View source for Slerp

{{Short description|Spherical linear interpolation in computer graphics}}
{{More footnotes needed|date=April 2021}}
In [[computer graphics]], '''slerp''' is shorthand for '''spherical linear interpolation''', introduced by Ken Shoemake<ref>{{Cite web|url=https://dl.acm.org/profile/81100026146|title = Ken Shoemake - Home}}</ref> in the context of [[quaternion]] [[interpolation]] for the purpose of [[animation|animating]] 3D [[rotation]]. It refers to constant-speed motion along a unit-radius [[great circle]] arc, given the ends and an interpolation parameter between 0 and&nbsp;1.

== 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 ''p''<sub>0</sub> and ''p''<sub>1</sub> be the first and last points of the arc, and let ''t'' be the parameter, {{nowrap|0 ≤ ''t'' ≤ 1}}. Compute Ω as the angle [[Subtended angle|subtended]] by the arc, so that {{nowrap|cos Ω {{=}} ''p''<sub>0</sub> ⋅ ''p''<sub>1</sub>}}, the ''n''-dimensional [[dot product]] of the unit vectors from the origin to the ends. The geometric formula is then
: <math> \operatorname{slerp}(p_0,p_1; t) = \frac{\sin {[(1-t)\Omega}]}{\sin \Omega} p_0 + \frac{\sin [t\Omega]}{\sin \Omega} p_1.</math>

The symmetry lies in the fact that {{nowrap|slerp(''p''<sub>0</sub>, ''p''<sub>1</sub>; ''t'')}} {{nowrap|{{=}} slerp(''p''<sub>1</sub>, ''p''<sub>0</sub>; 1 − ''t'')}}. In the limit as {{nowrap|Ω → 0}}, this formula reduces to the corresponding symmetric formula for [[linear interpolation]],
: <math> \operatorname{slerp}(p_0,p_1; t) = (1-t) p_0 + t p_1.</math>

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]].

[[Image:Slerp factor explanation.png|right|thumb|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 {{nowrap|''p''<sub>0</sub>cos ''θ'' + ''p''<sub>1</sub>sin ''θ''}}. Letting {{nowrap|''θ'' {{=}} ''t''{{pi}}/2}}, and applying the trigonometric identity {{nowrap|cos ''θ'' {{=}} sin({{pi}}/2 − ''θ'')}}, this becomes the slerp formula. The factor of {{nowrap|1/sin Ω}} in the general formula is a normalization, since a vector ''p''<sub>1</sub> at an angle of Ω to ''p''<sub>0</sub> projects onto the perpendicular ⊥''p''<sub>0</sub> with a length of only {{nowrap|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 [[Jack E. Bresenham|Bresenham]]'s [[Midpoint circle algorithm|circle algorithm]]. Evaluation at the special parameter values 0 and 1 trivially yields ''p''<sub>0</sub> and ''p''<sub>1</sub>, respectively; and bisection, evaluation at&nbsp;{{sfrac|1|2}}, simplifies to {{nowrap|(''p''<sub>0</sub> + ''p''<sub>1</sub>)/2}}, normalized. Another special case, common in animation, is evaluation with fixed ends and equal parametric steps. If ''p''<sub>''k''−1</sub> and ''p''<sub>''k''</sub> are two consecutive values, and if ''c'' is twice their dot product (constant for all steps), then the next value, ''p''<sub>''k''+1</sub>, is the reflection {{nowrap|''p''<sub>''k''+1</sub> {{=}} ''cp''<sub>''k''</sub> − ''p''<sub>''k''−1</sub>}}.

== Quaternion slerp ==
When slerp is applied to unit [[quaternion]]s, the quaternion path maps to a path through 3D rotations in a [[quaternions and spatial rotation|standard way]]. The effect is a rotation with uniform [[angular velocity]] around a fixed [[axis of rotation|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 [[covering map|universal covering group]] of unit quaternions, [[3-sphere|''S''<sup>3</sup>]]. 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, {{nowrap|cos Ω}}, is negative, thus ensuring that {{nowrap|−90° ≤ Ω ≤ 90°}}.

Slerp also has expressions in terms of quaternion algebra, all using [[exponentiation]]. [[real numbers|Real]] powers of a quaternion are defined in terms of the quaternion [[exponential function]], written as ''e''<sup>''q''</sup> and given by the [[power series]] equally familiar from calculus, complex analysis and matrix algebra:
: <math> e^q = 1 + q + \frac{q^2}{2} + \frac{q^3}{6} + \cdots + \frac{q^n}{n!} + \cdots .</math>

Writing a unit quaternion ''q'' in [[versor]] form, {{nowrap|cos Ω + '''v''' sin Ω}}, with '''v''' a unit 3-vector, and noting that the quaternion square '''v'''<sup>2</sup> equals −1 (implying a quaternion version of [[Euler's formula]]), we have {{nowrap|''e''<sup> '''v'''Ω</sup> {{=}} ''q''}}, and {{nowrap|''q''<sup>''t''</sup> {{=}} cos ''t''Ω + '''v''' sin ''t''Ω}}. The identification of interest is {{nowrap|''q'' {{=}} ''q''<sub>1</sub>''q''<sub>0</sub><sup>−1</sup>}}, so that the real part of ''q'' is {{nowrap|cos Ω}}, the same as the geometric dot product used above. Here are four equivalent quaternion expressions for slerp.

: <math>
\begin{align}
\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{align}
</math>

The [[derivative]] of {{nowrap|slerp(''q''<sub>0</sub>, ''q''<sub>1</sub>; ''t'')}} with respect to ''t'', assuming the ends are fixed, is log(''q''<sub>1</sub>''q''<sub>0</sub><sup>−1</sup>) 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 transport]]ed 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 (Riemannian geometry)|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's algorithm|de&nbsp;Casteljau algorithm]] for [[Bézier curve]]s. 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&nbsp;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,<ref>{{cite thesis |last1=Pennec |first1=Xavier |title=Computing the Mean of Geometric Features Application to the Mean Rotation |date=March 1998 |publisher=INRIA |url=https://hal.inria.fr/inria-00073318 |accessdate=19 June 2020|type=report }}</ref> but the extension loses the [[time complexity|fixed execution-time]] of the slerp algorithm.

== See also ==
* [[Circular interpolation]]
* [[Quaternions and spatial rotation]]
* [[Spherical mean (statistics)]]

== References ==
{{reflist}}

== External links ==
{{refbegin}}
* {{cite web|last1=Shoemake|first1=Ken|title=Animating Rotation with Quaternion Curves|url=https://www.cs.cmu.edu/~kiranb/animation/p245-shoemake.pdf|publisher=[[SIGGRAPH]] 1985}}
* {{cite web|last1=Erik B.|first1=Dam|last2=Martin|first2=Koch|last3=Lillholm|first3=Martin|title=Quaternions, Interpolation and Animation|url=http://web.mit.edu/2.998/www/QuaternionReport1.pdf|publisher=University of Copenhagen|archive-url=https://web.archive.org/web/20170830031501/http://web.mit.edu/2.998/www/QuaternionReport1.pdf|archive-date=2017-08-30|date=July 17, 1998|url-status=live}}
* {{cite web|last1=Blow|first1=Jonathan|title=Understanding Slerp, Then Not Using It|url=http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It|archive-url=https://web.archive.org/web/20170825184056/http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/|archive-date=2017-08-25|date=February 26, 2004|url-status=live}}
* {{cite web|last1=Martin|first1=Brian|title=Brian Martin on Quaternion Animation|url=http://theory.org/software/qfa/writeup/node12.html|archive-url=https://web.archive.org/web/20160324131048/https://theory.org/software/qfa/writeup/node12.html|archive-date=2016-03-24|date=June 23, 1999|url-status=live}}
{{refend}}

[[Category:Computer graphics algorithms]]
[[Category:Quaternions]]
[[Category:Interpolation]]
[[Category:Rotation in three dimensions]]
[[Category:Spherical curves]]