Akima spline

In applied mathematics, an Akima spline is a type of non-smoothing spline that gives good fits to curves where the second derivative is rapidly varying. The Akima spline was published by Hiroshi Akima in 1970.

Method

Given a set of "knot" points $(x_{i},y_{i})$, where the $x_{i}$ are strictly increasing, the Akima spline will go through each of the given points. At those points, its slope, $s_{i}$, is a function of the locations of the points $(x_{i-2},y_{i-2})$ through $(x_{i+2},y_{i+2})$. Specifically, we define $m_{i}$ as the slope of the line segment from $(x_{i},y_{i})$ to $(x_{i+1},y_{i+1})$, namely $(y_{i+1}-y_{i})/(x_{i+1}-x_{i})$. Then, $s_{i}$ is defined as the following weighted average of $m_{i-1}$ and $m_{i}$:

$s_{i}={\frac {|m_{i+1}-m_{i}|m_{i-1}+|m_{i-1}-m_{i-2}|m_{i}}{|m_{i+1}-m_{i}|+|m_{i-1}-m_{i-2}|}}$

The spline is then defined as the piecewise cubic function whose value between $x_{i}$ and $x_{i+1}$ is the unique cubic polynomial $P(x)$ that satisfies the four constraints: $P(x_{i})=y_{i}$, $P(x_{i+1})=y_{i+1}$, $P'(x_{i})=s_{i}$, and $P'(x_{i+1})=s_{i+1}$.

The Akima spline is a  C¹ differentiable function (that is, has a continuous first derivative) but, in general, will have a discontinuous second derivative at the knot points.

The advantage of the Akima spline is due to the fact that the Akima spline uses only values from neighboring knot points in the construction of the coefficients of the interpolation polynomial between any two knot points. This means that there is no large system of equations to solve and the Akima spline avoids unphysical wiggles in regions where the second derivative in the underlying curve is rapidly changing. The disadvantage of the Akima spline is that it has a discontinuous second derivative.

External links

• Online demo of Akima spline interpolation in TypeScript