Local Consistency and Higher-Order Structure of Spherical Interpolation

Abstract: Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation construction for data on the unit sphere $\mathbb{S}<sup>2$, built from repeated spherical linear interpolation (SLERP). This paper gives a local consistency analysis of SIDER for smooth spherical curves sampled at equally spaced parameter values. The analysis is carried out in geodesic normal coordinates, which allows the SIDER recursion to be compared with classical Neville interpolation while retaining the curvature-dependent corrections introduced by SLERP. We first derive local expansions of SLERP and show that SIDER2 has third-order accuracy; its leading error has the same shifted nodal structure as Euclidean quadratic interpolation. We then prove that the adjacent SIDER2 errors entering SIDER3 have a common leading coefficient, so that the SIDER3 recurrence cancels the cubic term and yields fourth-order accuracy. Carrying the expansion one order further gives the corresponding coefficient compatibility for SIDER3 and proves fifth-order accuracy of SIDER4. Finally, we introduce a degree-filtered formal expansion framework for the general SIDER recursion. This framework proves that, for each fixed $n$, SIDER-$n$ preserves the required polynomial degree structure in the normalized stencil variable. Together with the interpolation conditions at the $n+1$ nodes, this yields the local consistency estimate $$d_{\mathbb{S}<sup>2}\bigl(γ(θh),P_i<sup>{[n]}(θ;h)\bigr)=O(h<sup>{n+1})$$ under the stated smoothness and small-stencil assumptions.

• The paper proves that SIDER-n achieves O(h^{n+1}) local accuracy on S² for smooth curves and equally spaced samples, using a degree-filtered expansion to establish shift-invariant error cancellation at every fixed level.
• Explicit analysis shows SIDER2, SIDER3, and SIDER4 attain third-, fourth-, and fifth-order accuracy, respectively, with curvature first entering the refined fourth-order error coefficient.
• The paper separates Neville-style nodal cancellation from SLERP’s curvature corrections, while identifying limitations including local convexity assumptions, uniform grids, unverified AI-assisted derivations, and the lack of full SENO stability analysis.

Overview

This paper develops a local consistency analysis for the SIDER family of interpolants (Spherical Interpolation of orDER $n$) on the unit sphere $S^2$, a recursive high-order construction built entirely from spherical linear interpolation (SLERP) and previously introduced as the building block of the SENO (spherical essentially non-oscillatory) method (2606.12802). The central question is whether the Neville-type order-raising mechanism, which is automatic for Euclidean polynomial interpolation on equally spaced nodes, survives the replacement of affine interpolation by geodesic interpolation. The paper answers affirmatively for every fixed level $n$, proving the local estimate $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$ for smooth curves sampled at equally spaced parameters, together with explicit refined error formulas at the first three levels.

Geometric setup and the role of curvature

The analysis is conducted in geodesic normal coordinates centered at the exact evaluation point $y = \gamma(\theta h)$, so that the sampled data $x_i = Log_y(p_i)$ are $O(h)$ vectors in $T_yS^2$. A key structural observation drives the whole paper: SLERP is affine to leading order in these coordinates, but its first non-affine correction is cubic, and this cubic order coincides with the leading error of a quadratic reconstruction. A coarse estimate $$Log_y(SLERP(A,B,\lambda)) = (1-\lambda)U + \lambda W + O(\rho^3)$$ is therefore insufficient for proving fourth-order accuracy of SIDER3, since the $h^3$ terms must be tracked explicitly.

The paper derives the explicit cubic SLERP expansion, with correction term $S^2$0 involving combinations of $S^2$1, $S^2$2, $S^2$3 (where $S^2$4), and $S^2$5 terms. A concrete example shows the SLERP midpoint of two points at distance $S^2$6 along orthogonal directions deviates from the affine midpoint by $S^2$7 in each coordinate, confirming the correction is a genuine curvature effect. Two companion lemmas complete the toolkit: a high-order expansion showing no quadratic term ever appears, and an affine-stability lemma stating that when both SLERP endpoints are $S^2$8-close to $S^2$9 with $n$0, the non-affine part is $n$1 and hence harmless. This separation is the technical linchpin: in SIDER2, SLERP operates on $n$2 points and the cubic term must be retained; in recursive blending steps, the final SLERP acts on already high-order-accurate points, so its nonlinear contribution is negligible.

Third- and fourth-order results

The refined SIDER2 result states that, uniformly for $n$3 in bounded intervals,

$n$4

where $n$5 is the third Taylor coefficient of the normal-coordinate curve. The leading error carries the same shifted nodal factor as Euclidean quadratic interpolation. The proof shows the cubic SLERP corrections vanish at leading order because the relevant $n$6 endpoint coordinates are collinear multiples of the velocity vector $n$7, so SLERP reduces to one-dimensional affine interpolation along a single geodesic.

For SIDER3, the crucial point is that the two adjacent SIDER2 candidates share the same vector coefficient $n$8 (they expand the same smooth normal-coordinate curve at the same evaluation point) and differ only by the shifted nodal factor. The SIDER3 weight $n$9 then cancels the cubic term identically, yielding $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$0 uniformly for $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$1. The paper is explicit that this is not a consequence of third-order accuracy alone: averaging two generic third-order approximations would remain third order; the cancellation requires the common leading coefficient.

Fifth-order SIDER4 and the compatibility condition

Carrying the expansion one order further, the paper derives a fourth-order refinement of the SIDER2 error in which the $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$2 coefficient is the vector

$$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$3

combining the quartic Taylor coefficient $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$4 with the first curvature-dependent correction. The dependence on the stencil shift is entirely captured by the scalar factor $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$5, and the refined SIDER3 error takes the form $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$6, with a stencil-shift-invariant leading coefficient. The SIDER4 blend then cancels the $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$7 term exactly, proving fifth-order local accuracy. Notably, at this level the leading coefficient is no longer purely a derivative of the curve; curvature enters explicitly, and the paper acknowledges that a naive induction would require showing increasingly complicated curvature combinations remain shift-invariant at every level.

The general recurrence and its gap

For a general SIDER step combining two adjacent SIDER-$$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$8 candidates with expansions $$d_{S^2}(\gamma(\theta h), P_i^{[n]}(\theta;h)) = O(h^{n+1})$$9, the recurrence calculation gives a residual proportional to $y = \gamma(\theta h)$0. The paper isolates this precisely: the nodal-factor algebra of Neville interpolation survives automatically, but on the sphere the vector coefficients are generated by nonlinear SLERP compositions and may in principle depend on the stencil shift. A root-counting argument determines only the zeros of the leading error polynomial, not the coefficient identity required for cancellation. The paper accordingly states a conditional theorem: if the shift-invariant leading coefficient holds at level $y = \gamma(\theta h)$1, then SIDER-$y = \gamma(\theta h)$2 achieves $y = \gamma(\theta h)$3.

Degree-filtered all-order proof

The gap is closed by a degree-filtered formal expansion framework. Defining $y = \gamma(\theta h)$4 as expansions whose $y = \gamma(\theta h)$5 coefficients are polynomials in the shifted stencil variable $y = \gamma(\theta h)$6 of degree at most $y = \gamma(\theta h)$7, and $y = \gamma(\theta h)$8 as expansions one degree lower, the paper proves two structural facts. First, finite differences lower the degree, so the exact sampled curve and its stencil translates form compatible pairs. Second, and nontrivially, SLERP preserves the filtration: in a difference-filtered expansion $y = \gamma(\theta h)$9, every nonlinear monomial has $x_i = Log_y(p_i)$0-degree no larger than its number of $x_i = Log_y(p_i)$1-factors, which is exactly what is needed to keep polynomial degrees bounded after substituting filtered series. This is established by expanding the sine ratios and logarithm map about the first endpoint, using the fact that $x_i = Log_y(p_i)$2 is analytic in $x_i = Log_y(p_i)$3 with every monomial containing at least two factors of $x_i = Log_y(p_i)$4.

The closure of SIDER under this filtration follows by induction: the error of SIDER-$x_i = Log_y(p_i)$5 has a shift-covariant filtered expansion whose coefficients are universal polynomials in $x_i = Log_y(p_i)$6 depending on the local jet of $x_i = Log_y(p_i)$7 but not on the stencil shift. Since SIDER-$x_i = Log_y(p_i)$8 interpolates $x_i = Log_y(p_i)$9 nodes, all coefficients through order $O(h)$0 vanish identically, and the leading $O(h)$1 coefficient must be a multiple of the nodal polynomial with a shift-invariant vector multiplier. This yields the all-order consistency theorem and simultaneously supplies the compatibility condition required by the recurrence framework.

Limitations and open questions

The paper states its assumptions plainly. All results are local and asymptotic: data points, extrapolated control points, and intermediate SIDER values must remain in a common geodesically convex ball of radius $O(h)$2, ensuring single-valuedness of the logarithm map and uniqueness of geodesics. The paper notes this assumption is stronger than what practical implementations may require, and it excludes antipodal ambiguity and large angular separations. The analysis assumes equally spaced parameter values; extension to nonuniform grids would require modified Neville weights and nodal factors, which the paper does not develop. The results concern the smooth SIDER building blocks only; the full SENO method adds nonlinear stencil selection, and no stability or non-oscillation theory for SENO is provided here. Finally, the paper carries a disclosure stating that the derivations were produced with AI assistance, that the mathematical depth exceeds current manual verification capability, and that the findings are presented as AI-assisted hypotheses pending formal peer verification — a caveat that bears directly on the reliability of every result summarized above.

Conclusion

The paper establishes that SIDER interpolation on $O(h)$3 achieves $O(h)$4 local consistency for every fixed level $O(h)$5 under smoothness and small-stencil assumptions, with explicit third-, fourth-, and fifth-order error formulas for SIDER2, SIDER3, and SIDER4. The mechanism is a precise separation between the algebraic nodal-factor cancellation inherited from Neville interpolation and the curvature-dependent corrections introduced by SLERP, with the degree-filtered expansion providing the shift-invariance of leading coefficients that makes the recursion close. The remaining open problems identified in the paper are concrete: nonuniform node spacing, interpolation on more general Riemannian manifolds or Lie groups, and an analysis of the SENO stencil-selection mechanism.