San José State University 

appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 

for Lenslike Surfaces 
The GaussBonnet Theorem in 3D space says that for a smooth surface S with boundary ∂S the integral of the Gaussian curvature over S plus the integral of the geodesic curvature around that boundary ∂S is equal to 2π times the Euler characteristic of the surface. In symbols, for a surface S and its boundary ∂S
What is considered here are surfaces shaped like the surface of a lens and having a sharp edge where the two spherical surfaces meet. Such a lensshaped surface can be formed by the intersection of two spheres. Let R_{1} and R_{2} be the radii of the two spheres and hence also the radii of curvature of the of the two lens surfaces. In the geometric scheme used here a portion of a sphere is the surface between a latitudinal angle of −π/2. Let Θ_{1} and Θ_{2} be the latitudinal angles for the lens edge where the two spheres intersect. Both Θ_{1} and Θ_{2} will be negative for the usual lens shape.
The radius r of the circle of intersection is given by
This is the radius of the sharp edge of the lenslike surface.
Crucial variables for the analysis are the cutoff distances
For the usual lens shape a_{1} and a_{2} are negative.
The Gaussian curvatures on the two lens faces are 1/R_{1}² and 1/R_{2}². Therefore the integrals of Gaussian curvature over the lens faces are just the surface areas times these curvatures.
The areas A_{1} and A_{2} of the lens surfaces are given by
and thus the integrals of Gaussian curvature on the lens faces are
The geodesic curvature of a circle of radius r on a sphere of radius R is
Therefore the integral of geodesic curvature around the circumference of 2πr is
The sum of the integral of Gaussian curvature over the lens face of radius of curvature R_{1} and the integral of geodesic curvature of the circle of intersection within the sphere of radius R_{1} is then
and likewise for the second sphere; i.e.,
Therefore the sum of these two terms is 4π which is just 2πχ(lens). Thus the integral of Guassian curvature over the surface of the lens plus the two integrals of geodesic curvature around the edge with the geodesic curvature determined with respect to the two surfaces which contain it is equal to 2π times the Euler characteristic of the lens.
In the tabulation for the lens the parameters a_{1} and a_{2} were negative so the terms −2πsin(Θ_{1}) and −2πsin(Θ_{2}) are positive and add to the quantities on the LHS of the equation. If a_{1} and a_{2} are negative then the surface has an indented crease instead of a protruding edge and the contribution of the integrals of geodesic curvature are negative.
For more on aspects of the GaussBonnet Theorem.
