Closed-Form Solutions and Solvability of Line-Geometric Paden–Kahan Problems

TitleClosed-Form Solutions and Solvability of Line-Geometric Paden–Kahan Problems
Publication TypeJournal Article
Year of PublicationSubmitted
AuthorsBongardt B
JournalRobotics
IssueKinematics and Robot Design V (KaRD2022)
KeywordsComputational kinematics, dual-complex numbers, Lie theory, line geometry, Paden–Kahan problems, transference principle
Abstract

In the field of computational kinematics, the geometric problems by Paden and Kahan represent useful tools for developing analytical solutions to complex problems by following the strategy of divide-and-conquer. The original, point-wise problems by Paden and Kahan are generalized to corresponding problems of line geometry in this article. For establishing closed-form methods to solve the line-geometric variants in analogy to their point-wise counterparts, the principle of transference is applied to their direction-wise analogues. The solvability of the distinct problem classes is surveyed. In order to compute the closed-form solutions, three matrix-vector formalisms are stated and refined which permit to treat directions, points, and oriented lines in 3-space in a coherent manner. The problem solutions further rely on four trigonometric constraint equations and their analytic solutions as well as on two particular variants of the bilateration problem.