# Bertrand–Diguet–Puiseux theorem

In the mathematical study of the differential geometry of surfaces, the **Bertrand–Diguet–Puiseux theorem** expresses the Gaussian curvature of a surface in terms of the circumference of a geodesic circle, or the area of a geodesic disc. The theorem is named for Joseph Bertrand, Victor Puiseux, and Charles François Diguet.

Let *p* be a point on a smooth surface *M*. The geodesic circle of radius *r* centered at *p* is the set of all points whose geodesic distance from *p* is equal to *r*. Let *C*(*r*) denote the circumference of this circle, and *A*(*r*) denote the area of the disc contained within the circle. The Bertrand–Diguet–Puiseux theorem asserts that

The theorem is closely related to the Gauss–Bonnet theorem.

## ReferencesEdit

- Berger, Marcel (2004),
*A Panoramic View of Riemannian Geometry*, Springer-Verlag, ISBN 3-540-65317-1 - Bertrand, J; Diguet, C.F.; Puiseux, V (1848), "Démonstration d'un théorème de Gauss",
*Journal de Mathématiques*,**13**: 80–90 - Spivak, Michael (1999),
*A comprehensive introduction to differential geometry, Volume II*, Publish or Perish Press, ISBN 0-914098-71-3

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