Spline

[ValeryOchkov]

Why we have 3 functions lspline, pspline and cspline? Can we have only one spline?

 

[terryhendicott · Accepted solution]

Hi,

The second derivative at the endpoints for each function is as follows:

cspline—Determined by the straight line through the nearest two points.

lspline—Equals to 0.

pspline—Equals to the value of the next nearest point.

There is a fourth spline possibility.  Call it a dspline? A spline can be put through n points in the XY plane if you also specify the derivatives or slope at each end point of the spline.  I used to do this with roller coasters so the curved track could match slope of straight portions at start/finish.  Using this method no extrapolation is possible.

The four splines looking closely at first two points.  Each is subtly different.

[ChrisKaswer]

As "terryhendicott" points out, each option provides the flexibility to more closely fit a cubic spline representation to the function/response you may need to model, because of the end-point conditions. I've used 2 variations in the past to mimic the behavior of a system that requires each of those two conditions at their endpoints. It is very useful to have these options available to us.

[rgunwaldsen]

Very nice. I could have used this several times over the last few years (instead of my own home-brewed kludge).

[ValeryOchkov]

SMath has lspline and pspline. With 2 or 3 arguments

[ValeryOchkov]

Two picture from the chapter 15 Hydropower "Thoughts and Calculations when Looking at a Banknote or Spline Interpolation" of the book

STEM Problems with Mathcad and Python - 1st Edition - Valery Ochkov - (routledge.com)

[ValeryOchkov]

To be honest, I was expecting such an answer. I have a problem and I didn't solve it with the function lspline, but I solved it with a function cspline.

I have a problem and I didn't solve it with the function lspline, but I solved it with a function linterp.