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

${\displaystyle \operatorname {Slerp} (p_{0},p_{1};t)={\frac {\sin {[(1-t)\Omega }]}{\sin \Omega }}p_{0}+{\frac {\sin[t\Omega ]}{\sin \Omega }}p_{1}.}$

The symmetry lies in the fact that ${\displaystyle \operatorname {Slerp} (p_{0},p_{1};t)}$ ${\displaystyle =\operatorname {Slerp} (p_{1},p_{0};1-t)}$. In the limit as Ω → 0, this formula reduces to the corresponding symmetric formula for linear interpolation,

${\displaystyle \operatorname {Lerp} (p_{0},p_{1};t)=(1-t)p_{0}+tp_{1}.\,\!}$

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.

Oblique vector rectifies to Slerp factor.

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.

When Slerp is applied to unit quaternions, the quaternion path maps to a path through 3D rotations in a standard way. The effect is a rotation with uniform angular velocity around a fixed rotation axis. When the initial end point is the identity quaternion, Slerp gives a segment of a one-parameter subgroup of both the Lie group of 3D rotations, SO(3), and its universal covering group of unit quaternions, S³. Slerp gives a straightest and shortest path between its quaternion end points, and maps to a rotation through an angle of 2Ω. However, because the covering is double (q and −q map to the same rotation), the rotation path may turn either the "short way" (less than 180°) or the "long way" (more than 180°). Long paths can be prevented by negating one end if the dot product, cos Ω, is negative, thus ensuring that −90° ≤ Ω ≤ 90°.

Slerp also has expressions in terms of quaternion algebra, all using exponentiation. Real powers of a quaternion are defined in terms of the quaternion exponential function, written as e^q and given by the power series equally familiar from calculus, complex analysis and matrix algebra:

${\displaystyle e^{q}=1+q+{\frac {q^{2}}{2}}+{\frac {q^{3}}{6}}+\cdots +{\frac {q^{n}}{n!}}+\cdots .}$

Writing a unit quaternion q in versor form, cos Ω + v sin Ω, with v a unit 3-vector, and noting that the quaternion square v² equals −1 (implying a quaternion version of Euler's formula), we have e^v Ω = q, and q^t = cos t Ω + v sin t Ω. The identification of interest is q = q₁ q₀⁻¹, so that the real part of q is cos Ω, the same as the geometric dot product used above. Here are four equivalent quaternion expressions for Slerp.

${\displaystyle {\begin{aligned}\operatorname {Slerp} (q_{0},q_{1},t)&=q_{0}(q_{0}^{-1}q_{1})^{t}\\[6pt]&=q_{1}(q_{1}^{-1}q_{0})^{1-t}\\[6pt]&=(q_{0}q_{1}^{-1})^{1-t}q_{1}\\[6pt]&=(q_{1}q_{0}^{-1})^{t}q_{0}\end{aligned}}}$

The derivative of Slerp(q₀, q₁; t) with respect to t, assuming the ends are fixed, is log(q₁q₀⁻¹) times the function value, where the quaternion natural logarithm in this case yields half the 3D angular velocity vector. The initial tangent vector is parallel transported to each tangent along the curve; thus the curve is, indeed, a geodesic.

In the tangent space at any point on a quaternion Slerp curve, the inverse of the exponential map transforms the curve into a line segment. Slerp curves not extending through a point fail to transform into lines in that point's tangent space.

Quaternion Slerps are commonly used to construct smooth animation curves by mimicking affine constructions like the de Casteljau algorithm for Bézier curves. Since the sphere is not an affine space, familiar properties of affine constructions may fail, though the constructed curves may otherwise be entirely satisfactory. For example, the de Casteljau algorithm may be used to split a curve in affine space; this does not work on a sphere.

The two-valued Slerp can be extended to interpolate among many unit quaternions,[1] but the extension loses the fixed execution-time of the Slerp algorithm.

The below C++ code adapts the Slerp algorithm implemented by Eigen (C++ library) in Geometry/Quaternion.h that handles some common edge cases.

Quaternion slerp(const double& t, const Quaternion& v0, const Quaternion& v1) const
{
  // Only unit quaternions are valid rotations.
  // Normalize to avoid undefined behavior.
  v0.normalize();
  v1.normalize();

  const double one = 1.0 - numeric_limits<double>::epsilon();
  double d = v0.dot(v1);
  double absD = abs(d);

  double scale0;
  double scale1;

  // When the dot product of the quaternion vectors are within machine epsilon of 1,  
  // The two quaternions are almost representing the same spatial rotation. Hence 
  // use linear interpolation
  if(absD>=one) 
  {
    scale0 = 1.0 - t;
    scale1 = t;
  } 
  else 
  {
    // theta is the angle between the 2 quaternions
    double theta = acos(absD);
    double sinTheta = sin(theta);

    scale0 = sin((1.0 - t) * theta) / sinTheta;
    scale1 = sin((t * theta)) / sinTheta;
  }

  // If the dot product is negative, slerp won't take
  // the shorter path. Correct by negating the scale
  if(d<0.0) scale1 = -scale1;

  return Quaternion(scale0 * q0 + scale1 * q1);
}

The following is the same algorithm except in Python. It uses NumPy and can calculate multiple slerp interpolations between the two quaternions.

import numpy as np

def slerp(v0, v1, t_array):
    """Spherical linear interpolation."""
    # >>> slerp([1,0,0,0], [0,0,0,1], np.arange(0, 1, 0.001))
    t_array = np.array(t_array)
    v0 = np.array(v0)
    v1 = np.array(v1)
    dot = np.sum(v0 * v1)

    if dot < 0.0:
        v1 = -v1
        dot = -dot
    
    DOT_THRESHOLD = 0.9995
    if dot > DOT_THRESHOLD:
        result = v0[np.newaxis,:] + t_array[:,np.newaxis] * (v1 - v0)[np.newaxis,:]
        return (result.T / np.linalg.norm(result, axis=1)).T
    
    theta_0 = np.arccos(dot)
    sin_theta_0 = np.sin(theta_0)

    theta = theta_0 * t_array
    sin_theta = np.sin(theta)
    
    s0 = np.cos(theta) - dot * sin_theta / sin_theta_0
    s1 = sin_theta / sin_theta_0
    return (s0[:,np.newaxis] * v0[np.newaxis,:]) + (s1[:,np.newaxis] * v1[np.newaxis,:])

• Shoemake, Ken. "Animating Rotation with Quaternion Curves" (PDF). SIGGRAPH 1985.
• Erik B., Dam; Martin, Koch; Lillholm, Martin (July 17, 1998). "Quaternions, Interpolation and Animation" (PDF). University of Copenhagen. Archived (PDF) from the original on 2017-08-30.
• Blow, Jonathan (February 26, 2004). "Understanding Slerp, Then Not Using It". Archived from the original on 2017-08-25.
• Martin, Brian (June 23, 1999). "Brian Martin on Quaternion Animation". Archived from the original on 2016-03-24.