# Interpolation

## Introduction

An interpolation, by definition, passes through every data point, and as such, the correlation coefficient
will always be 1, and the standard error will always be zero.
CurveExpert Professional supports polynomial splines and tension splines.
All splines are defined in a piecewise fashion between data points.

Note

The dataset must be sorted (based on the independent variable) in order for any spline interpolation to work. Select *Data->Sort* from the main menu in order to sort your dataset if necessary. If CurveExpert Professional detects that your dataset is not sorted, it will not allow spline interpolations to be selected.

Note

Currently, CurveExpert Professional supports interpolation for datasets with one independent variable only. For
multivariate datasets, all interpolations will be disabled.

## Linear Splines

To calculate a linear spline, select *Calculate->Linear Spline* from the main menu. The linear spline is simply
a polynomial spline of order 1. It appears as a “dot-to-dot” connecting each point with a straight line
segment. Linear splines only guarantee continuity of the spline at the data points.

## Cubic Splines

To calculate a cubic spline, select *Calculate->Cubic Spline* from the main menu. The cubic spline is simply
a polynomial spline of order 3; cubic splines are the most common form of spline. Cubic splines guarantee
continuity in the spline, and continuity in the first and second derivatives of the spline at the data points.
At the endpoints, the second derivative is set to zero, which is termed a “natural” spline at the endpoints,
as the curvature goes to zero.

## Polynomial Splines

To calculate a generic polynomial spline, select *Calculate->Polynomial Spline* from the main menu. A prompt will
appear to ask for the degree of the polynomial spline; by way of example, a degree of 1 would be a linear spline,
and a degree of 3 would be a cubic spline.

## Tension Splines

To calculate a tension spline, select *Calculate->Tension Spline* from the main menu. A prompt will appear
to ask for the amount of tension desired. Tension splines are based on hyperbolic functions, and simulate a cord
being stretched with a defined tension (amount of force) between the data points. An extremely high tension
approaches a linear spline, and low tensions will appear correspondingly “loose” around the data points, resembling
a cubic spline.