A Quaternion--BCH Framework for the Local Accuracy of SIDER Interpolation

Abstract

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation method for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper develops a quaternion--Lie algebra framework for proving the local consistency of SIDER for smooth spherical curves sampled at equally spaced parameter values. Points on $\mathbb{S}$ are represented as pure unit quaternions, and interpolation errors are measured in fixed-base quaternion logarithmic coordinates. In this setting, each SLERP operation admits an exact Baker--Campbell--Hausdorff (BCH) representation, which converts the geometric interpolation problem into an algebraic problem involving filtered Lie-polynomial expansions. The BCH expansion shows that SLERP is affine to leading order, has no quadratic correction, and has a first nonlinear correction that is cubic and commutator-valued. Using this structure, we prove that SIDER2 has a third-order divided-error form with the same leading nodal factor as ordinary quadratic interpolation. We then show that the recursive SIDER step raises the order by one: the affine part gives the Neville-type finite-difference cancellation, while the nonlinear BCH remainder preserves the sharp filtered degree structure after the nodal factor is removed. Consequently, for every fixed $n\geq2$, $d_{\mathbb{S}^2}\bigl(γ(θh),P_i^{[n]}(θ;h)\bigr) = O(h^{n+1}) $under the stated smoothness and small-stencil assumptions. The proof also identifies the shift-invariance of the leading divided-error coefficient as the algebraic compatibility condition underlying the SIDER recurrence.