JP4000117B2 - Method for judging inside / outside points of area - Google Patents

Description

The present invention relates to a method for determining an internal / external point of an area using an internal / external point determination algorithm for determining an internal / external point of the area at high speed, which is applied to various fields such as computer numerical simulation using graphics.

  For example, when detecting an object such as a dangerous object buried in a remote area with a sensor mounted on a ship, in order to quickly determine whether the detected object, i.e., the detection point is inside the remote area, It is necessary to perform computer processing on many detection points and nautical charts. As a region inside / outside determination method for determining whether or not this detection point is inside a region such as a saddle region, for example, a half line is drawn in a certain direction from a point where the inside / outside should be determined to form a region. An internal / external point determination algorithm is disclosed in which the presence or absence of an intersection with each side of a polygon is confirmed including the direction of intersection, and whether or not the point is inside the region is determined based on the confirmation result (patent) Reference 1). In addition, from the area including the recess, first, the outer product of the adjacent side vectors of the polygon made up of this area is obtained, and a convex polygon made up of the vertex that encloses the area is formed. Are sequentially divided into convex polygons until there are no concave polygons, and based on the determination result whether the points such as detection points are inside or outside each convex polygon, An inside / outside point determination algorithm for determining whether or not a point is inside a region is also disclosed (see Patent Document 2).

JP 2003-85569 A ([0017] to [0025]) JP 2003-85577 A ([0014] to [0024])

  However, for the inside / outside point determination algorithm described in Patent Document 1, since it is necessary to check the presence / absence and the direction of intersection with all sides forming the region for each point, the determination algorithm becomes complicated, Many calculations are required, and there is a risk of omission of determination. Moreover, in the inside / outside point determination algorithm described in Patent Document 2, when a polygon includes a concave vertex, it is necessary to repeat the calculation for dividing the convex polygon for each vertex to determine the internal / external point. The algorithm is complex and requires many iterations.

  Accordingly, an object of the present invention is to provide an inside / outside point determination algorithm that can easily and quickly determine whether a large number of points such as the above-described detection points are inside or outside a polygon forming an area without omission. It is to be.

  In order to solve the above problems, the present invention employs the following configuration.

That is, the area inside / outside determination method according to claim 1 uses a computer having a CPU and a memory to determine whether or not a point P is inside a convex polygon. A storage step of storing, by the memory, the coordinates of each vertex of the polygon, the coordinates of the point P, and the coordinates of an example of an inner point in the polygon; all sides of the polygon, a primary expression generating step Wath table by a linear equation of the X-Y coordinate system by the CPU, each of the Y-coordinate Y0ij obtained by substituting the X-coordinate X0 of the point P on the linear expression for large and small relation of Y-coordinate Y0 of the point P is, each obtained by substituting the X-coordinate Xi of said points example the linear expression against the Y coordinate Yiij, the Y coordinate value Yi in said point example If all match the magnitude relationship, Point P is determined by the CPU and Ru interior point der of the polygon, if all do not agree, and out said point P by using the determination step by the CPU as the outer point of the polygon The point is determined . Here, the convex polygon means a polygon in which all inner angles are smaller than 180 °.

In this way, the number of inner point examples given in advance is only one, and if the number of inner side points (one) and the above point P are subjected to the primary calculation for the number of sides of the polygon that can be expressed by a linear expression, Therefore, it is possible to determine the inside / outside points by performing only the first calculation twice, and the calculation is very small compared with the conventional determination algorithm. Even if there are many points P, high-speed determination without omission of determination is possible. .

A method for determining whether an area is inside or outside according to claim 2 uses a computer having a CPU and a memory to determine whether a certain point P is inside a polygon having a concave vertex of an internal angle larger than 180 °. to a out point determination method of the area, the coordinates of each vertex of the polygon, a storage step of storing the coordinates of said point P by said memory, said extended edges forming the recess apex multi A polygon dividing step for dividing the polygon into a plurality of convex polygons by obtaining an intersection with another side of the square or an intersection between the extended lines by the CPU, and within each of the divided convex polygons The storage step of storing the coordinates of each of the internal points in the memory by the memory, and from the coordinates of the vertexes and the intersections of the polygons, all the sides of the convex polygons are converted into the XY coordinate system by the CPU . once In a primary expression generating step table to, with respect to the point each Y coordinate Y0ij obtained by substituting the X-coordinate X0 of P on the linear expression, the magnitude relation between the Y-coordinate Y0 of the point P, the respective convex polygons , The magnitude relationship of the Y coordinate Y0 of the point P with respect to the respective Y coordinate Y0ij obtained by substituting the X coordinate X0 of the point P into the primary equation is expressed as follows. against the respective Y coordinates Yiij obtained by substituting, if they match all the magnitude relationship of the Y coordinate values Yi in said point example, by the said point P is Ru interior point der of the polygon the CPU In the case where all the points do not coincide with each other, the inside and outside points are determined using a determination step in which the CPU determines that the points are outside points .

  Thus, even if the polygon is a polygon having the above-mentioned concave vertex, by dividing into convex polygons and giving a plurality of internal point examples, the primary calculation is performed for the number of sides of the convex polygon. Even if there are a large number of the points P, the inside / outside point can be determined without omission by simply performing the required number of times described above.

The region inside / outside determination method according to claim 3 is the method according to claim 1, wherein the determination step according to claim 1 is performed when the coordinates of the point P coincide with any of the coordinates of each vertex of the polygon, or the Y of the point P The coordinate value Y0 matches one of the Y coordinate values Y0ij on each side including the extension of the polygon corresponding to the X coordinate value X0, and the Y coordinate value Y0ij other than the matched Y coordinate value When the magnitude relationship of the Y coordinate value Y0 of the point P coincides with the magnitude relationship of the Y coordinate value Yj of the inner point example with respect to the Y coordinate value Yiij, the point P is on the side of the convex polygon. It has the function to determine that it exists.

  In this way, it is possible to determine the point P on the side of the polygon, that is, the boundary of the region, and it is only necessary to determine the inside / outside point for the point P other than the point on the boundary, thereby simplifying the calculation. This contributes to high-speed determination with no determination omission.

  In the present invention, whether or not a certain point P is inside a polygon, that is, the inside of the region, is given in advance the coordinates of an example of an inner point within the polygon, and the point is applied to all the sides of the polygon. When P is on the same side as the inner point example, it can be determined that the point P is the inner point. Therefore, the primary calculation is performed by the number of sides of the polygon that can be expressed by a linear expression. It is sufficient to perform the above, and the calculation is very small as compared with the conventional determination algorithm. Even if there are a large number of points P, it is possible to perform internal / external point determination without omission of determination at high speed.

Embodiments of the present invention will be described below with reference to FIGS. As described in [Technical field] (paragraph number [0001]), the present invention is applied to various fields such as computer numerical simulation using figures, and is used to determine the inner and outer points of a region at high speed. The present invention relates to an internal / external point determination method using a point determination algorithm. In the computer numerical simulation, a computer (including a personal computer) including an input unit, a calculation unit and a control unit such as a CPU, and a memory (main and auxiliary storage unit) such as a main memory and a hard disk is generally used. Therefore, the method for determining the internal / external points of the area according to the present invention also uses such normal computer hardware resources in a normal manner, that is, the storage step is performed by the memory, the primary expression generation step, The square division step and the determination step can be realized by the CPU.

FIG. 1 shows an example of an interior point P _i (X) when a region, that is, a polygon is a convex polygon 1, assuming that the convex polygon 1 is in the XY coordinate system (the same applies to the following polygons). _This shows the inside / outside point determination method that gives _i , Y _i ). First, the coordinates of the vertices A1 to A5 of the convex polygon 1, A1 (X1, Y1), A2 (X2, Y2), A3 (X3, Y3) are given as input data. As the coordinate system, an XY coordinate system can be used. Next, an interior point example P _i (X _i , Y _i ) is given inside the convex polygon 1. Each side A1-A2, A2-A3, A3-A4, A4-A5, A5-A1 is expressed by a linear expression. As an example, for the side A1-A2 (including the extended portion), this straight line can be expressed by a linear expression as follows.
Now, assuming that the point P to be judged inside / outside is a decision point P ₀ (X ₀ , Y ₀ ) (determination point 1) and a decision point P ₀ (X ₀ , Y ₀ ) (determination point 2), the point P ₀ It becomes the X coordinate X ₀ of as Y-coordinate value Y ₁₂ Hatsugi on formula sides obtained by substituting the (1) A1-A2 (including the extension).
Also, the interior point Example P _i (X _i, Y _i) and when the inner points Example P _i (X _i, Y _i) X coordinate X _i edges obtained by substituting the equation (1) of the A1-A2 The Y coordinate value Y _i12 on (including the extension) is as follows.

The Y coordinate value Y ₁₂ , Y ₂₃ , Y of the Y coordinate value Y ₀ of the determination point P ₀ (X ₀ , Y ₀ ) on each side (including the extension) corresponding to the X coordinate value X _{0 34} , Y ₄₅ , and Y ₅₁ have magnitude relations on the respective sides corresponding to the X coordinate value X _i of the Y coordinate value Y _i of the inner point example P _i (X _i , Y _i ) (including the extended portion). The decision point P ₀ consists of the inner point of the convex polygon 1, that is, the convex polygon 1 when the magnitude relations with respect to the Y coordinate values Y _i12 , Y _i23 , Y _i34 , Y _i45 , Y _i51 all match. The inner point of the region is determined. In the case shown in FIG. 1, the determination point P ₀ (X ₀ , Y ₀ ) (determination point 1) is determined as an inner point, and the determination point P ₀ (X ₀ , Y ₀ ) (determination point 2) is Since all the magnitude relations do not coincide with the inner point example P _i (X _i , Y _i ), it is determined as the outer point.

Note that the Y coordinate value Y ₀ of the determination point P ₀ (X ₀ , Y ₀ ) is the Y coordinate value on each side (including the extended portion) corresponding to the X coordinate value X ₀ , Y ₁₂ , Y _23. , Y ₃₄ , Y ₄₅ , Y ₅₁ , and a determination for a Y coordinate value other than the matched Y coordinate value (any one of Y ₁₂ , Y ₂₃ , Y ₃₄ , Y ₄₅ , Y ₅₁ ) When the magnitude relationship of the Y coordinate value Y ₀ of the point P ₀ (X ₀ , Y ₀ ) matches all the magnitude relationships in the case of the inner point example P _i (X _i , Y _i ), the decision point P ₀ (X _0, Y ₀₎ is judged to be on the side of the convex polygon 1. Further, when the determination point P ₀ matches the coordinates of the vertex A of the convex polygon, it can be determined in advance that the determination point P ₀ is immediately on the side. When the determination point P ₀ is on the side, it can be determined in advance whether or not it is determined as an internal point or an external point. Y-coordinate Y ₀ of the decision point P ₀ (X _0, Y ₀₎ is, Y coordinates or error range considered to coincide with the vertices of the Y coordinate, on the sides, i.e., the Y-coordinate value Y ₀ of the determination point P ₀ The difference from the Y coordinate of the side or vertex can also be determined in advance.

Further, when any one of the sides A1-A2, A2-A3, A3-A4, A4-A5, A5-A1 of the convex polygon 1 is parallel to the Y axis, it is parallel to the Y axis. The determination point P ₀ (X ₀ , Y ₀ ) and the inner point example P _i (X _i , Y _i ) are on the same side with respect to the side, that is, the determination point P ₀ with respect to the X coordinate of the parallel side The X coordinate X ₀ and the X coordinate X _i of the inner point example P _i have the same magnitude relationship, and the X coordinate X of the determination point P ₀ (X ₀ , Y ₀ ) on each side other than the parallel side The magnitude relationship of the Y coordinate value Y _{0 with} respect to the Y coordinate value on each side corresponding to ₀ indicates that the Y coordinate for each Y coordinate value corresponding to the X coordinate X _i of the inner point example P _i (X _i , Y _i ). When all the magnitude relationships of the values Yi match, the determination point P ₀ (X ₀ , Y ₀ ) (determination point 1) is determined as the inner point of the convex polygon 1. The determination point P ₀ (X ₀ , Y ₀ ) (determination point 2) is not determined as the inner point of the convex polygon 1 and is therefore determined as the outer point.

Similarly, when any one of the sides A1-A2, A2-A3, A3-A4, A4-A5, A5-A1 of the convex polygon 1 is parallel to the X-axis, The determination point P ₀ (X ₀ , Y ₀ ) and the inner point example P _i (X _i , Y _i ) are on the same side with respect to the parallel side, that is, the determination point P with respect to the Y coordinate of the parallel side. magnitude of the Y coordinate Y _{i 0} of the Y-coordinate Y ₀ and the inner point example P _i are the same, and on each side other than the parallel sides, X decision point P ₀ (X _0, Y ₀₎ The magnitude relationship of the Y coordinate value Y _{0 with} respect to the Y coordinate value on each side corresponding to the coordinate X ₀ corresponds to each Y coordinate value corresponding to the X coordinate X _i of the inner point example P _i (X _i , Y _i ). When all the magnitude relations of the Y coordinate values Yi match, the determination point P ₀ (X ₀ , Y ₀ ) is determined as the inner point of the convex polygon 1.

FIG. 2 shows an inside / outside point determination method for giving an inside point example Pi (Xi, Yi) when the region, that is, the polygon is the concave polygon 2 having the concave vertex A1. First, as in the case of the convex polygon 1, the coordinates of the vertices A1 to A7, A1 (X1, Y1), A2 (X2, Y2), A3 (X3, Y3), A4 (X4, Y4), A5 ( X5, Y5), A6 (X6, Y6), and A6 (X7, Y7) are given as input data. Since the polygon 2 is a concave polygon, the sides A1-A2 and A1-A7 that form the concave vertex A1 are extended to intersect points A8 (X8, Y8) and A9 (sides A4-A5 and A3-A4), respectively. X9, Y9). In this way, the concave polygon 2 is changed into a convex polygon 3 having vertices A1, A2, A3, A9, a convex polygon 4 having vertices A1, A9, A4, A8 and vertices A1, A8, A5, A6, divided into three convex polygons 3,4,5 convex polygon 5 having A7, one by one interior point example each convex polygon, i.e. a convex polygon 3 Uchitenrei P ₃ i (X ₃ i, Y ₃ i), an inner point example P ₄ i (X ₄ i, Y ₄ i) on the convex polygon 4, and an inner point example P ₅ i (X ₅ i, Y ₅ i on the convex polygon ₅ ).

Now, assuming that a point P that should be determined inside / outside is a determination point P ₀ (X ₀ , Y ₀ ), the inside / outside point determination is performed for the convex polygon 3 by the procedure shown in FIG. As described above, the determination point P ₀ (X ₀ , Y ₀ ) is an inner point example P ₃ i (X _{3) with} respect to each Y coordinate value on each side of the convex polygon 3 with respect to the X coordinate X ₀ . If the magnitude relations of i, Y ₃ i) do not all match and it is determined that the point is not an inner point but an outer point, then the inner and outer points are determined for the convex polygon 4 in the same manner. The magnitude relationship for the decision point P ₀ (X ₀ , Y ₀ ) does not all match the magnitude relationship for the inner point example P ₄ i (X ₄ i, Y ₄ i), and the decision point P ₀ (X ₀ , Y ₀ ) is determined not to be an inner point but to an outer point, the inner and outer points are further determined for the convex polygon 5 using the inner point example P ₅ i (X ₅ i, Y ₅ i). Do. In the example of FIG. 2, the determination point P ₀ (X _0, Y ₀₎ is, for each Y-coordinate values on each side of a convex polygon 5 for the X-coordinate X _0, examples interior point P ₅ i (X All the magnitude relationships are the same as in the case of ₃ i, Y ₃ i), and the determination point P ₀ (X ₀ , Y ₀ ) is determined to be the inner point of the convex polygon 5. For example, when the determination point P ₀ (X ₀ , Y ₀ ) is determined to be an inner point in the convex polygon 4, for example, the determination of the inner / outer point for the determination point P ₀ (X ₀ , Y ₀ ) is Then, the inside / outside point determination calculation for the convex polygon 5 is not performed.

Thus, even if the region is a concave polygon including a concave vertex, the side forming the concave vertex A1 is extended until it intersects with the other side, and is divided into a plurality of convex polygons. By giving one example of the inner points to the polygon one by one, the inside / outside point determination for the determination point P ₀ (X ₀ , Y ₀ ) is performed in each convex polygon as in the case shown in FIG. Can do.

FIG. 3 shows an inner / outer point determination method that gives an inner point example Pi (Xi, Yi) when a region, that is, a polygon is a concave polygon 6 having a plurality of concave vertices, such as concave vertices A1 and A5. It is a thing. First, as in the case of the convex polygon 1, the coordinates of the known vertices A1 to A8, A1 (X1, Y1), A2 (X2, Y2), A3 (X3, Y3), A4 (X4, Y4), A5 (X5, Y5), A6 (X6, Y6), A7 (X7, Y7), and A8 (X8, Y8) are given as input data. As in the case of the concave polygon 2, the sides A1-A2 and A1-A8 that form the concave vertex A1 are extended to intersect with the sides A5-A4, A3-A4 A10 (X10, Y10), A9 (X9). , Y9), and similarly, one side A5-A4 forming the concave vertex A5 is extended to obtain the intersection A11 (X11, Y11) with the side A7-A8. In this way, the concave polygon 6 is changed into a convex polygon 7 having vertices A1, A2, A3, A9, a convex polygon 8 having vertices A1, A9, A4, A10, vertices A5, A11, A8, A1, A convex polygon 9 having A10 and a convex polygon 10 having vertices A5, A6, A7, and A11 are divided into four convex polygons 7, 8, 9, and 10, and an inner point example is formed on each convex polygon. One by one, that is, the inner point example P7i (X7i, Y7i) for the convex polygon 7, the inner point example P8i (X8i, Y8i) for the convex polygon 8, and the inner point example P9i (X9i, Y9i) for the convex polygon 9. ), gives Uchitenrei P10i (X10i the convex polygon 10, Y10i) and the coordinate of the decision point _{P 0 (X 0, Y 0} ) as input data, respectively. The intersection points A9 to A11 between the straight line extending the side forming the concave vertex and the concave polygon 6 are automatically calculated.

Thus, even if the region is a concave polygon including a plurality of concave vertices, the sides forming the concave vertices are extended until they intersect with other sides and divided into a plurality of convex polygons. By giving one example of the internal points to the polygon one by one, the determination point P ₀ (X ₀ , Y ₀ ) is omitted in each convex polygon in the same manner as shown in FIGS. The inside / outside point determination can be performed. In this case, as in the case shown in FIG. 2, for example, from first convex polygon 7, the interior point Example P7i determination point _{P 0 (X 0, Y 0} ) is for each side of each convex polygon Whether or not all are on the same side is determined by numerical calculation, and until it is determined to be an inner point, it is determined whether the convex polygons 8 to 10 are sequentially on the same side as the inner point examples P8i to P10i. When the determination point P ₀ (X ₀ , Y ₀ ) is determined to be the inner point, the numerical calculation is completed, and the inner / outer point determination is performed for the next determination point by the same method.

  In the case of a concave polygon including a plurality of concave vertices, depending on the distribution situation of the determination points, as shown in FIG. 4, a convex polygon including the intersection of the extended lines of the sides forming the concave vertices, for example, Convex polygon 11 with vertices A1, A2, A3, A10, A9, Convex polygon 12 with vertices A1, A9, A12, Convex polygon 13 with vertices A10, A4, A11, A12, Vertex A1, A12, It can also be divided into a convex polygon 14 having A5, A13, A8, a convex polygon 15 having vertices A5, A6, A7, A13, and a convex polygon 16 having vertices A5, A12, A11. Therefore, since the division pattern of the concave polygon into the convex polygon is not necessarily one, it is convenient to determine in advance according to the distribution of determination points.

In addition, in the case of a polygonal polygon, or in the case of a concave polygon having a plurality of concave vertices, especially in the case of a concave polygon, after dividing into each convex polygon as described above, It is also possible to determine the inside / outside point of the decision point P ₀ by giving an example of each outside point. In this case, as compared with the case of giving the inner point example, more outer point examples are required. Therefore, it is desirable that the inner / outer determination is performed so that the outer point is determined if it is not the inner point. However, for example, as shown in FIG. 5, when it is determined only whether or not the determination point P ₀ is outside the side A1-A5 of the convex polygon 1, it is easier to determine by giving the outer point example P _E. In some cases, it can be done. For this reason, whether to give an example of an internal point or an example of an external point depends on which of the problems to be solved is easier to make an internal / external decision.

FIG. 6 and FIG. 7 show the calculation flow of the above-described inside / outside point determination algorithm when an inside point example is given and the region inside / outside point determination method using this algorithm. As described above, the above-described inside / outside point determination is performed by the number of actual data (determination points P ₀ ) and the number of defined areas such as the convex polygon 1, the convex polygons 3-5, or the convex polygons 7-10. By repeating the algorithm, the result of internal / external determination for each area is obtained for each actual data.

  This invention detects a removal object such as a dangerous object buried in the seabed, quickly determines whether or not this detection point is inside the anchorage area, and easily grasps how much the removal object is in advance. It can be used when it is necessary to determine the inside / outside points of a region in various fields such as computer numerical simulation using figures.

It is explanatory drawing of the internal point determination in a polygon when there is no recessed part vertex of embodiment. It is explanatory drawing of the internal point determination in a polygon with one recessed part vertex of other embodiment. It is explanatory drawing of the internal point determination in a polygon with a several recessed part vertex of other embodiment. It is explanatory drawing of the interior point determination in the case of the polygon shown in FIG. 3, when the division | segmentation pattern into a convex polygon differs. It is explanatory drawing of the external point determination in a polygon in case there is no recessed part vertex of other embodiment. It is a flowchart which shows the algorithm of the inside / outside point determination of embodiment. It is a flowchart of the inside / outside point determination method using the algorithm of FIG.

Explanation of symbols

1, 3 to 5, 7 to 15: Convex polygon 2, 6: Concave polygon A1 to A8: Vertex P, P ₀ : Determination point Pi: Inner point example P _E : Outer point example

Claims (3)

An internal / external point determination method of an area for determining whether or not a certain point P is inside a convex polygon using a computer having a CPU and a memory ,
Storing the coordinates of each vertex of the polygon, the coordinates of the point P, and the coordinates of an example of an inner point in the polygon by the memory;
All sides of the polygon from the coordinates of the vertices, and a linear equation generating step Wath table by a linear equation of the X-Y coordinate system by the CPU,
With respect to the point each Y coordinate Y0ij obtained by substituting the X-coordinate X0 of P on the linear expression, the magnitude relation of the Y-coordinate Y0 of the point P, assigns the X-coordinate Xi of said points example the linear equation against the respective Y coordinates Yiij obtained by, if they match all the magnitude relationship of the Y coordinate values Yi in said point example, the point P is determined by the CPU as the interior point of the polygon, An inside / outside point determination method for an area characterized in that, when all do not match, an inside / outside point is determined using a determination step in which the CPU determines that the point P is an outside point of the polygon. Using a computer having a CPU and memory, a certain point P is inside or outside the point determining process of determining areas whether inside a polygon having a recess apex of larger interior angle than 180 °,
The coordinates of each vertex of the polygon, a storage step of the coordinate of the point P is stored by said memory,
A polygon that divides the polygon into a plurality of convex polygons by extending the side forming the concave vertex and intersecting with the other side of the polygon or the intersection of the extended lines by the CPU A splitting step;
A storing step of storing by each of the memory the coordinates of the inner points example in the divided respective convex polygonal in shape,
From the polygon vertices and the intersection of coordinates, the all sides of each convex polygon, and tables to primary expression generating step by a linear equation in X-Y coordinate system by the CPU,
With respect to the point each Y coordinate Y0ij obtained by substituting the X-coordinate X0 of P on the linear expression, the magnitude relation between the Y-coordinate Y0 of the point P, the each convex polygon, the point P on the linear expression The magnitude relationship of the Y coordinate Y0 of the point P with respect to the respective Y coordinate Y0ij obtained by substituting the X coordinate X0 of Y is obtained by substituting the X coordinate Xi of the inner point example into the linear equation. against the coordinates Yiij, if they match all the magnitude relationship of the Y coordinate values Yi in said point example, the point P is determined by the CPU and Ru interior point der of the polygon, if all do not match An internal / external point determination method for an area, wherein an internal / external point is determined using a determination step of determining by the CPU that the point is an external point. The polygon is determined when the coordinate of the point P coincides with any of the coordinates of each vertex of the polygon, or the Y coordinate value Y0 of the point P corresponds to the X coordinate value X0. The magnitude relationship of the Y coordinate value Y0 of the point P with respect to the Y coordinate value Y0ij other than the matched Y coordinate value matches one of the Y coordinate values Y0ij on each side including the extension of the inner point example. The CPU has a function of determining that the point P is on a side of the convex polygon when all the Y-coordinate values of the Y-coordinate value Yj coincide with the Y-coordinate value Yiij. 2. A method for determining the inside / outside points of the region according to 1 .