Slerp

Shoemake [16] introduced the idea of interpolating rotations with quaternions, but the constructed curves (slerps) did not satisfy an obvious variational principle [21] as splines [2] do in flat space.

However, it should be possible to give more sophisticated spherical spline curves based on the de Castaljau method that are computed using multiple slerps between pairs of points and which work well for arbitrary knot positions (indeed, knot insertion methods for spline curves should suffice for this, cf Farin [1993])

Unlike the slerp techniques, this approach produces motions with rational point trajectories, the socalled rational motions. In kinematical geometry, these motions had been studied since the end of the 19th century (5,43,52].

The first describes the skeleton-driven deformation which is solved by spherical linear interpolation (slerp).

The quaternion spherical linear interpolation method (slerp), which many researchers have adopted for numerical control, is inadequate for tool-axisorientation planning in general five-axis computer numerical control machines. This paper explains why the quaternion slerp method fails to achieve constant rotational speed.

In other words, E.3-1 says that the slerped quaternion qk can be computed by multiplication of k-th power of a constant quaternion qc and q0, where β=α/n and u is a unit vector defined by components of q0 and qn as u = (w0un – wnu0 + u0×un)/sin(α).

Board and skater orientations were slerped, and the velocity was re-computed back from the positional Hermite curve rather than being interpolated.

More specifically, the quaternion representing the currently reported head position, Q∗(t), is slerped (linearly interpolated in quaternion space) with the most recent quater- nion used for rendering, Qt−1.