Roundness (object) - Calculation in Two-dimensions

Calculation in Two-dimensions

A single trace covering the full rotation is made and at each equally spaced angle, a measurement, of the radius or distance between the center of rotation and the surface point. A least-squares fit to the data gives the following estimators of the parameters of the circle:

$\hat{R} = \frac{1}{N}\sum\limits_{i=1}^N R_i$

$\hat{a} = \frac{2}{N} \sum\limits_{i=1}^N R_i \cos{\theta_i}$

$\hat{b} = \frac{2}{N} \sum\limits_{i=1}^N R_i \sin{\theta_i}$

The deviation is then measured as:

$\hat{\Delta} = R_i - \hat{R} - \hat{a} \cos{\theta_i} - \hat{b} \sin{\theta_i}$

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