Monotone cubic spline

A spline is a curve made of polynomial pieces joined smoothly. A cubic spline uses a third-degree polynomial between each pair of points. On each interval from $t_k$ to $t_{k+1}$, with $s = (t - t_k)/h$ and $h = t_{k+1} - t_k$, a cubic Hermite piece is set by the two end values $y_k, y_{k+1}$ and the slopes $m_k, m_{k+1}$ there:

$$p(s) = (2s^3 - 3s^2 + 1)\,y_k + (s^3 - 2s^2 + s)\,h\,m_k + (-2s^3 + 3s^2)\,y_{k+1} + (s^3 - s^2)\,h\,m_{k+1}$$

The overshoot problem

A standard smooth cubic spline chooses slopes to make the curve as smooth as possible. Near sharp changes this makes it overshoot: it swings past the data and back. For motion that is visible and can be harmful. A foot meant to rest at floor height dips below the floor, or a joint briefly passes its limit.

The monotone fix

Fritsch and Carlson (1980) choose the slopes $m_k$ so that the curve is monotone wherever the data is: if the points only rise, the curve only rises; if two neighbouring points are equal, the curve stays flat between them. Slopes are taken from the neighbouring secant slopes $\Delta_k = (y_{k+1} - y_k)/h$ and set to zero at peaks and valleys. This is often called PCHIP (piecewise cubic Hermite interpolating polynomial).

The cost is a little less smoothness: the second derivative can jump at the points. For keyframed motion that’s a good trade.

In this project

The keyframes of the recovery moves are joined with monotone cubic splines, so a foot that should stay planted, or a heel that should stay down, doesn’t drift between keys.