176 4. 3D Math for Games

To solve this problem, we can use a variant of the LERP operation known

as spherical linear interpolation, or SLERP for short. The SLERP operation uses

sines and cosines to interpolate along a great circle of the 4D hypersphere,

rather than along a chord, as shown in Figure 4.23. This results in a constant

angular speed when β varies at a constant rate.

The formula for SLERP is similar to the LERP formula, but the weights

(1 – β) and β are replaced with weights wp and wq involving sines of the angle

between the two quaternions.

where

The cosine of the angle between any two unit-length quaternions can

be found by taking their four-dimensional dot product. Once we know

cos(θ), we can calculate the angle θ and the various sines we need quite

easily:

qA (β = 0)

qLER P= LERP(qA, qB, 0.4)

qB ( β = 1)

qSLERP= SLERP (qA, qB, 0.4)

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cho
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(^0). 4
(^) alo
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(^) ar
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Figure 4.23. Spherical linear interpolation along a great circle arc of a 4D hypersphere.
SLERP(p,q, )β=wwpqp+ q,
sin((1 ) )
,
sin( )
sin( ).
sin( )
p
q
w
w
−β θ

θ
= βθ
θ
1
cos( ) p q ;
cos (p q).
pqxx pqyy pqzz p qww
−
θ= =+++
θ=
⋅
⋅