In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

study of the

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensiv ...

, the Bertrand–Diguet–Puiseux theorem expresses the

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

of a surface in terms of the

circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...

of a

geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...

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 :

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

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

References

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