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interpolation-on-manifolds

Abstract math for interpolating angles, vectors, and quaternions on curved manifolds. USE FOR: implementing Lerp (linear interpolation) for vectors; Nlerp (normalized lerp) for directions; Slerp (spherical linear interpolation) for quaternions and unit vectors; shortest-path sign flip when dot product is negative; choosing between Lerp/Nlerp/Slerp; angular lerp with wrap-around. DO NOT USE FOR: quaternion arithmetic (use quaternion-algebra); rotation representation conversions (use 3d-rotation-theory); angle unit conversions (use angles-and-circular-arithmetic).

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NetFabric/NetFabric.Numerics36 stars(GitHub 36 + cccskills 0)

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Files included (3)

SKILL.md2.3 KB
references/formulas.md2.0 KB
references/numerical-stability.md1.9 KB

SKILL.md (original)

Review the instructions the agent will receive before you install.

Interpolation on Manifolds

Lerp (Linear Interpolation)

lerp(a, b, t) = (1−t)·a + t·b Result NOT on unit sphere; constant Euclidean velocity.

Nlerp (Normalized Lerp)

nlerp(a, b, t) = normalize(lerp(a, b, t)) Cheaper than Slerp; breaks down at antipodal inputs.

Slerp (Spherical Linear Interpolation)

Great-circle arc at constant angular velocity:

Ω = arccos(a · b)
slerp(a, b, t) = sin((1−t)Ω)/sin(Ω) · a  +  sin(tΩ)/sin(Ω) · b

Fall back to Lerp when Ω ≈ 0. Undefined for antipodal inputs (Ω = π).

Shortest-Path Flip

For quaternions, q and −q represent the same rotation. If a · b < 0, interpolation takes the long path (> 180°). Fix:

if (a · b < 0):  b = −b    (before calling slerp/nlerp)

Which to Use

Situation	Method	Reason
Arbitrary vectors in ℝⁿ	Lerp	Correct; no manifold curvature
Unit vectors (approximate)	Nlerp	Fast; adequate for small angles
Unit quaternions (smooth)	Slerp	Constant angular velocity
Unit vectors (exact)	Slerp	Correct geodesic arc
Angles with wrapping	Angular lerp	Handles 350°↔10° via shortest arc

Angular Lerp (Wrap-Aware)

delta = reduce(b − a, Full)          // shortest signed difference ∈ [−Full/2, Full/2)
result = a + t · delta

This ensures the interpolation takes the short way around.

Reference Files

File	Load When
references/formulas.md	Slerp derivation, degenerate cases, angle lerp proof
references/numerical-stability.md	Near-identical inputs, antipodal inputs, sin(Ω)→0 handling

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Install

Install with a command

npx skills add https://github.com/NetFabric/NetFabric.Numerics --skill interpolation-on-manifolds

Paste it into your terminal and run it (requires Node.js). You can choose which agents to install it for, such as Claude Code or Codex.

Ask your agent

Please install the following Agent Skill.
https://github.com/NetFabric/NetFabric.Numerics/tree/main/.agents/skills/interpolation-on-manifolds

First review SKILL.md and the bundled files and confirm there is nothing suspicious (such as sending data externally or reading credentials). Then place it in ~/.claude/skills/interpolation-on-manifolds/ for Claude Code, or ~/.agents/skills/interpolation-on-manifolds/ for Codex.

Send this text to Claude Code or Codex as is. It checks the contents before installing.

View source on GitHub

Repository: NetFabric/NetFabric.Numerics
Stars: 36
Forks: 1
Category: Documents
License: MIT
Original language: English
Last updated: Aug 10, 2026

A skill is a set of instructions and scripts for an agent. cccskills does not guarantee that skills are safe. Review the contents before you install.