2. Measurement Process Characterization
2.3. Calibration
2.3.4. Catalog of calibration designs
2.3.4.4. Roundness measurements

## Single-trace roundness design

Low precision measurements Some measurements of roundness do not require a high level of precision, such as measurements on cylinders, spheres, and ring gages where roundness is not of primary importance. The diagram of the measurement method shows the trace and Y, the distance from the spindle center to the trace at the angle. A least-squares circle fit to data at equally spaced angles gives estimates of P - R, the noncircularity, where R is the radius of the circle and P is the distance from the center of the circle to the trace.
Single trace method For this purpose, a single trace covering exactly 360° is made of the workpiece and measurements $$Y_i$$ at angles $$\theta_i$$ of the distance between the center of the spindle and the trace, are made at $$\theta_i \{ i=1, \, \ldots, \, N\}$$ equally spaced angles. A least-squares circle fit to the data gives the following estimators of the parameters of the circle. \begin{array} $$\widehat{R} = \frac{1}{N} \sum_{i=1}^{N} Y_i \\ \widehat{a} = \frac{2}{N} \sum_{i=1}^{N} Y_i \, \mbox{cos}(\theta_i) \\ \widehat{b} = \frac{2}{N} \sum_{i=1}^{N} Y_i \, \mbox{sin}(\theta_i) \\ \end{array} Noncircularity of workpiece The deviation of the trace from the circle at angle \( \theta_i$$, which defines the noncircularity of the workpiece, is estimated by: $$\widehat{\Delta} = Y_i - \widehat{R} - \widehat{a} \, \mbox{cos}(\theta_i) - \widehat{b} \, \mbox{sin}(\theta_i) \, .$$
Weakness of single trace method The weakness of this method is that the deviations contain both the spindle error and the workpiece error, and these two errors cannot be separated with the single trace. Because the spindle error is usually small and within known limits, its effect can be ignored except when the most precise measurements are needed.