds, and that $L_2$ convergence to a geodesic is asymptotic in general. A pseudospectral solver based on Chebyshev polynomials discretizes the coordinate form $(1/\alpha)\partial_\tau x_i = \partial_s^2 x_i + \sum\Gamma^i_{jk} \partial_s x_j \partial_s x_k$, enabling geodesics to be computed in milliseconds and validated against gradient-descent approaches on classic surfaces. The method demonstrates speed and accuracy advantages in 2D surface geodesics and in contraction-based control loops, highlighting its practical impact for real-time planning and control in geometric settings.

Abstract

We present an analysis on the convergence properties of the so-called geometric heat flow equation for computing geodesics (shortest-path~curves) on Riemannian manifolds. Computing geodesics numerically in real-time has become an important capability in several fields, including control and motion planning. The geometric heat flow equation involves solving a parabolic partial differential equation whose solution is a geodesic. In practice, solving this PDE numerically can be done efficiently, and tends to be more numerically stable and exhibit a better rate of convergence compared to numerical optimization. We prove that the geometric heat flow equation is globally exponentially stable in $L_2$ if the curvature of the Riemannian manifold is not too positive, and that asymptotic convergence in $L_2$ is always guaranteed. We also present a pseudospectral method that leverages Chebyshev polynomials to accurately compute geodesics in only a few milliseconds for non-contrived manifolds. Our analysis was verified with our custom pseudospectral method by computing geodesics on common non-Euclidean surfaces, and in feedback for a contraction-based controller with a non-flat metric for a nonlinear system.

Figures (2)

• Figure 1: Convergence rate tests on spherical surface. (a): The proposed method has a high exponential convergence rate as $\alpha$ increases. (b): The exponential convergence rate of the proposed method is reduced as the radius of the sphere becomes smaller (curvature becomes larger).
• Figure 2: Comparison of the PDE pseudospectral method (ours) and gradient descent with a contraction-based controller. (a): The proposed method can be more than 3x faster than numerical optimization. (b) The Riemannian energy evolves nearly identically for both methods.

Theorems & Definitions (14)

• Proposition 1
• proof
• Proposition 2
• proof
• Corollary 1
• proof
• Lemma 1
• proof
• Lemma 2: Poincaré Inequality
• Theorem 1