Mathematical Background

This document explains some of the mathematical terms used throughout the Aspherix® Calibration documentation.

Squared Curve Distance

The squared curve distance ( $D_{sq}$ ) is often employed in computation of quality functions. A curve is a list of points $(x,y)$ , ordered by increasing $x$ value. It is assumed that the $\left\{x\right\}$ are unique within a curve. Now, assume two curves $A = \left\{ \left(x^A,y^A \right) \right\}$ and $B = \left\{ \left(x^B,y^B \right) \right\}$ with $N_A$ and $N_B$ points, respectively. The distance from $A$ to $B$ $D_{sq}\left(A,B\right)$ is then computed as follows:

For each point in $A$ , $\left( x^A_i, y^A_i \right)$ , the quantity

$d_i = y_i^A - \hat{y}^B \left( x_i^A \right)$

where $\hat{y}^B \left( x \right)$ is the linear interpolation of the curve $B$ to the point $x$ . The squared curve distance is then given by

$D_{sq} \left(A,B\right) = \frac{\sum_{i=1}^{N_A} d_i}{N_A \max(\left\{\left|y^A\right|\right\})^2}$

where $\max(\left\{\left|y^A\right|\right\})$ is the maximum absolute $y$ value in $A$ .

Note

Generally, $D_{sq}\left(A,B\right) \neq D_{sq}\left(B,A\right)$ , although both values are a proper measurement for the similarity of the curves.