San José State University

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Thayer Watkins
Silicon Valley
& Tornado Alley
USA

A Gauss-Bonnet-type Theorem
for Lens-like Surfaces

The Gauss-Bonnet 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

SK(p)dA + ∫∂Skgds = 2πχ(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 lens-shaped surface can be formed by the intersection of two spheres. Let R1 and R2 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 Θ1 and Θ2 be the latitudinal angles for the lens edge where the two spheres intersect. Both Θ1 and Θ2 will be negative for the usual lens shape.

The radius r of the circle of intersection is given by

r = R1cos(Θ1) = R1cos(Θ1)

This is the radius of the sharp edge of the lens-like surface.

Crucial variables for the analysis are the cutoff distances

a1 = R1sin(Θ1)
a2 = R2sin(Θ2)

For the usual lens shape a1 and a2 are negative.

The Gaussian curvatures on the two lens faces are 1/R1² and 1/R2². Therefore the integrals of Gaussian curvature over the lens faces are just the surface areas times these curvatures.

The areas A1 and A2 of the lens surfaces are given by

A11) = 2πR1²(1 + sin(Θ1))
and
A22) = 2πR2²(1 + sin(Θ1))

and thus the integrals of Gaussian curvature on the lens faces are

A11)/R1² = 2π(1 + sin(Θ1))
and
A22)/R2² = 2π(1 + sin(Θ1))

Geodesic Curvature on
the Circle of Intersection

The geodesic curvature of a circle of radius r on a sphere of radius R is

kg = −a/(rR)

Therefore the integral of geodesic curvature around the circumference of 2πr is

−2πa/R
and since a=Rsin(Θ)
this reduces to
−2πsin(Θ)

The sum of the integral of Gaussian curvature over the lens face of radius of curvature R1 and the integral of geodesic curvature of the circle of intersection within the sphere of radius R1 is then

S1K(p)dA + ∫S1kg1ds
= 2π(1 + sin(Θ1)) − sin(Θ1) = 2π

and likewise for the second sphere; i.e.,

S2K(p)dA + ∫S2kg2ds
= 2π(1 + sin(Θ2)) − 2πsin(Θ2) = 2π

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 a1 and a2 were negative so the terms −2πsin(Θ1) and −2πsin(Θ2) are positive and add to the quantities on the LHS of the equation. If a1 and a2 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 Gauss-Bonnet Theorem.


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