Robert L. Foote
Wabash College

Planimeters and Isoperimetric Inequalities on Constant Curvature Surfaces

A planimeter is a mechanical instrument used to determine the area of a region in the plane. As the boundary of the region is traced, a wheel attached to the instrument partially rolls and partially slides, recording a component of its motion on the plane. The area of the region is a simple function of the net roll of the wheel. We show how the analogue of this instrument works on the sphere and the hyperbolic plane, and then use the results to give a simple proof of the isoperimetric inequality for these surfaces. If M is S^2, E^2, or H^2, the planimeter defines a relation among certain one-forms on its configuration space, which is a submanifold of MxM. Integrating along a path determined by the region in M results in one of the well-known Bonnesen isoperimetric inequalities.