Slerp - Geometric Slerp

Geometric Slerp

Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends. Let p₀ and p₁ be the first and last points of the arc, and let t be the parameter, 0 ≤ t ≤ 1. Compute Ω as the angle subtended by the arc, so that cos Ω = p₀ ∙ p₁, the n-dimensional dot product of the unit vectors from the origin to the ends. The geometric formula is then

The symmetry can be seen in the fact that Slerp(p₀, p₁; t) = Slerp(p₁, p₀; 1 − t). In the limit as Ω → 0, this formula reduces to the corresponding symmetric formula for linear interpolation,

A Slerp path is, in fact, the spherical geometry equivalent of a path along a line segment in the plane; a great circle is a spherical geodesic.

More familiar than the general Slerp formula is the case when the end vectors are perpendicular, in which case the formula is p₀ cos θ + p₁ sin θ. Letting θ = t π/2, and applying the trigonometric identity cos θ = sin (π/2 − θ), this becomes the Slerp formula. The factor of 1/sin Ω in the general formula is a normalization, since a vector p₁ at an angle of Ω to p₀ projects onto the perpendicular ⊥p₀ with a length of only sin Ω.

Some special cases of Slerp admit more efficient calculation. When a circular arc is to be drawn into a raster image, the preferred method is some variation of Bresenham's circle algorithm. Evaluation at the special parameter values 0 and 1 trivially yields p₀ and p₁, respectively; and bisection, evaluation at ½, simplifies to (p₀ + p₁)/2, normalized. Another special case, common in animation, is evaluation with fixed ends and equal parametric steps. If p_k−1 and p_k are two consecutive values, and if c is twice their dot product (constant for all steps), then the next value, p_k+1, is the reflection p_k+1 = c p_k − p_k−1.