Hermite interpolation is used in computer graphics and animation to provide a smooth path between points. The figure below shows the four Hermite basis functions, labelled h00(t), h01(t), h10(t) and h11(t), where t lies in the interval [0, 1].

A smooth curve can be drawn between any two points as an appropriate linear combination of the basis functions on the left. The first step is to normalise the horizontal distance between the two points into the interval [0, 1]. Thus, if we want a point (x, y) on a smooth path between (x0, y0) and (x1, y1), we calculate the four Hermite spline values using:

t = (x-x0) / (x1-x0).

The curve starts from (x0, y0), so the coefficient of h00(t) is y0. The curve ends at (x1, y1), so the coefficient of h01(t) is y1. In general, the curve traces a path of all points (x, y) where x lies in the interval [x0, x1] and y is the following linear combination of the Hermite basis functions:

y = h00(t)y0 + h01(t)y1 + h10(t)m0 + h11(t)m1

where m0 is the gradient (direction) of the curve leaving (x0, y0) and m1 is the gradient (direction) of the curve approaching (x1, y1).

For mortality we model log(hazard) as a smooth curve between ages x0 and x1, which is an age range set by the actuary. For pensioner mortality we might have x0=50 and x1=120. The basic shape of the Hermite curve for log(mortality) can be explored here.

The Hermite model for log(mortality) does not have to be static. We can add a selection effect, an age-related time trend or even cyclic seasonal variation.