TY - JOUR

T1 - Relative geodesics in bi-invariant Lie groups

AU - Zhang, E.

AU - Noakes, L.

PY - 2017/5/1

Y1 - 2017/5/1

N2 - Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f1 and f2 in G/K from one to the other. An (f1, f2)-relative geodesic (Holm et al. 2013 Proc. R. Soc. A 469, 20130297. (doi:10.1098/rspa.2013.0297)) is defined as a critical point of the total energy E(g) as g varies in the set of all (f1, f2)-admissible curves. We obtain the Euler-Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler-Lagrange equation is simplified for the case where G/K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S2, we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.

AB - Motivated by registration problems, this paper deals with a curve matching problem in homogeneous spaces. Let G be a connected finite-dimensional bi-invariant Lie group and K a closed subgroup. A smooth curve g in G is said to be admissible if it can transform two smooth curves f1 and f2 in G/K from one to the other. An (f1, f2)-relative geodesic (Holm et al. 2013 Proc. R. Soc. A 469, 20130297. (doi:10.1098/rspa.2013.0297)) is defined as a critical point of the total energy E(g) as g varies in the set of all (f1, f2)-admissible curves. We obtain the Euler-Lagrange equation, a first-order differential equation, satisfied by a relative geodesic. Furthermore, the Euler-Lagrange equation is simplified for the case where G/K is globally symmetric. As a concrete example, relative geodesics are found for special cases where G is SO(3) and K is SO(2). As an application of discrepancy for curves in S2, we construct and study a new measure of non-congruency for constant speed curves in Euclidean 3-space. Numerical examples are given to illustrate results.

KW - Bi-invariant Lie groups

KW - Euler-Lagrange equation

KW - Relative geodesics

KW - The special orthogonal group

UR - http://www.scopus.com/inward/record.url?scp=85020071458&partnerID=8YFLogxK

U2 - 10.1098/rspa.2016.0619

DO - 10.1098/rspa.2016.0619

M3 - Article

AN - SCOPUS:85020071458

VL - 473

JO - Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences

JF - Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences

SN - 1364-5021

IS - 2201

M1 - 20160619

ER -