The Prytz connections

Geir Bogfjellmo; Charles Curry; Sylvie Vega-Molino

doi:10.3934/jcd.2024016

Article Contents

2024, Volume 11, Issue 3: 318-335. Doi: 10.3934/jcd.2024016

This issue Previous Article Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations Next Article A new Lagrangian approach to control affine systems with a quadratic Lagrange term

The Prytz connections

1.
Norwegian University of Life Sciences, Norway
2.
Norwegian University of Science and Technology, Norway
3.
University of Bergen, Norway

^*Corresponding author: Geir Bogfjellmo
^*Corresponding author: Geir Bogfjellmo

Received: August 2023

Revised: February 2024

Early access: March 14, 2024

Published online: March 14, 2024

Sylvie Vega-Molino is supported by the grant GeoProCo from the Trond Mohn Foundation Grant TMS2021STG02 (GeoProCo).

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Abstract

The Prytz planimeter is a simple mechanical device that historically was used to approximate areas of plane regions. It also serves as a starting point to explain topics in differential geometry and sub-Riemannian geometry. In this article, we present a mathematical description and analysis of the planimeter and show how the analysis of its motion leads to concepts such as connections and sub-Riemannian geodesics.

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Mathematics Subject Classification: Primary: 53Z30, 53B15, 53C17.

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Figure 1. An illustration of the moving segment theorem. The area between the two shapes is swept out twice, once in the positive direction and once in the negative direction

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Figure 2. A Prytz planimeter

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Figure 3. Some examples of planimeter paths. The blue line follows the tracer end $ p(t) $ and the red line the chisel end $ q(t) $. The circular arc segment between $ q(0) $ and $ q(T) $ with center in $ p(0) = p(T) $ is shown as a dashed line. The exact values are $ A = \pi\approx 3.14159 $ for the circle, and $ A = 2\sin\left(\frac{\pi}{5}\right)\approx 1.17557 $ for the star

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Figure 4. An illustration of the planimeter and the coordinates used. The chisel end $ q $ can only move parallel to the line $ p $-$ q $

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Figure 5. Examples of sub-Riemannian geodesics

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