## 1 Introduction  
Scanning methods are widely used in the measurement and reconstruction of straightness profiles. The motion trajectory of the measuring device always plays the role of the measurement reference, of which the error will have an impact on the final reconstruction result. Especially for the on-machine measurement, as the machine tool spindle is used as a scanning stage, the error of the measurement reference is at the same level as the accuracy of the measured profiles. Therefore, the errors of the reference must be eliminated to avoid affecting the reconstruction result. Several multi-probe methods were developed to remove the influence of reference errors. The two-point method can eliminate the straightness error of the measurement reference [1; 2; 3; 4; 5]. Besides, the yaw error of the measuring device during scanning also affects the reconstruction result. Therefore, the three-point method [6; 7; 8], four-point method (or the method using more than four probes) [9; 10], the traceable multiple sensor (TMS) method [11], etc. were invented to overcome the influence of the straightness error and eliminate the yaw error at the same time. With more probes, abundant information makes exact reconstruction possible. In some cases, the reconstruction error caused by yaw error is too small compared with that caused by straightness error, and the probe zero-adjustment error will have a significant impact on the final result due to the cumulative effect when using three or more probes. Therefore, the two-point method has not been replaced and still widely used by now. Besides, other errors that impact the reconstruction result most include the zero-adjustment error, the data processing error, and the measurement error of displacement sensors. When using scanning methods to measure and reconstruct profiles, these errors are worth considering. Using the height difference between adjacent measurement points, the two-point method, including the sequential two-point method (STP), the generalized two-point method (GTP), and the combined two-point method (CTP), can obtain the reconstruction profile by a recursive process. In the STP method, the step distance of the measurement device is the same as the spacing between two probes, which is equal to the interval between adjacent sampling points, and the reconstruction result is not affected by the data processing error. However, due to the size of probes, the number of sampling points could be small, which means the lateral resolution could not be high enough and there is a higher probability that some key features of the measured profile will be missed. In the GTP method, the density of the sampling points is higher, as the interval between sampling points can be smaller than the probe spacing. Nevertheless, the reconstruction result suffers from data processing error on account of the higher-order harmonic component distortion. The effect of the data processing error is especially evident when there are stepwise variations in measured profiles. The CTP method [3] combines the above two methods and achieves theoretically exact reconstruction of measured profiles. In many cases, its accuracy is hard to meet due to the assumption of a completely smooth part of the measured profile to obtain the relative relations of different groups of reconstruction points, which makes it difficult to implement in practice. Different systems and methods have been developed to improve the lateral resolution in the absence of the data processing error [12; 13; 14; 15; 16; 17]. The common methods are using additional probes or additional scanning to obtain more measurement information. While some methods among them are affected by the zero-adjustment error of the probes placed abreast in the system. The above methods can be used to reconstruct a single straightness profile under different conditions. In practical applications, the parameters of a pair of parallel profiles also need to be monitored.