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We identify three shortcomings in Schr\u00f6dinger's approach and propose solutions to them. First, to encompass the Bezold\u2010Br\u00fccke effect, we replace the straight\u2010line definition of stimulus quality between a color and black with the geodesic path in perceptual color space. Second, to model diminishing returns in color perception, we employ a non\u2010Riemannian perceptual metric, which introduces a potential ambiguity in defining lightness, but our experiments show that this ambiguity is inconsequential. Third, we provide a geometric definition of the neutral axis as the closest color to black within each equal\u2010lightness surface\u2014a definition feasible only in a non\u2010Riemannian framework. Collectively, our solutions provide the first comprehensive realization of Helmholtz's vision: formal geometric definitions of hue, saturation, and lightness derived entirely from the metric of perceptual similarity, without reliance on external constructs.<\/jats:p>","DOI":"10.1111\/cgf.70136","type":"journal-article","created":{"date-parts":[[2025,5,23]],"date-time":"2025-05-23T06:22:48Z","timestamp":1747981368000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Geometry of Color in the Light of a Non\u2010Riemannian Space"],"prefix":"10.1111","volume":"44","author":[{"given":"Roxana","family":"Bujack","sequence":"first","affiliation":[{"name":"Los Alamos National Laboratory"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emily N.","family":"Stark","sequence":"additional","affiliation":[{"name":"Los Alamos National Laboratory"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Terece L.","family":"Turton","sequence":"additional","affiliation":[{"name":"Los Alamos National Laboratory"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonah M.","family":"Miller","sequence":"additional","affiliation":[{"name":"Los Alamos National Laboratory"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David H.","family":"Rogers","sequence":"additional","affiliation":[{"name":"Los Alamos National Laboratory"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,5,23]]},"reference":[{"issue":"560","key":"e_1_2_9_2_2","first-page":"120","article-title":"On the change in hue of spectrum colours by dilution with white light","volume":"83","author":"Abney W. 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