Question

I’m attempting to calculate a specific y-value from a natural cubic spline given an x-value, using the natural spline form from http://mathworld.wolfram.com/CubicSpline.html and coded in Java at http://www.cse.unsw.edu.au/~lambert/splines/natcubic.html.

Unfortunately this spline form takes a t-value on the range from 0->1 for each segment of the spline in order to compute x and y coordinates along each segment. I’d like to input a given x-coordinate and get the corresponding y-coordinate, but i’m not sure how to get a t-value from my x-coordinate. I.e., the coefficients of the spline are stored as y=f(t) and x=f(t), but i’d like to get y=f(x). Short of creating a look-up table, are there any exact solutions to get y=f(x) from this form of spline?

I’ve tried other implementations that take an x-value and return a y-value (e.g., apache commons math, Flanagan’s scientific library), but the spline version from the websites above appear to behave favorably in my application.

Thanks.

Answer 1

What you want is not always possible.
There can be a single segment of the spline, which passes through two different points for a single value of x.

If you really need it as a function of x, you can solve the cubic equation X(t) = x0 for t (using for example Cardano‘s formulas) and then substitute the value of t in Y(t).