## 3 Errors Identification Principle
### Calibration of the Relative Position  
There is a point on every revolutionary paraboloid surface at which the tangent plane is parallel to the plane mirror around the revolutionary paraboloid. This point is defined as the feature point of the revolutionary paraboloid. To quantitatively determine the relative positioning of paraboloids, it can be defined as the relative positioning of the feature points of the corresponding paraboloids. Like the computation of the relative angle, the position of other targets was also found relative to the reference target, as shown in Figure 5. The displacement values \(X_{Fi}\) and \(Y_{Fi}\) are the relative position of the feature point of the \(i\)th target along the \(X\)-axis and \(Y\)-axis, respectively. As Figure 6a shows, when the angle sensor moves along the \(X\)-axis, the reflection point moves from \(p_{s1}\) to \(p_{s2}\) and the spot on the image sensor moves from \(p_{c1}\) to \(p_{c2}\). In the same paraboloid, the translational displacement of the angle sensor, as mentioned above, is linearly related with the displacement of the spot on the image sensor. Besides this, the coefficient of linearity can be obtained by the measurement of the angle sensor on the revolutionary paraboloid. To conduct a large-scale measurement, the distances of the feature points of different revolutionary paraboloids should be determined. In order to calculate the relative positioning of feature points in the workpiece coordinate system, the machine drives an angle sensor from the measurement point \(M_{1}\) to \(M_{2}\) along the \(X\)-axis, as shown in Figure 6b. The points \(F_{1}\) and \(F_{2}\) are defined as the feature points of two revolutionary paraboloids. Figure 5: Sketch of the relative angle and position of targets. Figure 6: (**a**) Relative displacement measurement in one revolutionary paraboloid; and (**b**) relative position measurement of feature points on different revolutionary paraboloids. Spot images of the feature point and the measurement point are shown in Figure 7a,b. The position of the spot center is obtained by ellipse fitting, which can realize subpixel spot extraction. It was found by calibration that each pixel in the image corresponds to 2.27 arc seconds or 1.6 \(\upmu\)m, and the resolutions of the angle and position can reach, respectively, 0.2 arc seconds and 0.16 \(\upmu\)m by subpixel spot extraction. Actually, the spot image of a feature point cannot be obtained directly, so it is obtained by measuring the corresponding plane mirror. As illustrated in Figure 6b, the vector relation between two feature points \(F_{1}\) and \(F_{2}\) is described by the following equation:  
\[F_{1}^{\rightarrow}F_{2}=F_{1}^{\rightarrow}M_{1}+M_{1}^{\rightarrow}M_{2}+ M_{2}^{\rightarrow}F_{2}. \tag{2}\]  
In the \(X\) direction, we have:  
\[(X_{F2}-X_{F1})=(X_{M1}-X_{F1})+(X_{M2}-X_{M1})+(X_{F2}-X_{M2}), \tag{3}\]  
where \(X_{F2}\) and \(X_{F1}\) are the \(X\) coordinates of feature points \(F_{1}\) and \(F_{2}\) in the object ordinate system, and \(X_{M2}\) and \(X_{M1}\) are the \(X\) coordinates of measuring points \(M_{1}\) and \(M_{2}\) in the object ordinate system. Through measurement by laser interferometer with a position lens or machine tools with higher precision, \(lx_{M2M1}\), defined as the distance between measuring points \(M_{1}\) and \(M_{2}\), can be calculated. In addition, the distance between the measurement point and the feature point in a revolutionary paraboloid can be determined based on the spot displacement in an image by the following equation:  
\[(X_{Mi}-X_{Fi})=k(x_{Mi}-x_{Fi}), \tag{4}\]  
where \(k\) is the conversion coefficient from spot displacement to actual displacement calibrated by a laser interferometer; \(X_{Mi}\) and \(X_{Fi}\) are the \(X\) coordinates of the measurement point and feature point of the \(i\)th revolutionary paraboloid in the object coordinate system; and \(x_{Mi}\) and \(x_{Fi}\) are the \(X\) coordinates of the measurement point and feature point of the \(i\)th revolutionary paraboloid in the image system.