A parallel plate refraction imaging binocular vision three-dimensional measurement method

Abstract

The present invention discloses a parallel plate refraction imaging binocular vision three-dimensional measurement method, the method adopts equipment including a binocular camera, a parallel plate, a calibration plate and a data processing system, the binocular camera is set up and fixed at a position where the image of the calibration plate can be collected through the parallel plate, and after calibration, it is used to collect the image of the calibration plate after the parallel plate is placed and measure; after the binocular camera is calibrated, the parallel plate is set up and fixed between the calibration plate and the binocular camera, and is used to isolate the measurement system and the object to be measured; the calibration plate is used for the calibration of the binocular camera and the analysis of the calibration plate image before and after refraction. The present invention establishes a binocular camera parallel plate refraction model, calculates the thickness d and refractive index n of the parallel plate, and restores the real world coordinates of the measured point p according to the established binocular camera parallel plate refraction model and the calculated parallel plate normal vector η, thickness d and refractive index n.

Description

Technical Field

[0001] The invention relates to the field of visual three-dimensional measurement, and in particular to a parallel plate refraction imaging binocular visual three-dimensional measurement method. Background Art

[0002] With the development of visual technology, visual 3D measurement is widely used in the field of non-contact 3D measurement, and has the characteristics of high measurement accuracy and fast measurement speed. The binocular vision 3D measurement system shoots the object at two angles and uses parallax to obtain the depth information of the object, which can further reduce the complexity of the measurement system. Although its algorithm is complex, with the improvement of the performance of the computing unit, the binocular vision 3D measurement system has irreplaceable advantages in the field of online detection and non-contact measurement of industrial products.

[0003] In order to cope with various measurement environments, parallel plates are added to the visual 3D measurement system: for example, in industrial production, parallel plates are used to isolate the measurement system from corrosive, toxic, high temperature, low temperature and other environments; in dangerous environments, explosion-proof glass is used to protect the measurement system; in museums, shopping malls and other environments, glass covers or other transparent media are used to protect valuables, and the measurement system performs measurements through these media.

[0004] In the existing binocular measurement method through a parallel plate, the influence of the parallel plate on the measurement is usually ignored, and the influence of the birefringence effect caused by the parallel plate on the imaging of the binocular system is not analyzed, resulting in poor measurement accuracy. Summary of the invention

[0005] To solve the above problems, the present invention proposes a parallel plate refraction imaging binocular vision 3D measurement method, establishes a binocular camera parallel plate refraction model, analyzes the birefringence effect caused by the parallel plate, and has high measurement accuracy.

[0006] The technical solution adopted by the present invention to solve the problem is:

[0007] A parallel plate refraction imaging binocular vision three-dimensional measurement method, the method using equipment including a binocular camera, a parallel plate, a calibration plate and a data processing system, including the following steps:

[0008] S1. Calibrate the binocular camera without setting up a parallel plate, and use the world coordinates of the calibration plate target as the standard reference point P1…P before refraction. n The coordinates of the two optical centers O1 and O2 of the binocular camera are calculated. According to the pixel size d of the binocular camera x d y And the quantized focal length f of the camera in the pixel coordinate system X and Y axes obtained by calibration x 、f y Calculate the focal length f of the binocular cameral 、f r ;

[0009] S2, set up a fixed parallel plate between the binocular camera and the calibration plate, and each binocular camera collects an image of the calibration plate after being refracted by the parallel plate;

[0010] S3, the target point P on the calibration plate image collected by the binocular camera after refraction l1 …P ln , P r1 …P rn The pixel coordinates are converted to world coordinates, and the P l1 …P ln and the straight line l between the optical center O1 of the left camera l1 …l ln and P r1 …P rn The straight line l with the right camera optical center O2 r1 …l rn ;

[0011] S4, establishing a parallel plate refraction model for binocular cameras;

[0012] S5, calculated P1…P n of and l l1 …l ln , l r1 …l rn The normal vectors η1…η of the intersecting lines are n , using η1…η n Calculate the normal vector η of the parallel plate;

[0013] S6, calculated P1…P n The straight line with normal vector η and l l1 …l ln , l r1 …l rn The intersection point P' l1 …P' ln and P' r1 …P' rn to P1…P n The distance h l1 …h ln 、h r1 …h rn And the parallel plate normal vector η and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn ;

[0014] S7, h l1 …h ln 、h r1 …h rn and α l1 …α ln , α r1 …α rn The parallel plate refraction model of the binocular camera established in step S4 is introduced to calculate the thickness d and refractive index n of the parallel plate;

[0015] S8. Use a binocular camera to collect images through a parallel plate, convert the pixel coordinates of the measured point into world coordinates, and calculate the straight lines l1 and l2 passing through the corresponding optical centers O1 and O2 respectively. According to the binocular camera parallel plate refraction model established in step S4 and the calculated parallel plate normal vector η, thickness d and refractive index n, restore the real world coordinates of the measured point p.

[0016] Furthermore, after the binocular camera is calibrated in step S1, the calibration plate is kept stationary at the position when the last calibration plate image is taken during the binocular camera calibration process, and the calibration plate plane at this time is used as the XOY plane of the world coordinate system to establish the world coordinate system; the conversion relationship between the quantized focal length and the camera focal length is given by the following formula:

[0017] f=(f x d x +f y d y ) / 2.

[0018] Furthermore, step S2 performs distortion correction on the refracted calibration plate image.

[0019] Furthermore, in step S3, a pinhole imaging model is established with O1 and O2 as the center, and P l1 …P ln , P r1 …P rn Pixel coordinates are converted to world coordinates, straight line l l1 …l ln , l r1 …l rn The calculation of is given by the following formula, where λ is the proportionality factor:

[0020]

[0021] Furthermore, in the binocular parallel plate refraction model of step S4, through P1 ...P n The straight line whose normal vector is equal to the normal vector of the parallel plate is l l1 …l ln and l r1 …l rn Corresponding intersection, intersection point P' l1 …P' lnand P' r1 …P' rn to P1…P n The distance h l1 …h ln 、h r1 …h rn with the normal vector of the parallel plate and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn The relationship is given by the following formula:

[0022]

[0023] Where d is the thickness of the parallel plate and n is the refractive index of the parallel plate.

[0024] Further, step S5 takes η1…η n The mean The normal vector η of the parallel plate is calculated as the initial value of the normal vector of the parallel plate by the nonlinear least squares iteration method. The cost function is given by the following function:

[0025]

[0026] Furthermore, in step S7, the thickness d and the refractive index n of the parallel plate are calculated by a nonlinear least squares iteration method, and the following function is used as a cost function:

[0027]

[0028] The rough measured values ​​of the refractive index n and thickness d of the parallel plate are used as the initial values ​​of the nonlinear least squares iteration method.

[0029] Furthermore, in step S8, straight lines l1 and l2 intersect straight line l at points p1 and p2, the normal vector of l is equal to the normal vector η of the parallel plate, and the distance between points p1 and p2 is h 12 It is given by the following formula:

[0030]

[0031] Among them, α1, α2 are the angles between l1, l2 and η, then p is on the straight line l, and the vector In the same direction as η, the distances h1 and h2 between p and p1 and p2 are given by the following formulas:

[0032]

[0033] The technical effects of the present invention are as follows:

[0034] Compared with the prior art, the present invention has the following beneficial technical effects: the present invention establishes a binocular camera parallel plate refraction model, analyzes the double refraction effect caused by the parallel plate, and has high measurement accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 It is the overall flow chart of the present invention.

[0036] Figure 2 This is a pinhole imaging model diagram built with the camera optical center as the center.

[0037] Figure 3 This is a diagram of the parallel plate refraction model of a binocular camera.

[0038] Figure 4 It is a schematic diagram of a binocular camera performing three-dimensional measurement through a parallel plate.

[0039] Figure 5 It is a graph showing the deviation of the target point translation distance of the calibration plate before and after the binocular camera measures the target point translation through the parallel plate. DETAILED DESCRIPTION

[0040] In order to make the above-mentioned features and advantages of the present invention better understood by those skilled in the art, the present invention is described in detail below in conjunction with specific implementation cases and drawings. Based on the examples of the present invention, other embodiments obtained by those skilled in the art without making any creative work shall all fall within the scope of protection of the present invention.

[0041] The invention discloses a parallel plate refraction imaging binocular vision three-dimensional measurement method. The equipment used in the method comprises a binocular camera, a parallel plate, a calibration plate and a data processing system. The binocular camera is set up and fixed at a position where the image of the calibration plate can be collected through the parallel plate, and is used for collecting the image of the calibration plate after the parallel plate is placed and for measurement after calibration. After the binocular camera is calibrated, the parallel plate is set up and fixed between the calibration plate and the binocular camera, and is used for isolating the measurement system and the object to be measured. The calibration plate is used for the calibration of the binocular camera and the analysis of the image of the calibration plate before and after refraction. The data processing system is used for processing images and equation and function operations.

[0042] Reference Figure 1 As shown, a parallel plate refraction imaging binocular vision three-dimensional measurement method of the present invention specifically comprises the following steps:

[0043] S1. Calibrate the binocular camera without setting up a parallel plate. After the binocular camera is calibrated, keep the calibration plate still at the position when the last calibration plate image is taken during the binocular camera calibration process. Use the calibration plate plane at this time as the XOY plane of the world coordinate system to establish the world coordinate system; use the world coordinates of the calibration plate target as the standard reference point P1…P before refraction nThe coordinates of the binocular camera are obtained by using the rotation matrix R1, R2 and translation matrix T1, T2 of the binocular camera obtained after calibration to obtain the external parameter matrix of the binocular camera. The world coordinates of the optical centers O1 and O2 of the two cameras are calculated according to the conversion relationship between the binocular camera coordinate system and the world coordinate system. The conversion relationship between the camera coordinate system and the world coordinate system is given by the following formula:

[0044]

[0045] Where R, T are R1, T1 or R2, T2, (x c ,y c ,z c ) is the camera coordinate system coordinate, (X w ,Y w ,Z w ) are the world coordinate system coordinates.

[0046] Secondly, according to the pixel size d of the binocular camera x d y And the quantized focal length f of the camera on the X and Y axes obtained by calibration x 、f y Calculate the focal length f of the binocular camera separately l 、f r The conversion relationship between quantized focal length and camera focal length is given by the following formula:

[0047] f=(f x d x +f y d y ) / 2.

[0048] S2. A fixed parallel plate is set up between the binocular camera and the calibration plate. Each binocular camera collects an image of the calibration plate after being refracted by the parallel plate.

[0049] S3, according to the conversion relationship between the pixel coordinate system and the world coordinate system, the target point P on the calibration plate image collected by the binocular camera after refraction is converted to l1 …P ln , P r1 …P rn The pixel coordinates are converted to world coordinates. The conversion relationship between the world coordinate system and the pixel coordinate system is given by the following formula:

[0050]

[0051] Among them, (u, v) is the pixel coordinate system, and (u0, v0) is the coordinate of the origin of the image coordinate system in the pixel coordinate system.

[0052] Calculated P l1 …P ln and the straight line l between the optical center O1 of the left camera l1 …lln and P r1 …P rn The straight line l with the right camera optical center O2 r1 …l rn , the calculation method is: refer to Figure 2 As shown in the figure, taking a camera as an example, a pinhole imaging model is established with O1 and O2 as the center, and P l1 …P ln , P r1 …P rn Pixel coordinates are converted to world coordinates, straight line l l1 …l ln , l r1 …l rn The calculation of is given by the following formula, where λ is the proportionality factor:

[0053]

[0054] S4. Establish a parallel plate refraction model for binocular cameras. Figure 3 As shown, in the binocular parallel plate refraction model, through P1…P n The straight line whose normal vector is equal to the normal vector of the parallel plate is l l1 …l ln and l r1 …l rn Corresponding intersection, intersection point P' l1 …P' ln and P' r1 …P' rn to P1…P n The distance h l1 …h ln 、h r1 …h rn with the normal vector of the parallel plate and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn The relationship is given by the following formula:

[0055]

[0056] Where d is the thickness of the parallel plate and n is the refractive index of the parallel plate.

[0057] S5, calculated P1…P n of and l l1 …l ln , l r1 …l rn The normal vectors η1…η of the intersecting lines aren , using η1…η n Calculate the parallel plate normal vector η. Specifically, take η1…η n The mean The normal vector η of the parallel plate is calculated as the initial value of the normal vector of the parallel plate by the nonlinear least squares iteration method. The cost function is given by the following function:

[0058]

[0059] S6, calculated P1…P n The straight line with normal vector η and l l1 …l ln , l r1 …l rn The intersection point P' l1 …P' ln and P' r1 …P' rn to P1…P n The distance h l1 …h ln 、h r1 …h rn And the parallel plate normal vector η and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn At this time, the distance h is the distance between points P' and P at the shortest common perpendicular segment of the straight line l and the normal vector η passing through point P.

[0060] S7, h l1 …h ln 、h r1 …h rn and α l1 …α ln , α r1 …α rn Substitute the binocular camera parallel plate refraction model established in step S4 to calculate the thickness d and refractive index n of the parallel plate. Specifically, the thickness d and refractive index n of the parallel plate are calculated by nonlinear least squares iteration method, and the following function is used as the cost function:

[0061]

[0062] The rough measured values ​​of the refractive index n and thickness d of the parallel plate are used as the initial values ​​of the nonlinear least squares iteration method.

[0063] S8. Use a binocular camera to collect images through a parallel plate, convert the pixel coordinates of the measured point into world coordinates, and calculate the straight lines l1 and l2 passing through the corresponding optical centers O1 and O2 respectively. According to the binocular camera parallel plate refraction model established in step S4 and the calculated parallel plate normal vector η, thickness d and refractive index n, restore the real world coordinates of the measured point p.

[0064] Reference Figure 4 As shown, in step S8, straight lines l1 and l2 intersect straight line l at points p1 and p2, the normal vector of l is equal to the normal vector η of the parallel plate, and the distance between points p1 and p2 is h 12 It is given by the following formula:

[0065]

[0066] Among them, α1, α2 are the angles between l1, l2 and η, then p is on the straight line l, and the vector In the same direction as η, the distances h1 and h2 between p and p1 and p2 are given by the following formulas:

[0067]

[0068] In summary, the core of the present invention lies in the extraction and calculation of the calibration plate image before and after the parallel plate is placed, that is, extracting the target coordinates of the calibration plate image before and after refraction and using the binocular camera parallel plate refraction model established in step S4 to calculate the parallel plate's own parameters, that is, the thickness, normal vector and refractive index of the parallel plate. Taking an experiment as an example, the specific steps and data of the experiment are given below:

[0069] First, the resolution of the binocular cameras selected in the experiment is 2592×2048, the pixel size is 4.8μm×4.8μm, a 75mm lens is selected, and the baseline length of the binocular camera is 200mm; the calibration plate uses a 9×9 circular directional target with a center distance of 12mm.

[0070] Adjust the lens focus to a camera working distance of 1300mm. Do not place a parallel plate. Use Zhang's calibration method to calibrate the camera. Fix the position of the calibration plate at the last calibration acquisition position and set the calibration plate surface as the XOY plane of the world coordinate system. Figure 5 As shown, the target center, i.e., the target point coordinates, are used as the standard reference point coordinates.

[0071] Second, the world coordinates of the optical centers O1 and O2 of the left and right cameras are calculated using the calibrated external parameters, which are given by the following formula:

[0072]

[0073] Calculate the focal length of the lens, multiply the calibrated quantized focal lengths of the left and right cameras by the pixel size and take the average, and the focal lengths of the two cameras are 72.112335mm and 71.775466mm respectively.

[0074] Third, the selected parallel plate is customized quartz glass. The rough measured thickness of the quartz glass is 15.03mm, the refractive index is 1.40, and the parallelism tolerance of the upper and lower surfaces of the quartz glass is 0.1mm. The quartz glass is placed between the working distance and the camera and close to the camera. The binocular camera collects the image of the calibration plate fixed at the final calibration position, and the calibration parameters are used to perform distortion correction on the collected image.

[0075] Fourth, extract the pixel coordinates of the target point of the calibration plate image collected after refraction, referring to Figure 2 As shown, it is converted to world coordinates. In the world coordinate system, the normal vectors η1…η of 81 straight lines passing through the standard reference point and passing through the optical centers of the left and right cameras and the corresponding target points after refraction are calculated. 81 .

[0076] The model established in step S4 is an ideal model. Due to the manufacturing tolerance and calculation error of the quartz glass, the 81 normal vectors are not equal. Therefore, the mean of the 81 normal vectors is used as the initial value of the Levenberg-Marquardt iterative method (which is the most widely used nonlinear least squares algorithm, the Levenberg-Marquardt method in Chinese) to calculate the normal vector of the quartz glass, where the cost function is:

[0077]

[0078] Fifth, in the world coordinate system, the obtained parallel plate normal vector is used to calculate the straight line l between 81 pairs of standard reference points and the corresponding left and right camera target points and the optical center after refraction. l1 …l l81 , l r1 …l r81 The distance h between the two intersection points of the straight line passing through the standard reference point and whose normal vector is the normal vector of the quartz glass l1 …h l81 、h r1 …h r81 and l1 …l l81 , l r1 …l r81 The angle α with the normal vector of the quartz glass l1 …α l81 , α r1 …α r81 Due to the manufacturing tolerance of quartz glass and calculation errors, there is no l1 …l l81 , lr1 …l r81 The straight line that intersects and passes through the standard reference point and whose normal vector is the normal vector of the quartz glass, therefore, the distance h at this time l1 …h l81 、h r1 …h r81 is a straight line passing through the standard reference point with a normal vector equal to the normal vector of the quartz glass and l l1 …l l81 , l r1 …l r81 The distance between the point at the shortest common perpendicular segment and the standard reference point.

[0079] Sixth, h l1 …h l81 、h r1 …h r81 and α l1 …α l8 , α r1 …α r81 As the parameters of the Levenberg-Marquardt iteration method, the roughly measured parallel plate thickness and refractive index are used as the initial values ​​of the Levenberg-Marquardt iteration method. The calculated quartz glass thickness is 15.0341 mm and the refractive index is 1.3910. The cost function is given by the following function:

[0080]

[0081] Seventh, fix the calibration plate on the translation stage, use the translation stage to translate the calibration plate 5mm, use the binocular camera to collect the images of the calibration plate before and after the translation through the parallel plate, extract the pixel coordinates of the target point in the left and right images, convert the pixel coordinates into world coordinates, and calculate the straight line l that intersects the two straight lines passing through the target point and the corresponding optical center and the distance h between the two intersections satisfies the following formula in the world coordinates:

[0082]

[0083] Due to the manufacturing tolerance and calculation error of quartz glass, there is no straight line that completely intersects the two straight lines l1 and l2 passing through the target point and the corresponding optical center, and the distance between the intersection points is h. Therefore, the distance between l and the intersection points of the common perpendicular lines l1 and l2 is taken as h, and then the real world coordinates of the target point are restored by any of the following equations:

[0084]

[0085] Calculate the difference between the displacement distance and translation distance of the 81 pairs of calibration plate targets before and after translation, as shown in Table 1:

[0086] Table 1

[0087]

[0088] The root mean square error of the measurement result of the translation distance of the calibration plate by the binocular camera through the quartz glass is 0.018422mm.

[0089] In summary, good measurement results can be obtained by adopting the above measurement method. Due to the manufacturing tolerance of quartz glass, the parallelism of the parallel plate itself and its material uniformity will be affected, so the measured value will jump slightly at some points, and the order of the jump points will change when measuring at other positions.

[0090] The examples given above are preferred examples for implementing the present invention. The present invention is not limited to the above embodiments. Other embodiments obtained by those skilled in the art based on the features and principles of the present invention without making any creative work shall fall within the scope of protection of the present invention.

Claims

1. A parallel plate refraction imaging binocular vision three-dimensional measurement method, characterized in that: The method uses a device including a binocular camera, a parallel plate and a calibration plate, and includes the following steps: S1. Calibrate the binocular camera without setting up a parallel plate, and use the world coordinates of the calibration plate target as the standard reference point P1…P before refraction. n The coordinates of the two optical centers O1 and O2 of the binocular camera are calculated. According to the pixel size d of the binocular camera x ,d y And the quantized focal length f of the camera in the pixel coordinate system X and Y axes obtained by calibration x 、f y Calculate the focal length f of the binocular camera l 、f r ; S2, set up a fixed parallel plate between the binocular camera and the calibration plate, and each binocular camera collects an image of the calibration plate after being refracted by the parallel plate; S3, the target point P on the calibration plate image collected by the binocular camera after refraction l1 …P ln , P r1 …P rn The pixel coordinates are converted to world coordinates, and the P l1 …P ln and the straight line l between the optical center O1 of the left camera l1 …l ln and P r1 …P rn The straight line l with the right camera optical center O2 r1 …l rn ; S4, establishing a parallel plate refraction model for binocular cameras; S5, calculated P1…P n of and l l1 …l ln , l r1 …l rn The normal vectors η1…η of the intersecting lines n , using η1…η n Calculate the normal vector η of the parallel plate; S6, calculated P1…P n The straight line with normal vector η and l l1 …l ln , l r1 …l rn The intersection point P′ l1 …P′ ln and P′ r1 …P′ rn to P1…P n The distance h l1 …h ln 、h r1 …h rn And the parallel plate normal vector η and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn ; S7, h l1 …h ln 、h r1 …h rn and α l1 …α ln , α r1 …α rn The parallel plate refraction model of the binocular camera established in step S4 is introduced to calculate the thickness d and refractive index n of the parallel plate; S8. Use a binocular camera to collect images through a parallel plate, convert the pixel coordinates of the measured point into world coordinates, and calculate the straight lines l1 and l2 passing through the corresponding optical centers O1 and O2 respectively. According to the binocular camera parallel plate refraction model established in step S4 and the calculated parallel plate normal vector η, thickness d and refractive index n, restore the real world coordinates of the measured point p.

2. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: Step S1: After the binocular camera is calibrated, keep the calibration plate still at the position when the last calibration plate image is taken during the binocular camera calibration process, and use the calibration plate plane at this time as the XOY plane of the world coordinate system to establish the world coordinate system; the conversion relationship between the quantized focal length and the camera focal length is given by the following formula: f=(f x d x +f y d y ) / 2。 3. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: Step S3 establishes pinhole imaging models with O1 and O2 as the center, and l1 …P ln , P r1 …P rn Pixel coordinates are converted to world coordinates, straight line l l1 …l ln , l r1 …l rn The calculation of is given by the following formula, where λ is the proportionality factor:

4. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: In the binocular parallel plate refraction model of step S4, through P1…P n The straight line whose normal vector is equal to the normal vector of the parallel plate is l l1 …l ln and l r1 …l rn Corresponding intersection, intersection point P′ l1 …P′ ln and P′ r1 …P′ rn to P1…P n The distance h l1 …h ln 、h r1 …h rn with the normal vector of the parallel plate and l l1 …l ln , l r1 …l rn The angle α l1 …α ln , α r1 …α rn The relationship is given by the following formula: Where d is the thickness of the parallel plate and n is the refractive index of the parallel plate.

5. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: Step S5: take η1…η n The mean The normal vector η of the parallel plate is calculated as the initial value of the normal vector of the parallel plate by the nonlinear least squares iteration method. The cost function is given by the following function:

6. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: Step S7 calculates the thickness d and refractive index n of the parallel plate by nonlinear least squares iteration method, using the following function as the cost function: The rough measured values ​​of the refractive index n and thickness d of the parallel plate are used as the initial values ​​of the nonlinear least squares iteration method.

7. The parallel plate refraction imaging binocular vision three-dimensional measurement method according to claim 1, characterized in that: In step S8, straight lines l1 and l2 intersect straight line l at points p1 and p2, the normal vector of l is equal to the normal vector η of the parallel plate, and the distance between points p1 and p2 is h. 12 It is given by the following formula: Among them, α1, α2 are the angles between l1, l2 and η, then p is on the straight line l, and the vector In the same direction as η, the distances h1 and h2 between p and p1 and p2 are given by the following formulas:

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