A ship encounter fan-shaped space danger degree space analysis method

Abstract

The application is a ship encounter sector space danger degree space analysis method, comprising the following steps: taking the ship as the center and the detection distance R set by the ship-borne radar as the radius, a ship encounter sector space model is established; a sea space threatened by the target ship to the main ship is established according to the time gradient range, and the imminent coefficient when the two ship fields are tangent, i.e. the scaling multiple when the distance is closest, is used for judgment, when the scaling multiple when the distance is closest is <1, the following steps are continued; the encounter sector space set GP after the target ship G is divided, and the encounter sector space set QP after the main ship Q is divided are determined; the relative position of the two ships is preliminarily judged according to the DCPA of the ship and the target ship; the minimum vector distance of the two ships is determined according to the relative heading of the two ships, and the spatial imminent degree of the calculated combined sector GP is realized. The method is an analysis method which can simplify the ship encounter space vector structure, accurately mine and extract the space situation and space features of the ship encounter space in different directions.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of full automation, and relates to a ship encounter sector space danger degree space analysis method. BACKGROUND

[0002] With the continuous progress of intelligent navigation technology and intelligent maritime safety guarantee technology, intelligent ship navigation safety technology has become a key technology in the future ocean that attracts global attention. Under this background, the problem of ship navigation encounter safety is becoming increasingly important. At present, one of the key technologies that urgently needs to be broken through in the field of intelligent ships is the ship navigation encounter danger analysis method. When ships sail in various waters, they inevitably encounter various complex situations, especially in busy waters, where there are many surrounding ships or navigation-obstructing objects, and the navigation space presents the characteristics of complex and high-density distribution. In this environment, the navigation encounter space relationship becomes complex and chaotic and variable, and the change law and unknown nonlinear space characteristics of the encounter dynamic space are difficult to fully understand, which makes it difficult to accurately analyze the navigation danger. Therefore, improving the ship navigation safety space analysis capability is of great significance to improving the intelligent navigation technology level of ships and even promoting the construction of intelligent ship system. The improvement of this capability will help the intelligent ship to better cope with the complex and variable navigation environment and ensure the safety of navigation, and inject new vitality into the development of intelligent navigation technology.

[0003] Therefore, it is necessary to design a ship encounter sector space danger degree space analysis method to solve the above problems. SUMMARY

[0004] To solve the above problems, the technical scheme adopted by the present application is: a ship encounter sector space danger degree space analysis method, comprising the following steps:

[0005] S1: establishing a ship encounter sector space model with the ship as the center and the detection distance R set by the ship-borne radar as the radius;

[0006] S2: combining the calculation results of the ship approach rate, the encounter danger parameters and the relative spatial position distribution, establishing the sea space threatened by the target ship to the main ship according to the time gradient range, i.e. the ship encounter sector space-time domain;

[0007] S3: calculating the encounter parameters of the ship encounter sector space-time domain: the DCPA of the ship and the target ship and the TCPA of the ship and the target ship;

[0008] S4: when TCPA<0, the target ship has passed the closest encounter point of the two ships; return to calculate the TCPA of the next target ship and the ship;

[0009] When TCPA is greater than or equal to 0, the target ship has not arrived or has just arrived at the closest encounter point of the two ships, and the approaching coefficient when the two ship fields are tangent is calculated;

[0010] S5: According to the approaching coefficient when the two ship fields are tangent, that is, the scaling multiple when the distance is closest, when the scaling multiple when the distance is closest <1, continue the following steps;

[0011] S6: Determine the encounter sector space set GP of the target ship G after segmentation, and the encounter sector space set QP of the main ship Q after segmentation;

[0012] S7: Preliminarily judge the relative position of the two ships according to the DCPA of the ship and the target ship;

[0013] S8: Determine the minimum distance of the two ships according to the relative heading of the two ships, and realize the space approaching degree of the calculated combined sector GP.

[0014] The ship encounter sector space danger degree space analysis method provided by the application has the following advantages:

[0015] The core of the method is to construct a multi-time-space, multi-angle and multi-pose double-unit encounter sector dynamic space model, analyze and calculate the spatial geometric characteristics and space-time variation law of potential encounter danger in the multi-time-space sequence process, and provide an innovative new idea for improving the safety analysis capability of the unmanned ship intelligent navigation system.

[0016] The method is based on the ship navigation encounter state under the multi-time-space scene, establishes a double-unit encounter sector dynamic space model, realizes the GIS vectorization conversion expression of the encounter space, and then based on the constructed encounter sector space-time domain, analyzes and extracts the potential space-time danger characteristics and space approaching degree in depth with small sectors as units, and effectively extracts the geometric space-time characteristics of potential encounter danger in the multi-time sequence.

[0017] The method is an analysis method that can simplify the vector structure of the ship encounter space, accurately mine and extract the encounter space situation and spatial characteristics of ships in different directions, and has high analysis capability based on space-time geometric position information. It is a high-order space analysis method that can solve the problems of navigation space relationship reasoning, encounter safety space feature extraction and navigation situation space neighboring feature mining. Especially in the complex scenes such as multi-obstacle and high density encountered in intelligent navigation, it performs well. BRIEF DESCRIPTION OF DRAWINGS

[0018] In order to more clearly illustrate the technical solutions in the embodiments of the application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiment or prior art description. Obviously, the drawings in the following description are some embodiments of the application, and those skilled in the art can also obtain other drawings according to these drawings without creative labor.

[0019] Figure 1 is the schematic diagram of the equal sector and 48 sector division principle of the method of the present application;

[0020] Figure 2 is the vector diagram of the dangerous parameter calculation based on the ship encounter sector of the present application;

[0021] Figure 3 represents the vector diagram of the collision danger information calculation based on the ship encounter sector space model under the multi-target ship scene;

[0022] Figure 4 The approaching relationship calculation and analytical vector diagram of the ship encounter space are intuitively shown. (a) Equal sector schematic diagram, (b) 48 sector division principle schematic diagram. DETAILED DESCRIPTION

[0023] A ship encounter sector space danger degree space analysis method based on the ship encounter sector space, comprising the following steps:

[0024] S1: Establish a ship encounter sector space model with the own ship as the center and the detection distance R set by the shipborne radar as the radius;

[0025] S2: Combine the calculation results of the ship approach rate, the encounter danger parameter, and the relative spatial position distribution, and establish the sea space threatened by the target ship to the main ship according to the time gradient range, i.e. the ship encounter sector space-time domain; the approach rate is the relative speed, the encounter danger parameter is the minimum encounter time and the minimum encounter distance, and the relative position distribution is the assumed spatial coordinates

[0026] S3: Calculate the encounter parameters of the ship encounter sector space-time domain: the DCPA of the own ship and the target ship and the TCPA of the own ship and the target ship;

[0027] S4: When TCPA<0, the target ship has passed the closest encounter point of the two ships; return to calculate the TCPA of the next target ship and the own ship;

[0028] When TCPA≥0, the target ship has not arrived or has just arrived at the closest encounter point of the two ships, and the approaching coefficient when the two ship fields are tangent is calculated;

[0029] S5: According to the approaching coefficient when the two ship fields are tangent, i.e. the scaling multiple when the distance is closest, when the scaling multiple when the distance is closest <1, continue the following steps;

[0030] S6: Determine the encounter sector space set GP of the target ship G after segmentation and the encounter sector space set QP of the main ship Q after segmentation;

[0031] S7: Preliminarily judge the relative position of the two ships according to the DCPA of the own ship and the target ship;

[0032] S8: According to the relative heading of two ships, the minimum distance between two ships is determined, and the spatial approaching degree of the calculated combined sector GP is realized.

[0033] The dynamic space model of the encounter sector is established with the center of the main ship as the origin. We use [0, 1] to represent the boundary of the encounter sector space-time domain as the range of collision danger perceived by the driver. Secondly, the time gradient range of the encounter sector space-time domain is constructed by using the time when the own ship invades the target ship encounter sector space model and the time when the own ship leaves the target ship encounter sector space model.

[0034] In order to calculate the dynamic and static parameters when two ships encounter, this method assumes that the ship speed of the own ship S0 is V0, the heading is θ0, the position coordinates of the main ship are (X0, Y0). The ship speed of the target ship S1 is V1, the heading is θ1, and the position coordinates of the target ship are (X1, Y1). The Euclidean distance between the two ships is D. Figure 1 The encounter danger information calculation vector diagram based on the dynamic space model of the ship encounter sector is shown in the figure. The components of the own ship speed on the coordinate axis are: Similarly, the components of the target ship speed on the coordinate axis are: The components of the target ship relative speed on the coordinate axis are: The relative speed is:

[0035] The Euclidean distance and relative angle between the main ship and the target ship can be represented as:

[0036]

[0037] Where, in the formula, α1——relative speed azimuth angle, The true bearing of the target ship relative to the own ship is:

[0038]

[0039] The calculation method of DCPA and TCPA of the own ship and the target ship is shown in the figure:

[0040] DCPA=D·sin(ψ1-α1-π) TCPA=D·cos(ψ1-α1-π) / v R

[0041] When TCPA>0, it means that the target ship has not reached the closest encounter point of the two ships; on the contrary, TCPA<0, the target ship has passed the closest encounter point of the two ships; TCPA=0, the target ship just reaches the closest encounter point of the two ships. When the target ship passes the bow of the own ship from the port side, the DCPA value is negative, and when it passes the stern of the own ship, the value is positive; when the target ship passes the bow of the own ship from the starboard side, the value is positive, and when it passes the stern of the own ship, the value is negative.

[0042] This method comprehensively considers the target vessel's course, speed, and size. The introduction of the encounter uncertainty parameter U improves the ability to identify hazards in moving targets. It exhibits higher vigilance towards approaching vessels in cross-encounter situations and is more in line with the relevant provisions of the International Regulations for Preventing Collisions at Sea (COLREGS). It expands the vessel's sector area with the main vessel as the subject, performs spatial analysis of nearby vessels, and predicts different course trajectories for different vessels in different scenarios. Figure 2 This is a vector diagram representing the calculation of collision hazard information based on a sector-shaped spatial model of ship encounters in a multi-target ship scenario.

[0043] Furthermore, the formula for obtaining the scaling factor of the ship encounter sector space model is as follows:

[0044] First, assume the ship's heading is θ0, the dimensions of the sector of the ship are a (major radius) and b (minor radius), the ship's speed is V0, and the ship's position coordinates are (X0, Y0), (X′) 01 ,Y′ 01 Establish a rectangular coordinate system with the ship's course as the Y-axis. The ship's position coordinates in the new coordinate system are (X′0, Y′0); the target ship's course is θ1, its speed is V1, and its position coordinates are (X1, Y1), (X′0, Y′0). 11 ,Y′ 11 Let (X′1, Y′1) be the initial coordinates of the target ship, and let (X′1, Y′1) be the new coordinates of the target ship. Assume that the distance between the target ship and the target ship changes with time as a function U(i).

[0045] With the ship as the center, the radii of the sector of the ship's domain are a and b. The relationship between the two coordinate systems is as follows:

[0046]

[0047] The relative headings of the two ships are:

[0048] θ = θ1 - θ0

[0049] When the target ship is located on the boundary of the vessel's own territory, in other words, when the vessel's own territory multiplied by the proximity factor U makes the target ship exactly located on the boundary of the vessel's own territory, it is expressed by the following formula:

[0050] (X′1-X′0) 2 +(Y′1-Y′0) 2 =(Ur) 2

[0051] The new coordinates of this vessel in the new coordinate system are (X′0, Y′0); the new coordinates of the target vessel in the new coordinate system are (X′1, Y′1);

[0052] Assume that when the ship is the overtaking ship, R1 = R3 = 4L, where R1 is the radius of the sector formed by the bow inside the combined sector, R3 is the radius of the sector formed by the stern inside the combined sector, and L is the length of the ship

[0053] When the ship and the other ship form a meeting situation, R1 = 6.4L and R3 = 1.6L;

[0054] Considering the maximum safe water area radius of the ship and the saving of the meeting water area range, the long radius of the sector of the ship is 6.4L and the short radius is 1.6L, then we have:

[0055]

[0056] By simplifying, it is equivalent to:

[0057]

[0058] U represents the scaling factor of the meeting sector space, and U is less than 1, which means that the meeting sector space is dangerous;

[0059] The transformation into the initial coordinate expression is:

[0060]

[0061] When the ship and the target ship sail at the original heading and speed, the coordinate change values of the two ships are respectively:

[0062]

[0063] Bringing the above formula into the expression of the coefficient U can obtain:

[0064] U(i) 2 = Ni 2 + Pi+Q

[0065] i represents time, N, P, and Q are proportional coefficients of the functions of i, and according to the physical meaning of each parameter of the ship in the coordinate system, we can obtain:

[0066] v 0x = v0sinθ0

[0067] v 0y = v0cosθ0

[0068] Similarly, we can obtain:

[0069] v 1x = v1sinθ1

[0070] v 1y = v1cosθ1

[0071] Let the directional component coefficient be

[0072] k1 = sin θ0, k2 = cos θ0

[0073] The relationship between each coordinate can be simplified as:

[0074] x = X 11 -X 01

[0075] y = Y 11 -Y 01

[0076] We can get:

[0077]

[0078] Take the derivative of the function of U, and according to the derivative operation, we can get:

[0079]

[0080] Let the derivative of the above formula be zero, and derive:

[0081]

[0082] When , the function U has a minimum value, U min represents a kind of sailing state of the target ship relative to the own ship, which is the minimum value of the scaling multiple of the ship encounter sector space model.

[0083] The ship encounter sector space dynamic approach analysis method is to use the space approach theory to establish a ship encounter sector dynamic approach relationship analysis model Q b , which is used to reason and calculate the space approach relationship of the combined sector in the ship encounter space. Based on the ship encounter space approach theory, an analytical method suitable for representing the approach relationship of the combined sector ship field is created. This model can accurately depict the approach relationship of the ship in the encounter space. Using the space information platform, the main ship is taken as a point layer, and the irregular polygon composed of two different radius sectors in the free space is taken as the approach object to be analyzed. Then, the dynamic approach degree analysis model is used to calculate the approach relationship between the encounter space formed by different target ships and the main ship one by one. Figure 3 Ship encounter sector space dynamic approach relationship calculation and analytical vector diagram;

[0084] The minimum distance between the two ships is determined according to the relative heading of the two ships, and the space approach degree of the calculated combined sector GP is as follows:

[0085] Assume that the information of target ship G and host ship Q, GP = {gp1, gp1', gp2",...} represents the divided encounter space set of target ship G, and QP = {qp1, qp1', qp2",...} represents the divided encounter space set of host ship Q; QP and GP include two-dimensional point set P = {p1, p2,..., pn, p1', p2',..., pn,...}, which is composed of all two-dimensional point sets of the encounter space sector; in order to calculate Figure 4 The space approaching relationship between the host ship and the GP convex hull set is first defined as the calculation method of the sector GP and the nearest point:

[0086]

[0087] The above formula NN GP (q) represents the nearest point subset of the host ship q and the GP convex hull set, D(q, gp i ) represents the Euclidean distance between the host ship q and the GP convex hull set, gp i represents the element in the GP convex hull set;

[0088] The shortest distance of a point to a point on the edge of a combined sector is calculated, and any point on each edge has its own shortest distance to the center of the combined sector, that is, the corresponding shortest sector radius The specific space approaching size can be calculated next. According to Figure 4 the ship encounter space polygon QP and GP, the space approaching distance of the target point to the ship encounter space polygon and the space approaching degree S GP (x) is calculated as follows

[0089] S GP (x) = Min: [D(x, gp1), D(x, gp2),..., D(x, gp i )]

[0090] Q b = S(x) t2 -S(x) t1 / △t

[0091] In the above formula, S GP (x) represents the minimum distance of the target point X to the visible edge set T, that is, the space approaching distance; Q b represents the space approaching degree of the combined sector GP, S(x) t1 represents the sector at the first time, S(x) t2 represents the sector at the second time, and△t represents a time period; if the value of Q b is larger, it means that the encounter space is more dangerous to the X point.

[0092] The process of judging the relative position of the two ships according to the DCPA of the target ship and the own ship is as follows:

[0093] If DCPA < 0, the target ship passes the bow or stern of the own ship from the left side of the own ship; if DCPA > 0, the target ship passes the bow or stern of the own ship from the right side of the own ship.

[0094] Further, when the relative heading θ of the two ships is in [67.5°, 292.5°], S GP (x) = r = 6.4L; if θ ∈ [0°, 67.5°) ∪ (292.5°, 360°], S GP (x) = r = 4L.

[0095] Embodiment 1: This embodiment proposes a dangerous degree spatial analysis method based on ship encounter sector space, Figure 4 The encounter space is divided into a series of equiangular sector regions, (a) equiangular sector, (b) schematic diagram of 48 sector division principle, each sector region has a common center point, and the center point connects the spatial radiation lines of constant mathematical distance. The position points on each radiation line are associated with the mathematical distance of the center point. When ship drivers navigate in crowded and dense waters (such as: nearshore navigation, inland navigation), the navigation waters contain obstacles, target ships, etc., which cause the navigation to have large encounter danger coefficient, high spatial ambiguity, discrete and complex spatial relationship, etc. Then, according to the principle of Figure 4The figure is taken as the center of the main ship as a circle point, and three concentric circles are drawn with 1 / 3r meters, 2 / 3r meters and r meters as the radius (R is the detection distance set by the ship-borne radar). Take 11.25 degrees north by west as the starting point, draw 16 sectors with equal angles, and divide them into 48 sectors. Take the angle bisector of each sector as the direction of the sector. The directions corresponding to each sector are as follows: the N1, N2 and N3 sectors represent the north direction, the NNE1, NNE2 and NNE3 sectors represent the northeast by north direction (22.5 degrees north by east), the NE1, NE2 and NE3 sectors represent the northeast direction (45 degrees north by east), the NEE1, NEE2 and NEE3 sectors represent the northeast by east direction (67.5 degrees north by east); the E1, E2 and E3 sectors represent the east direction, the SEE1, SEE2 and SEE3 sectors represent the southeast by east direction (22.5 degrees south by east), the SE1, SE2 and SE3 sectors represent the southeast direction (45 degrees south by east), the SSE1, SSE2 and SSE3 sectors represent the southeast by south direction (67.5 degrees south by east); the S1, S2 and S3 sectors represent the south direction, the SSW1, SSW2 and SSW3 sectors represent the southwest by south direction (22.5 degrees south by west), the SW1, SW2 and SW3 sectors represent the southwest direction (45 degrees south by west), the SWW1, SWW2 and SWW3 sectors represent the southwest by west direction (67.5 degrees south by west); the W1, W2 and W3 sectors represent the west direction, the NWW1, NWW2 and NWW3 sectors represent the northwest by west direction (22.5 degrees north by west), the NW1, NW2 and NW3 sectors represent the northwest direction (45 degrees north by west), and the NNW1, NNW2 and NNW3 sectors represent the northwest by north direction.

[0096] A ship encounter sector space danger degree spatial analysis method based on ships, comprising the following steps:

[0097] S1: initialize all ship navigation motion parameters V n ,C n , spatial position X n ,Y n ;

[0098] S2: initialize the number of ships N in a certain range, and set the relative coordinates X n (t),Y n (t) of the surrounding ships;

[0099] S3: when N=1, execute;

[0100] S4: calculate the relative heading θ=θ1-θ0 of the two ships;

[0101] S5: calculate the distance between the two ships, and calculate the ship relative motion encounter parameters TCPA and DCPA;

[0102] S6: If TCPA < 0, the target ship has passed the closest point of approach, then the calculation is finished and the analysis result is stored;

[0103] If TCPA ≥ 0, it means that the target ship has not arrived or just arrived at the closest point of approach, then the next step is calculated;

[0104] S7: Calculate the approaching coefficient U when the two ships are tangent to each other;

[0105] S8: The distance between the target ship and the own ship changes with time,

[0106] U(i) 2 = Ni 2 + Pi+ Q

[0107] Scaling factor when the distance is closest > 1, not intruded; = 1, just intruded / intrusion finished; < 1, already intruded

[0108] S9: Determine the target ship G segmented encounter sector space set GP = {gp1, gp1', gp2"…} and the main ship Q segmented encounter sector space set QP = {qp1, qp1', qp2"…};

[0109] S10: According to the DCPA, the relative position of the two ships is preliminarily judged. If DCPA < 0, the target ship passes the bow or stern of the own ship from the left side; if DCPA > 0, the target ship passes the bow or stern of the own ship from the right side.

[0110] S11: According to the relative heading of the two ships, the minimum distance S GP (x) between the two ships is determined. If

[0111] θ ∈ [67.5°, 292.5°], then S GP (x) = r = 6.4L; if θ ∈ [0°, 67.5°) ∪ (292.5°, 360°], then S GP (x) = r = 4L;

[0112] S12: Calculate the space approaching degree Q b = S(x) t2 - S(x) t1 / △t;

[0113] S13: When N ≥ 2, execute; according to formulas (5)-(14), loop calculation is performed; the danger degree between other target ships and the own ship is obtained.

Claims

1. A method for analyzing the danger degree of a ship encounter sector space based on a ship encounter sector space, characterized in that: The method comprises the following steps: S1: establishing a ship encounter sector space model with the ship as the center and the detection distance R set by the ship-borne radar as the radius; S2: combining the calculation results of the ship approach rate, the encounter danger parameters and the relative spatial position distribution, establishing a sea space threatened by the target ship to the main ship according to the time gradient range, i.e. the ship encounter sector space-time domain; S3: calculating the encounter parameters of the ship encounter sector space-time domain, i.e. the DCPA of the main ship and the target ship and the TCPA of the main ship and the target ship; S4: when TCPA<0, the target ship has passed the closest point of approach of the two ships; returning to calculate the TCPA of the next target ship and the main ship; when TCPA≥0, the target ship has not arrived or has just arrived at the closest point of approach of the two ships, calculating the approach coefficient when the two ship domains are tangent to each other; S5: judging according to the approach coefficient when the two ship domains are tangent to each other, i.e. the scaling factor when the distance is the closest; when the scaling factor when the distance is the closest is less than 1, continuing the following steps; S6: determining the encounter sector space set GP of the target ship G after segmentation and the encounter sector space set QP of the main ship Q after segmentation; S7: preliminarily judging the relative position of the two ships according to the DCPA of the main ship and the target ship; S8: determining the minimum distance of the two ships according to the relative heading of the two ships, and realizing the spatial approach degree of the calculated combined sector GP.

2. The method according to claim 1, wherein the method is characterized by: The expression of the scaling factor U of the ship encounter sector space model is as follows: U is less than 1, indicating that the encounter sector space is dangerous at this time; wherein the ship position coordinates are (X0, Y0), and the new ship position coordinates of the new coordinate system are (X'1, Y'1); wherein: θ is the relative heading of the two ships: θ=θ1-θ0 wherein: θ0 is the heading of the main ship, and θ1 is the heading of the target ship.

3. The method according to claim 1, wherein the method is characterized by: The spatial approach degree of the calculated combined sector GP is realized according to the minimum distance of the two ships determined according to the relative heading of the two ships as follows: Assuming that the information of the target ship G and the main ship Q, GP={gp1, gP1', gP2"...} represents the encounter space set of the target ship G after segmentation, and QP={qp1, qp1', qp2"...} represents the encounter space set of the main ship Q after segmentation; QP and GP include a two-dimensional point set P={p1, p2...pn, p1', p2'...pn,...}, which is composed of all two-dimensional point sets constituting the encounter space sector; in order to calculate the spatial approach relationship of the main ship and the GP convex hull set, the calculation method of the sector GP and the nearest point is defined first: The above equation NN GP (q) denotes the subset of nearest points of the GP convex hull set to the master vessel q, D(q, gP i ) denotes the Euclidean distance of the master vessel q to the GP convex hull set, gP i denotes an element in the GP convex hull set; The shortest distance from the point to a point on the edge of the combined sector is calculated, and any point on the edge has its own shortest distance to the center of the combined sector, that is, its corresponding shortest sector radius According to the ship encounter space polygon QP and GP, the target point and the space approaching distance and the space approaching degree S of the ship encounter space polygon GP The calculation of (x) is as follows S GP (x) = Min: [D(x, gpl), D(x, gp2),..., D(x, gpK)] (2) i )] Q b = S(x) t2 - S(x) t1 / Δt In the above formula, S Gp (x) represents the minimum distance from the target point X to the set of visible edges T, i.e. the spatial proximity distance; Q b represents the spatial proximity degree of the combined sector GP, S(x) t1 represents the sector at the first time, S(x) t2 represents the sector at the second time, and Δt represents a time period; if Q b The greater the value of Q, the greater the danger of the encounter space to the point X.

4. The method according to claim 1, wherein the method is characterized by: Also included is when the relative heading of the two vessels θ ∈ [67.5°, 292.5°], then S Gp (x) = r = 6.4 L; If θ ∈ [0°, 67.5°) ∪ (292.5°, 360°], then S GP (x) = r = 4L.

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