# Robert L. FooteWabash 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.