Fundamental theory and R-linear convergence of stretch energy minimization for spherical equiareal parameterization

Tsung Ming Huang*, Wei Hung Liao, Wen Wei Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Here, we extend the finite distortion problem from bounded domains in R2 to closed genus-zero surfaces in R3 by a stereographic projection. Then, we derive a theoretical foundation for spherical equiareal parameterization between a simply connected closed surface M and a unit sphere S2 by minimizing the total area distortion energy on C. After the minimizer of the total area distortion energy is determined, it is combined with an initial conformal map to determine the equiareal map between the extended planes. From the inverse stereographic projection, we derive the equiareal map between M and S2. The total area distortion energy is rewritten into the sum of Dirichlet energies associated with the southern and northern hemispheres and is decreased by alternatingly solving the corresponding Laplacian equations. Based on this foundational theory, we develop a modified stretch energy minimization function for the computation of spherical equiareal parameterization between M and S2. In addition, under relatively mild conditions, we verify that our proposed algorithm has asymptotic R-linear convergence or forms a quasi-periodic solution. Numerical experiments on various benchmarks validate that the assumptions for convergence always hold and indicate the efficiency, reliability, and robustness of the developed modified stretch energy minimization function.

Original languageEnglish
Pages (from-to)1-25
Number of pages25
JournalJournal of Numerical Mathematics
Issue number1
Publication statusPublished - 2024 Mar 1


  • R-linear convergence
  • spherical equiareal parameterization
  • stretch energy minimization

ASJC Scopus subject areas

  • Computational Mathematics
  • Numerical Analysis


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