Akima spline

Method

Given a set of "knot" points ${\displaystyle (x_i,y_i{)}_{i=1\dots n}}$, where the ${\displaystyle x_i}$ are strictly increasing, the Akima spline will go through each of the given points. At those points, its slope, ${\displaystyle s_i}$, is a function of the locations of the points ${\displaystyle (x_{i-2},y_{i-2})}$ through ${\displaystyle (x_{i+2},y_{i+2})}$. Specifically, if we define ${\displaystyle m_i}$ as the slope of the line segment from ${\displaystyle (x_i,y_i)}$ to ${\displaystyle (x_{i+1},y_{i+1})}$, namely

${\displaystyle m_i=\frac{y_{i+1}-y_i}{x_{i+1}-x_i},}$

then the spline slopes ${\displaystyle s_i}$ are defined as the following weighted average of ${\displaystyle m_{i-1}}$ and ${\displaystyle m_i}$,

${\displaystyle s_i=\frac{|m_{i+1}-m_i|m_{i-1}+|m_{i-1}-m_{i-2}|m_i}{|m_{i+1}-m_i|+|m_{i-1}-m_{i-2}|}.}$

If the denominator equals zero, the slope is given as

${\displaystyle s_i=\frac{m_{i-1}+m_i}{2}.}$

The first two and the last two points need a special prescription, for example,

${\displaystyle s_1=m_1,s_2=\frac{m_1+m_2}{2},s_{n-1}=\frac{m_{n-2}+m_{n-1}}{2},s_n=m_{n-1}.}$

The spline is then defined as the piecewise cubic function whose value between ${\displaystyle x_i}$ and ${\displaystyle x_{i+1}}$ is the unique cubic polynomial ${\displaystyle P_i(x)}$,

${\displaystyle P_i(x)=a_i+b_i(x-x_i)+c_i(x-x_i{)}^2+d_i(x-x_i{)}^3,}$

where the coefficients of the polynomial are chosen such that the four conditions of continuity of the spline together with its first derivative are satisfied,

${\displaystyle P(x_i)=y_i,P(x_{i+1})=y_{i+1},P^{\prime }(x_i)=s_i,P^{\prime }(x_{i+1})=s_{i+1}.}$

which gives

${\displaystyle a_i=y_i,}$

${\displaystyle b_i=s_i,}$

${\displaystyle c_i=\frac{3m_i-2s_i-s_{i+1}}{x_{i+1}-x_i},}$

${\displaystyle d_i=\frac{s_i+s_{i+1}-2m_i}{(x_{i+1}-x_i{)}^2}.}$

Due to these conditions the Akima spline is a C¹ differentiable function, that is, the function itself is continuous and the first derivative is also continuous. However, in general, the second derivative is not necessarily continuous.

An advantage of the Akima spline is due to the fact that it 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. A possible disadvantage of the Akima spline is that it has a discontinuous second derivative.