{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T02:28:58Z","timestamp":1790303338374,"version":"4.1.0"},"reference-count":19,"publisher":"The Royal Society","issue":"2326","license":[{"start":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:00:00Z","timestamp":1786579200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,8,13]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We revisit the geometric foundations of mesh representation through the lens of plane-based geometric algebra (PGA), investigating its efficiency and expressiveness for discrete geometry. We find how k-simplices (vertices, edges, faces,\u00a0\u2026) and k-complexes (point clouds, line complexes, meshes,\u2026) can be written compactly as joins of vertices and their sums, respectively. We show how a single formula for their k-magnitudes (amount, length, area,\u2026) follows naturally from PGA's Euclidean and ideal norms. This idea is then extended to produce unified coordinate-free formulae for classical results, such as volume, centre of mass (c.o.m.) and moments of inertia for simplices and complexes of arbitrary dimensionality. Finally, we demonstrate the practical use of these ideas on some real-world examples.<\/jats:p>\n                  <jats:p>This article is part of the theme issue \u2018Modern applications of geometric algebra\u2019.<\/jats:p>","DOI":"10.1098\/rsta.2025.0151","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T10:46:51Z","timestamp":1786618011000},"source":"Crossref","is-referenced-by-count":3,"title":["Clean up your mesh! Part 1: plane and simplex"],"prefix":"10.1098","volume":"384","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6870-1714","authenticated-orcid":false,"given":"Steven","family":"De Keninck","sequence":"first","affiliation":[{"name":"Universiteit van Amsterdam 1 , Amsterdam, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin","family":"Roelfs","sequence":"additional","affiliation":[{"name":"Universiteit Antwerpen 2 , Antwerp, Flanders, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leo","family":"Dorst","sequence":"additional","affiliation":[{"name":"Universiteit van Amsterdam 1 , Amsterdam, The Netherlands"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Eelbode","sequence":"additional","affiliation":[{"name":"Universiteit Antwerpen 2 , Antwerp, Flanders, Belgium"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"2026081306464660000_bib1","first-page":"1","article-title":"Geometric algebra for computer science","author":"Dorst","year":"2007"},{"key":"2026081306464660000_bib2","first-page":"1","article-title":"Geometric algebra and computer graphics","author":"Gunn","year":"2019"},{"key":"2026081306464660000_bib3","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1002\/mma.8639","article-title":"Normalization, square roots, and the exponential and logarithmic maps in geometric algebras of less than 6D","volume":"47","author":"De Keninck","year":"2024","journal-title":"Math. 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Solution published November 1866. Reprinted in mathematical questions with solutions","author":"Clifford","year":"1866"},{"key":"2026081306464660000_bib16","volume-title":"Elements of algebraic topology","author":"Munkres","year":"1984"},{"key":"2026081306464660000_bib17","article-title":"Clean up your Mesh \u2014 companion page","author":"De Keninck"},{"key":"2026081306464660000_bib18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-031-55985-3_2","volume-title":"Physical geometry by plane-based geometric algebra","author":"Dorst","year":"2024"},{"key":"2026081306464660000_bib19","doi-asserted-by":"crossref","first-page":"8","DOI":"10.3844\/jmssp.2005.8.11","article-title":"Explicit exact formulas for the 3-D tetrahedron inertia tensor in terms of its vertex coordinates","volume":"1","author":"Tonon","year":"2005","journal-title":"J. Math. 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