In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

study of the

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspective ...

, the Bertrand–Diguet–Puiseux theorem expresses the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . ...

of a surface in terms of the

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...

of a

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...

circle, or the area of a geodesic disc. The theorem is named for

Joseph Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was ...

, 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 :

K(p) = \lim_ 3\frac = \lim_ 12\frac.

The theorem is closely related to the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...

References

* * * Differential geometry of surfaces Theorems in differential geometry {{differential-geometry-stub