Title page for ETD etd-04152011-110946


Type of Document Dissertation
Author StClair, Jessica Lindsey
Author's Email Address jessicab@vt.edu
URN etd-04152011-110946
Title Geometry of Spaces of Planar Quadrilaterals
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Haskell, Peter E. Committee Chair
Day, Martin V. Committee Member
Floyd, William J. Committee Member
Thomson, James E. Committee Member
Keywords
  • Holonomy
  • Robotics
  • Riemannian Metric
  • Moduli Space
  • Pre-Moduli Space
  • Differential Geometry
Date of Defense 2011-04-14
Availability unrestricted
Abstract
The purpose of this dissertation is to investigate the geometry of spaces of planar quadrilaterals.

The topology of moduli spaces of planar quadrilaterals (the set of all distinct planar

quadrilaterals with fi xed side lengths) has been well-studied [5], [8], [10]. The symplectic

geometry of these spaces has been studied by Kapovich and Millson [6], but the Riemannian

geometry of these spaces has not been thoroughly examined. We study paths in the moduli

space and the pre-moduli space. We compare intraplanar paths between points in the moduli

space to extraplanar paths between those same points. We give conditions on side lengths

to guarantee that intraplanar motion is shorter between some points. Direct applications of

this result could be applied to motion-planning of a robot arm. We show that horizontal lifts

to the pre-moduli space of paths in the moduli space can exhibit holonomy. We determine

exactly which collections of side lengths allow holonomy.

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