Geometry of the Prytz Planimeter

Abstract

The Prytz planimeter is a simple example of a system governed by a non-holonomic constraint. It is unique among planimeters in that it measures something more subtle than area, combining the area, centroid and other moments of the region being measured, with weights depending on the length of the planimeter. As a tool for measuring area, it is most accurate for regions that are small relative to its length. The configuration space of the planimeter is a non-principal circle bundle acted on by SU(1,1), (isom. to SL(2,R)). The motion of the planimeter is realized as parallel translation for a connection on this bundle and for a connection on a principal SU(1,1)-bundle. The holonomy group is SU(1,1). As a consequence, the planimeter is an example of a system with a phase shift on the circle that is not a simple rotation. There is a qualitative difference in the holonomy when tracing large regions as opposed to small ones. Generic elements of SU(1,1) act on S^1 with two fixed points or with no fixed points. When tracing small regions, the holonomy acts without fixed points. Menzin's conjecture states (roughly) that if a planimeter of length L traces the boundary of a region with area A > pi L^2, then it exhibits an asymptotic behavior and the holonomy acts with two fixed points, one attracting and one repelling. This is obvious if the region is a disk, and intuitively plausible if the region is convex and A >> pi L^2. A proof of this conjecture is given for a special case, and the conjecture is shown to imply the isoperimetric inequality.