|Title||Closed-Form Solutions and Solvability of Line-Geometric Paden–Kahan Problems|
|Publication Type||Journal Article|
|Year of Publication||Submitted|
|Issue||Kinematics and Robot Design V (KaRD2022)|
|Keywords||Computational kinematics, dual-complex numbers, Lie theory, line geometry, Paden–Kahan problems, transference principle|
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.