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 formula_1, where the formula_2 are strictly increasing, the Akima spline will go through each of the given points. At those points, its slope, formula_3, is a function of the locations of the points formula_4 through formula_5. Specifically, we define formula_6 as the slope of the line segment from formula_1 to formula_8, namely formula_9. Then, formula_3  is defined as the following weighted average of formula_11 and formula_6: The spline is then defined as the piecewise cubic function whose value between formula_2 and formula_15 is the unique cubic polynomial formula_16 that satisfies the four constraints: formula_17, formula_18, formula_19, and formula_20. The Akima spline is a  C differentiable function (that is, has a continuous first derivative) but, in general, will have a discontinuous second derivative at the knot points. 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.