Evolute
centers of curvature of a curve
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve.
The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let M be a smooth, regular submanifold in Rn. For each point p in M and each vector v, based at p and normal to M, we associate the point p + v. This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of M.
Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.
History
Apollonius (c.
Begin with the source’s own compact description: “Evolute” is centers of curvature of a curve. The dossier treats that line as a proposition to test through Evolute, centers and curvature, not as a finished interpretation.
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The citation trail is more important than the brevity of the summary: it shows where individual claims can be examined in context. The source revision retrieved here is dated Aug 13, 2026. The linked authority identifier is Q658654. None of the 0 selected statements returned an explicit reference.
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This entry incorporates text from “Evolute” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.