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