## 1 Introduction  
Ultra-precision displacement measurement technology has become a research focus in the field of measurement technology [1; 2; 3]. Displacement measurement methods have a wide range of application requirements in different fields, such as structural health monitoring [4; 5], civil engineering [6; 7; 8], and manufacturing industries [9; 10]. In particular, with the development of science and technology, the demand for precision displacement measurement technology has increased in precision instrument manufacturing, precision machine tools, and precision movement platforms [11; 12]. Displacement measurement technology is mainly divided into non-optical measurement technology and optical measurement technology. On the one hand, non-optical displacement measurement methodsinclude capacitance measurement and inductance measurement [13; 14]. On the other hand, laser triangulation [15], optical levers [16], grating displacement measurement [17], fiber displacement measurement [18], and laser interferometry [19] belong to the optical displacement measurement methods. The capacitance measurement method can achieve high measurement resolution, but its range is shorter, generally less than 1 mm [20]. The measurement accuracy of the inductance measurement method can be up to \(\pm 0.1\) mm, but it has a poor dynamic response and the structure is more sophisticated [21]. Laser interferometry has been widely used in the field of precision displacement measurement [22; 23]. It has the advantages of high measurement precision and large range. However, it is sensitive to environmental interference, high cost, and complex instrument adjustments. In addition, large volumes also make it difficult to integrate into an industrial system for automated online measurement. Grating measurement has high precision and is not sensitive to the environment, but it is restricted by its sophisticated manufacturing level [24]. The measurement of two-dimensional displacement requires the use of planar grating, however, the planar grating is difficult to manufacture and the cost is higher. An optical freeform surface has a complex surface and more degrees to be controlled, so that it is capable of achieving optical mapping relationships [25]. Currently, optical freeform surfaces are widely applied in various fields. In particular, because of the large amount of geometric information in an optical freeform surface, it has broad application prospects in the area of displacement measurement [26; 27]. For instance, [PERSON] realized the nanometer scale measurement of two-dimensional displacement based on an optical freeform surface with a double sinusoidal surface [28; 29]. The surface structure of the double sinusoidal optical freeform surface is made up of the superposition of a periodic sinusoid in the directions of x and y. The space wavelength of the sinusoid is 150 pm and the amplitude is 100 nm. However, the range of the periodic sinusoid in the x and y directions is only 50 mm. Similarly, the motion range of straightness error measurement using the 3D sinusoidal-groove linear reflective grating by [PERSON] was also shorter than 100 mm. This measurement range could be suitable for the error detection of a small precise multi-axis system. As for the linear axis motion in a large space, the detection method is ineffective. Of course, in the measurement case, non-optical sensors, such as capacitive displacement sensors with high accuracy, also appear to be powerless because of their short measuring range [30; 31]. In order to overcome this difficulty, we need to improve the measuring range of the instrument. Our laboratory has completed the measurement of 2D micro-displacement on a single rotating paraboloid. In this paper, a 2D displacement measurement method based on a rotating paraboloid array is introduced. To improve the measuring range of the method based on a rotating paraboloid, this paper presents a two-dimensional displacement measurement system, which is composed of a slope sensor and a rotating paraboloid array.