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
K(p)=
\lim | 3 | |
r\to0+ |
2\pir-C(r) | |
\pir3 |
=
\lim | 12 | |
r\to0+ |
\pir2-A(r) | |
\pir4 |
.
The theorem is closely related to the Gauss–Bonnet theorem.