Irregular circular domain area calculation method
Through the irregular circle boundary judgment and geometric calculation method, the area of the irregular circle is directly calculated, which solves the problems of calculation accuracy and speed in the Monte Carlo method and achieves high-precision and fast calculation.
Patent Information
- Application Number
- CN202411873918.7
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-19
- Publication Date
- 2025-01-21
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
In the prior art, when calculating the irregular circle area, the small number of samples leads to poor calculation accuracy, and large number of samples reduces the calculation speed of the algorithm, and complex judgment conditions are prone to missed judgments or wrong judgments.
By judging the boundary of the irregular circle domain, the area is divided, and the geometric calculation method is used to directly calculate the area of the irregular circle domain accurately without sampling.
It achieves high-precision and fast calculation of the area of irregular circles, avoids the problems of sampling quantity and calculation speed in the Monte Carlo method, and reduces the complexity of judgment conditions.
Smart Images
Figure CN119338895A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to a method for calculating the area of an irregular circular domain, and belongs to the technical field of area calculation methods. Background Art
[0002] When calculating a certain indicator, it is necessary to calculate the precise area of the intersection of the circular domain and the triangular region, which involves the precise calculation of the area of the irregular circular domain. The calculation accuracy of the area determines the accuracy of the indicator capability characterization, and the speed of the algorithm calculation determines the timeliness of the subsequent task planning. It is necessary to construct an actuarial model algorithm based on the analytical method to achieve high-precision and fast calculation of irregular circular domains.
[0003] At present, among the methods for calculating the area of irregular circular domains, the Monte Carlo random simulation method is more common. The calculation accuracy of this method is related to the scale of random sampling and the complexity of the judgment conditions of the irregular circular domain. When the sampling scale is small, the calculation accuracy is poor. When the sampling scale is large, the calculation speed of the algorithm is reduced while the calculation accuracy is improved. At the same time, when the complexity of the judgment conditions is high, it is easy to miss or misjudge the conditions, resulting in calculation errors. Summary of the invention
[0004] The purpose of the present invention is to solve the shortcomings of the Monte Carlo random simulation method in the prior art, such as small sampling number, poor calculation accuracy, large sampling module, low calculation speed and complex judgment conditions, and to provide a method for calculating the area of an irregular circular domain. The method does not require sampling, but judges the boundary of the irregular circular domain and divides the area into regions, and realizes accurate calculation of the area of the irregular circular domain through geometric calculation.
[0005] In order to solve the above problems, the present application is implemented through the following technical solutions: A method for calculating the area of an irregular circular region, which is special in that it comprises the following steps: Step 1: Determine the relationship between the target point and the circular region: if the target point is within the circular region, execute step 2; if the target point is outside the circular region, execute step 3; Step 2: Divide the irregular circular domain, calculate the area of each part separately, and then calculate the sum of all areas; Step 3: Determine the analytical judgment condition of the boundary of the irregular circular domain, determine the overlap between the circular domain and the triangle range according to the judgment condition, and calculate the area of the irregular circular domain respectively.
[0006] Preferably, the step 2 specifically comprises the following steps: Step 2.1, split the irregular circular domain into a triangle and an arc; Step 2.2, calculate the area of the triangle: find the intersection of the south and north boundaries of the triangle range and the area of the circle, calculate the length of the chord formed by the two intersection points and the distance from the target to the south and north boundaries of the triangle range, and calculate the area of the triangle according to Heron's formula; Step 2.3, calculate the area of the arc: calculate the central angle corresponding to the chord based on the length of the chord obtained in step 2.2, and calculate the area of the sector and the area of the triangle based on the central angle: the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 2.4, add the area of the triangle in step 2.2 and the area of the arc in step 2.3; Preferably, the specific calculation process of step 2.2 is as follows: The coordinates of the northernmost and southernmost deployment points N, S and the target O in the triangle range are known and are recorded as , , , calculate the distances of NS, NO, and SO respectively The coordinates of the center of the circle are , the distance between NB and NS is , the parameter equation of the circle boundary curve is: { x = R cos θ + x 0 , y = R sin θ + y 0 , θ ∈ ( 0 , 2 π ] , The equation of the northern boundary line of the triangle range is: , then the coordinates of the north node satisfy: , So we have: , Right now , Right now , Find Roots in the range (0,1) , then the coordinates of the north node are: , Length from target to north node: The coordinates of the South Node satisfy: , Find Roots in the range (0,1) , then the coordinates of the north and south points are: , Length from target to South Node: , string The length is: , According to Heron's formula, the area of triangle △OMK is: , In the formula, .
[0007] Preferably, the specific calculation process of step 2.3 is as follows: Calculate the central angle of a circle by its length: , The area of sector MBK is: , The area of △MBK is: , The area of the arc is: .
[0008] Preferably, the step 3 specifically includes the following steps: Step 3.1, according to the relationship between the center point of the circular area and the north and south boundaries of the triangle range, determine the type of overlap between the circular area and the triangle range; Step 3.2, determine the calculation method of the area according to the overlap type in step 3.1; Preferably, the step 3.1 specifically includes the following steps: Step 3.1.1, calculate the distance from the center point of the circular area to the north and south boundaries of the triangle range; Step 3.1.2, judging the specific position of the center point of the circle in the triangle range according to the distance in step 3.1.1; Step 3.1.3: Based on the projection of the horizontal coordinate of the center point of the circular domain on the x-axis of the triangle range and step 3.1.2, determine the type of overlap between the circular domain area and the triangle range.
[0009] Preferably, the step 3.1.1 is as follows: Using Heron's formula, calculate the distance from the center point B of the circle to the south and north boundaries of the triangle range: , , Where: , ; Preferably, the step 3.1.2 is specifically as follows: The maximum radius of the protection circle is R. When the circle center point B is close to the southern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side; when When the circle center point B is close to the northern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side; Preferably, the step 3.1.3 is as follows: , , when , , Use step 3.2.1 to calculate the area; when , , when , , Use step 3.2.2 to calculate the area; when , , , Use step 3.2.3 to calculate the area; when , , , or , , , Use step 3.2.4 to calculate the area; when , , , When the area is the circular area; Preferably, the step 3.2 specifically includes the following steps: Step 3.2.1, when the center point of the circular domain is located on the north and south boundaries of the triangular range, the overlapped part of the circular domain area and the triangular range is half of the circular domain area; When the center point of the circular domain is not on the north or south boundary of the triangle range, the area of the irregular circular domain is the arc area, which is calculated as follows: first, find the intersection point of the south boundary or north boundary of the triangle range and the circular domain area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord based on the length of the chord, calculate the area of the sector and the area of the triangle based on the central angle, and the arc area = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.2, the area of the irregular circular domain = the area of the circular domain - the area of the arc. The specific method for calculating the area of the arc is as follows: find the intersection point of the southern boundary or the northern boundary of the triangle range and the area of the circular domain, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.3, the area of the irregular circular domain = the area of the circular domain - the area of the south arc - the area of the north arc, The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary and the north boundary of the triangle range with the circular area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.4, when the center point of the circular domain is close to the northern boundary of the triangle range: the area of the irregular circular domain = the area of the southern arcuate shape - the area of the northern arcuate shape; When the center point of the circular domain is close to the southern boundary of the triangle range: the area of the irregular circular domain = the area of the northern arc-the area of the southern arc; The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary, the north boundary and the circular area of the triangle range, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc.
[0010] The present application derives analytical judgment conditions for the boundaries of irregular circular domains based on vector operations. The area of irregular circular domains can be calculated based on geometric calculations. The design method has no errors except computer rounding errors, and has high calculation accuracy and fast calculation speed. BRIEF DESCRIPTION OF THE DRAWINGS
[0011] Figure 1 is a flow chart of the present invention; Figure 2 It is a schematic diagram of the area of the target point in the circular domain; Figure 3It is a schematic diagram of the area where the center of the circular area is outside the southern boundary of the triangular range and the area of the circular area is within the northern boundary of the triangular range; Figure 4 It is a schematic diagram of the area where the center of the circular domain is outside the northern boundary of the triangular range and the area of the circular domain is within the southern boundary of the triangular range; Figure 5 It is a schematic diagram of the area where the center of the circular region is within the southern boundary of the triangular range and the area of the circular region is within the northern boundary of the triangular range; Figure 6 It is a schematic diagram of the area where the center of the circular region is within the northern boundary of the triangular range and the area of the circular region is within the southern boundary of the triangular range; Figure 7 A schematic diagram of the area of the circular domain with the circular domain center within the north and south boundaries of the triangle range; Figure 8 It is a schematic diagram of the area where the center of the circular domain is within the north and south boundaries of the triangular range and part of the circular domain area is outside the north and south boundaries of the triangular range; Fig. 9 It is a schematic diagram of the area where the center of the circular area is outside the southern boundary of the triangular range and part of the circular area is outside the northern boundary of the triangular range; Fig.10 This is a schematic diagram of the area where the center of the circular domain is outside the northern boundary of the triangular range and part of the circular domain is outside the southern boundary of the triangular range. DETAILED DESCRIPTION
[0012] The following is a specific implementation of the present invention with reference to the accompanying drawings to further illustrate the composition of the present invention.
[0013] Embodiment 1. A method for calculating the area of an irregular circular region, the process is as follows Figure 1 As shown, the following steps are included: Step 1, determine the relationship between the target point and the circular domain: if the target point is in the circular domain, execute step 2; if the target point is outside the circular domain, execute step 3; Step 2: Divide the irregular circular domain, calculate the area of each part separately, and then calculate the sum of all areas; Step 3: Determine the analytical judgment condition of the boundary of the irregular circular domain, determine the overlap between the circular domain and the triangle range according to the judgment condition, and calculate the area of the irregular circular domain respectively.
[0014] Among them, Figure 2 As shown, the step 2 specifically includes the following steps: Step 2.1, split the irregular circular domain into a triangle and an arc; Step 2.2, calculate the area of the triangle: find the intersection of the south and north boundaries of the triangle range and the area of the circle, calculate the length of the chord formed by the two intersection points and the distance from the target to the south and north boundaries of the triangle range, and calculate the area of the triangle according to Heron's formula; Step 2.3, calculate the area of the arc: calculate the central angle corresponding to the chord based on the length of the chord obtained in step 2.2, and calculate the area of the sector and the area of the triangle based on the central angle: the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 2.4, add the area of the triangle in step 2.2 and the area of the arc in step 2.3; The specific calculation process of step 2.2 is as follows: The coordinates of the northernmost and southernmost deployment points N, S and the target O in the triangle range are known and are recorded as , , , calculate the distances of NS, NO, and SO respectively The coordinates of the center of the circle are , the distance between NB and NS is , the parameter equation of the protection circle boundary curve is: { x = R cos θ + x 0 , y = R sin θ + y 0 , θ ∈ ( 0 , 2 π ] , The equation of the northern boundary line of the triangle range is: , then the coordinates of the north node satisfy: , So we have: , Right now , Right now , Find Roots in the range (0,1) , then the coordinates of the north node are: , Length from target to north node: , The coordinates of the South Node satisfy: , Find Roots in the range (0,1) , then the coordinates of the north and south points are: , Length from target to South Node: , string The length is: , According to Heron's formula, the area of triangle △OMK is: , In the formula, .
[0015] The specific calculation process of step 2.3 is as follows: Calculate the central angle of a circle by its length: , The area of sector MBK is: , The area of △MBK is: , The area of the arc is: .
[0016] Preferably, the step 3 specifically includes the following steps: Step 3.1, according to the relationship between the center point of the circular area and the north and south boundaries of the triangle range, determine the type of overlap between the circular area and the triangle range; Step 3.2, determine the calculation method of the area according to the overlap type in step 3.1; Preferably, the step 3.1 specifically includes the following steps: Step 3.1.1, calculate the distance from the center point of the circular area to the north and south boundaries of the triangle range; Step 3.1.2, judging the specific position of the center point of the circle in the triangle range according to the distance in step 3.1.1; Step 3.1.3: Based on the projection of the horizontal coordinate of the center point of the circular domain on the x-axis of the triangle range and step 3.1.2, determine the type of overlap between the circular domain area and the triangle range.
[0017] Preferably, the step 3.1.1 is specifically as follows: Using Heron's formula, calculate the distance from the center point B of the circle to the south and north boundaries of the triangle range: , , Where: , ; Preferably, the step 3.1.2 is specifically as follows: The maximum radius of the protection circle is R. When the circle center point B is close to the southern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side; when When the circle center point B is close to the northern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side; Preferably, the step 3.1.3 is as follows: when , , when , , Use step 3.2.1 to calculate the area; when , , when , , Use step 3.2.2 to calculate the area; when , , , Use step 3.2.3 to calculate the area; when , , , or , , , Use step 3.2.4 to calculate the area; when , , , When the area is the area of the circle, Figure 7 shown.
[0018] Wherein, the step 3.2 specifically includes the following steps: Step 3.2.1, when the center point of the circular domain is located on the north and south boundaries of the triangular range, the overlapped part of the circular domain area and the triangular range is half of the circular domain area; When the center point of the circular domain is not on the north or south boundary of the triangle range, the area of the irregular circular domain is the arc area, which is calculated as follows: first, find the intersection point of the south boundary or north boundary of the triangle range and the circular domain area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord based on the length of the chord, calculate the area of the sector and the area of the triangle based on the central angle, and the arc area = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; like Figure 3 or Figure 4 As shown, the specific calculation method is as follows: The parameter equation of the protection circle boundary curve is: { x = R cos θ + x 0 , y = R sin θ + y 0 , θ ∈ ( 0 , 2 π ] , In the formula, is the coordinate of the center B of the circular area, and the equation of the boundary line on the south side of the triangle range is: , Then the coordinates of the two intersection points satisfy: , So we have: , Right now: , Right now: , Find Roots in the range (0,1) , , then the coordinates of the two intersection points on the south side are: , , So we can calculate the chord Length: , The area of the required arc is a sector and Difference: According to the string Calculate the central angle of a circle by its length: , sector The area is: , △ The area is: , The area of the arc is: .
[0019] Step 3.2.2, the area of the irregular circular domain = the area of the circular domain - the area of the arc. The specific method for calculating the area of the arc is as follows: find the intersection point of the southern boundary or the northern boundary of the triangle range and the area of the circular domain, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; like Figure 5 or Figure 6 As shown, the specific calculation is as follows: The area is: , △ The area is: , The area of the arc is: , The area of the irregular circle = .
[0020] Step 3.2.3, the area of the irregular circular domain = the area of the circular domain - the area of the south arc - the area of the north arc, The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary and the north boundary of the triangle range with the circular area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; like Figure 8 As shown, the south side arch area and the north side arch area are calculated according to the method in step 3.2.1. , , .
[0021] Step 3.2.4, when the center point of the circular domain is close to the southern boundary of the triangle range: the area of the irregular circular domain = the area of the northern arcuate shape - the area of the southern arcuate shape; The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary and the north boundary of the triangle range with the circular area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; When the center point of the circular region is close to the northern boundary of the triangle range, such as Fig. 9 As shown, the south side arch area and the north side arch area are calculated according to the method in step 3.2.1. , , .
[0022] When the center point of the circular region is close to the northern boundary of the triangle range, such as Fig.10 As shown, the south side arch area and the north side arch area are calculated according to the method in step 3.2.1. , , .
[0023] Finally, it should be noted that the above are only preferred embodiments of the present application and are not intended to limit the present application. Although the present application has been described in detail with reference to the aforementioned embodiments, it is still possible for those skilled in the art to modify the technical solutions described in the aforementioned embodiments, or to make equivalent substitutions for some of the technical features therein. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present application shall be included in the protection scope of the present application.
Claims
1. A method for calculating the area of an irregular circular region, characterized in that: The following steps are involved: Step 1: Determine the relationship between the target point and the circular region: if the target point is within the circular region, execute step 2; if the target point is outside the circular region, execute step 3; Step 2: Divide the irregular circular domain, calculate the area of each part separately, and then calculate the sum of all areas; Step 3: Determine the analytical judgment condition of the boundary of the irregular circular domain, determine the overlap between the circular domain and the triangle range according to the judgment condition, and calculate the area of the irregular circular domain respectively.
2. The method for calculating the area of an irregular circular region according to claim 1, characterized in that: The step 2 specifically includes the following steps: Step 2.1, split the irregular circular domain into a triangle and an arc; Step 2.2, calculate the area of the triangle: find the intersection of the south and north boundaries of the triangle range and the area of the circle, calculate the length of the chord formed by the two intersection points and the distance from the target to the south and north boundaries of the triangle range, and calculate the area of the triangle according to Heron's formula; Step 2.3, calculate the area of the arc: calculate the central angle corresponding to the chord based on the length of the chord obtained in step 2.2, and calculate the area of the sector and the area of the triangle based on the central angle: the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 2.
4. Add the area of the triangle in step 2.2 and the area of the arc in step 2.
3.
3. The method for calculating the area of an irregular circular region according to claim 2, characterized in that: The specific calculation process of step 2.2 is as follows: The coordinates of the northernmost and southernmost deployment points N, S and the target O in the triangle range are known and are recorded as , , , calculate the distances of NS, NO, and SO respectively The coordinates of the center of the circle are , the distance between NB and NS is , the parameter equation of the protection circle boundary curve is: , The equation of the northern boundary line of the triangle range is: , then the coordinates of the north node satisfy: , So we have: , Right now , Right now , Find Roots in the range (0,1) , then the coordinates of the north node are: , Length from target to north node: , The coordinates of the South Node satisfy: , Find Roots in the range (0,1) , then the coordinates of the north and south points are: , Length from target to South Node: , string The length is: , According to Heron's formula, the area of triangle △OMK is: , In the formula, .
4. The method for calculating the area of an irregular circular region according to claim 2, characterized in that: The specific calculation process of step 2.3 is as follows: Calculate the central angle of a circle by its length: , The area of sector MBK is: , The area of △MBK is: , The area of the arc is: .
5. The method for calculating the area of an irregular circular region according to claim 1 or 2, characterized in that: The step 3 specifically includes the following steps: Step 3.1, according to the relationship between the center point of the circular area and the north and south boundaries of the triangle range, determine the type of overlap between the circular area and the triangle range; Step 3.2: Determine the area calculation method based on the overlap type in step 3.
1.
6. The method for calculating the area of an irregular circular region according to claim 5, characterized in that: The step 3.1 specifically includes the following steps: Step 3.1.1, calculate the distance from the center point of the circular area to the north and south boundaries of the triangle range; Step 3.1.2, judging the specific position of the center point of the circle in the triangle range according to the distance in step 3.1.1; Step 3.1.3: Based on the projection of the horizontal coordinate of the center point of the circular domain on the x-axis of the triangle range and step 3.1.2, determine the type of overlap between the circular domain area and the triangle range.
7. The method for calculating the area of an irregular circular region according to claim 6, characterized in that: The step 3.1.1 is as follows: Using Heron's formula, calculate the distance from the center point B of the circle to the south and north boundaries of the triangle range: , , Where: , .
8. The method for calculating the area of an irregular circular region according to claim 7, characterized in that: The step 3.1.2 is as follows: The maximum radius of the protection circle is R. When the circle center point B is close to the southern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side; when When the circle center point B is close to the northern boundary of the triangle range, the plane vector , exist The following is expressed as: , , but , , B in Left side, , B in Right side.
9. The method for calculating the area of an irregular circular region according to claim 8, characterized in that: The step 3.1.3 is as follows: when , , when , , Use step 3.2.1 to calculate the area; when , , when , , Use step 3.2.2 to calculate the area; when , , , Use step 3.2.3 to calculate the area; when , , , or , , , Use step 3.2.4 to calculate the area; when , , , When the area is the circular area.
10. The method for calculating the area of an irregular circular region according to claim 5, characterized in that: The step 3.2 specifically includes the following steps: Step 3.2.1, when the center point of the circular domain is located on the north and south boundaries of the triangular range, the overlapped part of the circular domain area and the triangular range is half of the circular domain area; When the center point of the circular domain is not on the north or south boundary of the triangle range, the area of the irregular circular domain is the arc area, which is calculated as follows: first, find the intersection point of the south boundary or north boundary of the triangle range and the circular domain area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord based on the length of the chord, calculate the area of the sector and the area of the triangle based on the central angle, and the arc area = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.2, the area of the irregular circular domain = the area of the circular domain - the area of the arc. The specific calculation method of the arc area is as follows: find the intersection point of the southern boundary or the northern boundary of the triangle range and the circular domain area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.3, the area of the irregular circular domain = the area of the circular domain - the area of the south arc - the area of the north arc, The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary and the north boundary of the triangle range with the circular area, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc; Step 3.2.4, when the center point of the circular domain is close to the southern boundary of the triangle range: the area of the irregular circular domain = the area of the northern arcuate shape - the area of the southern arcuate shape; When the center point of the circular domain is close to the northern boundary of the triangle range: the area of the irregular circular domain = the area of the southern arcuate area - the area of the northern arcuate area; The method for calculating the area of the arc is as follows: respectively obtain the intersection points of the south boundary, the north boundary and the circular area of the triangle range, calculate the length of the chord formed by the two intersection points, calculate the central angle corresponding to the chord according to the length of the chord, calculate the area of the sector and the area of the triangle according to the central angle, and the area of the arc = the area of the sector corresponding to the same arc - the area of the triangle corresponding to the same arc.