Intrinsic Barycentric Coordinates and Homeomorphic Domain Inversions on the Sp(1) Lie Group
Abstract
We construct a unified theoretical framework for spherical barycentric coordinates initialized on the two-dimensional sphere S 2 and generalized to the three-dimensional sphere S 3 embedded within the non-commutative skew-field of quaternions H. This work resolves the structural divergence between extrinsic linear projective formulations and intrinsic Riemannian geodesic minimizations by establishing an exact bilateral transition map. By parameterizing the localized tangent spaces through pure quaternionic algebra, we prove that the projective center of mass defines a canonical initial state that ensures the global convergence of the intrinsic geodesic gradient flow lines. To achieve continuous directional interpolation without subsequent radial normalization operators, we introduce an optimal spherical Procrustes alignment on the compact Lie group Sp(1). This isometric decomposition isolates the global adjoint actions and constrains the manifold deviation to a purely quadratic deformation scale O(∥δΓ∥ 2 ). The resulting bilateral operator preserves the non-commutative structural symmetries of the manifold, establishing a globally continuous C 0 framework for invariant harmonic integration, surface shape inversions, and global geometric analysis on compact submanifolds.
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BibTeX
@misc{aitelhad2026intrinsic,
title = {{Intrinsic Barycentric Coordinates and Homeomorphic Domain Inversions on the Sp(1) Lie Group}},
author = {Abdellatif Aitelhad},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.13140/rg.2.2.12931.87843},
url = {https://hal.science/hal-05840142v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Intrinsic Barycentric Coordinates and Homeomorphic Domain Inversions on the Sp(1) Lie Group
AU - Abdellatif Aitelhad
PY - 2026
DA - 2026/10/26
PB - HAL
DO - 10.13140/rg.2.2.12931.87843
UR - https://hal.science/hal-05840142v1
AB - We construct a unified theoretical framework for spherical barycentric coordinates initialized on the two-dimensional sphere S 2 and generalized to the three-dimensional sphere S 3 embedded within the non-commutative skew-field of quaternions H. This work resolves the structural divergence between extrinsic linear projective formulations and intrinsic Riemannian geodesic minimizations by establishing an exact bilateral transition map. By parameterizing the localized tangent spaces through pure quaternionic algebra, we prove that the projective center of mass defines a canonical initial state that ensures the global convergence of the intrinsic geodesic gradient flow lines. To achieve continuous directional interpolation without subsequent radial normalization operators, we introduce an optimal spherical Procrustes alignment on the compact Lie group Sp(1). This isometric decomposition isolates the global adjoint actions and constrains the manifold deviation to a purely quadratic deformation scale O(∥δΓ∥ 2 ). The resulting bilateral operator preserves the non-commutative structural symmetries of the manifold, establishing a globally continuous C 0 framework for invariant harmonic integration, surface shape inversions, and global geometric analysis on compact submanifolds.
ER -