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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<jats:italic>M<\/jats:italic>and \ud835\udd4a<jats:sup>2<\/jats:sup>. 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<jats:italic>M<\/jats:italic>and \ud835\udd4a<jats:sup>2<\/jats:sup>. 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.<\/jats:p>","DOI":"10.1515\/jnma-2022-0072","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T16:41:01Z","timestamp":1692895261000},"source":"Crossref","is-referenced-by-count":0,"title":["Fundamental Theory and R-linear Convergence of Stretch Energy Minimization for Spherical Equiareal Parameterization"],"prefix":"10.1515","volume":"0","author":[{"given":"Tsung-Ming","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei-Hung","family":"Liao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wen-Wei","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"key":"2023090112092843667_j_jnma-2022-0072_ref_001","unstructured":"D. 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