JPH11110569A - Inner point discriminating method for graphic - Google Patents

Abstract

PROBLEM TO BE SOLVED: To discriminate the inner point of a graphic in a short time by checking whether or not all the outer products of two vectors whose terminating point is successive two points of the sequenced vertexes of a projected polygon and starting point is the point P are the same sign and discriminating whether or not the point P is present inside the projected polygon when the polygon is the projected polygon. SOLUTION: Whether or not all the outer products of the two vectors corresponding to two sides holding the polygon formed by the set of the directed vertexes on a plane and the sequenced vertexes of the polygon to the point P on the plane there between are of the same sign is checked, and whether or not the polygon is the projected polygon is discriminated. Then, when it is the projected polygon occording to the discrimination, whether or not all the outer products of the two vectors whose terminating point is successive two points of the sequenced vertexes of the projected polygon and starting point is a point P on the plane are of the same sign is checked, and whether or not the point P on the plane is present inside the projected polygon is discriminated. Thus, calculation force required for discriminating the inner point of the graphic is reduced and the inner point of the graphics is discriminated in a short time.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for determining an inner point of a graphic used in a numerical simulation using a computer, a graphic drawing device, or the like.

[0002]

BACKGROUND OF THE INVENTION The problem of whether a given point on a plane lies within a closed curve on the same plane has long been studied due to mathematical interest. At present, it is possible to express it most strictly and abstractly in terms of topology called the fundamental group, but it is also an analytical problem related to the residue theorem.

When a plane is regarded as a complex plane by introducing natural complex coordinates into a given plane, a given closed curve is represented by Γ, and a given point is represented by z0, a function f
(Z) = 1 / (z−z0), and the following one-round integration

[0004]

(Equation 1) By calculating, it can be determined whether or not the given point is the inner point of the closed curve. In other words, according to the residue theorem, if the value of this integral is zero, z0 is outside the curve, and if it is not zero, that is, if ± 2πi, the point is a point in the closed curve. Prove.

Dudenet, "History of Mathematics 1700-1900"
According to I "(Iwanami Shoten, 1985), it seems that Cauchy discovered the residue theorem. 1826 S
urun nouveaugenre de calc
ul analoge au calcul inf
A paper entitled "initesmal" (Exercise)
It is said that in s de mathematics, the infinitesimal analysis around the singularity was performed to find the residue theorem. Cauchy later calculated that the integral of the complex function around the rectangle was the sum of the residues inside it, and for the next 20 years he was heavily involved in this subject. It is no exaggeration to say that the calculation of the residue linked geometry with function theory. In other words, a purely geometric fact is revealed by the sum (integration) of the values of the analytical functions.

[0006] In actual calculations, the closed curve is not arbitrary, and a piecewise straight line is more practically important. That is, when a polygon and a point P are given, it is often determined whether or not the point P is inside the polygon.

The inside / outside judgment algorithm according to the above-described residue theorem needs to calculate the angle of each side of the polygon from the point P and check whether or not the sum of the angles is ± 2π. However, since the calculation of the angle uses an inverse trigonometric function, which is a transcendental function, the calculation time is long.

In consideration of this point, there has been proposed a method of performing inside / outside determination without using the inside / outside determination according to the above-mentioned residue theorem in many cases (inside / outside determination method for a figure composed of a broken line).

This prior art is a combination of the above-described residue theorem and the inside / outside judgment of a circumscribed rectangle using the characteristic of xy coordinates, and requires a long calculation time for the residue. In order to avoid the problem of the inside / outside judgment using the theorem, an algorithm according to a flowchart as shown in FIG. 6 is provided.

As shown in FIG. 7A, when a polygon A and a point P are given, a rectangle containing the polygon A is defined by the maximum value and the minimum value of the x-axis and the y-axis of the coordinate of the polygon A. Use, Figure 7
It is formed as shown in FIG.

In the flowchart of FIG. 6, first, an inside / outside judgment is made by a size judgment in advance for a rectangle D formed by the maximum value and the minimum value of the x and y axes of the xy coordinates in which the figure is embedded. (Step S
101) Thereafter, when it is determined that the object is within the rectangle D (step S102), the inside and outside of the polygon are determined with respect to the points in the rectangle D (steps S103 to S103).
S106).

It is considered that the necessity of determining whether or not such a given point is an inner point of a figure has increased significantly in recent years with the development of computer performance. In a recent graphic drawing apparatus or a basic problem related thereto, it is necessary to determine whether or not a point designated by a mouse or the like is an inner point of a given figure. In numerical calculations such as the Monte Carlo method, it is required to determine whether or not many points are inside points of one polygon.

The situation in which the inside / outside judgment assumed in the present invention is made is
For example, a polygon is fixed in an initial state, and thereafter, when about 1000 or more points are randomly given, it is determined whether or not each point is an inner point of the polygon. Therefore, even when supplementary information is added to the polygon as the initial information, the total determination calculation time for inside / outside determination for all points is sufficiently small.

Furthermore, in the above prior art, since the interior point determination for a general polygon requires a great deal of calculation time, after performing a preliminary interior point determination for a rectangle in which a polygon is embedded, if the preliminary If the point exists outside the rectangle, it is determined that the point exists outside the polygon. Conversely, if the point exists inside the rectangle, the polygon and the point, which take a long time to calculate, Was judged inside and outside.

[0015]

However, the above-mentioned prior art has the following problems.

When a polygon A 'as shown in FIG. 8 extends in a direction oblique to the xy axis, the corresponding x axis, y
The rectangle D 'formed from the maximum and minimum values of the axis has a large area. Therefore, the area of a rectangle for performing preliminary interior point determination according to the flowchart of FIG. 6 is very large compared to the area of a polygon originally desired, and when considering a random point distribution, the preliminary interior point The judgment is not functioning enough.

That is, even if the preliminary interior point determination is performed, it is necessary in many cases to calculate the following detailed interior point determination, and the speed cannot be increased.

The present invention has been made in view of the above-described conventional problems, and has as its object to provide a method for determining an inner point of a figure, which can determine an inner point of the figure in a short time.

[0019]

In order to achieve the above object, a first aspect of the present invention relates to a polygon formed by a set of oriented vertices on a plane and a point P on a plane. Checking whether or not the outer products of the two vectors corresponding to the two sides sandwiching the ordered vertices of the polygon are all the same sign, and determining whether the polygon is a convex polygon,
In the case of a convex polygon, two consecutive points of ordered vertices of the convex polygon are set as end points and the point P is set as a start point.
It is determined whether or not the outer products of two vectors have the same sign to determine whether or not the point P exists inside the convex polygon.

According to a second aspect of the present invention, in the first aspect, if it is determined that the polygon is not a convex polygon, it is determined that the polygon is not a convex polygon. The sum of the angles formed by the two vectors starting from the point P as the end point and the point P as the starting point is checked, and the point P
Is determined whether or not exists inside the convex polygon.

In the third aspect of the present invention, with respect to a convex polygon formed by a set of oriented vertices on a plane and a point P on the plane, a sequence of the ordered vertices of the convex polygon is continuous. By examining whether or not the outer products of two vectors having two points as end points and the point P as a starting point have the same sign, it is determined whether or not the point P exists inside the convex polygon. It was done.

In the fourth invention, a triangular DEF on a plane
And a point P on the plane, a vector cross product PD × P
It is determined whether or not the point P exists inside the triangle ABC by checking whether E, PE × PF, PF × PD all have the same sign.

In the fifth invention, a polygon formed by a set of oriented vertices on a plane and a point P on the plane
With respect to 2 that sandwich the ordered vertices of the polygon
First determining means for determining whether or not the outer products of the two vectors corresponding to the sides have the same sign and determining whether or not the polygon is a convex polygon; and the first determining means When it is determined that the convex polygon is a convex polygon, whether the outer products of the two vectors starting from the continuous point of the ordered vertices of the convex polygon and the starting point of the point P are all the same sign A program for determining whether or not the point P exists inside the convex polygon by checking whether the point P exists inside the convex polygon.

In a sixth aspect based on the fifth aspect, if the first determining means determines that the convex polygon is not a convex polygon, the sequence of the ordered vertices of the convex polygon is continuous. A third determining unit that determines whether or not the point P exists inside the convex polygon by examining the sum of angles formed by two vectors having the two points as end points and the point P as a starting point; It is a program that stores a program.

According to the seventh aspect, the ordered vertices of the convex polygon are continuous with respect to a convex polygon formed by a set of oriented vertices on the plane and a point P on the plane. A means for checking whether or not the outer products of two vectors starting from the two points and having the point P as the starting point have the same sign to determine whether or not the point P exists inside the convex polygon. Is stored.

In the eighth invention, a triangular DEF on a plane
And a point P on the plane, a vector cross product PD × P
A program having means for checking whether E, PE × PF, and PF × PD all have the same sign to determine whether or not the point P exists inside the triangle ABC is stored. .

[0027]

Embodiments of the present invention will be described below with reference to the drawings.

[First Embodiment] First, the outline of the first embodiment will be described.

When the assumed polygon is an oriented convex polygon, the interior point can be determined with a smaller amount of calculation than in a general case. That is, the cross product formed by the given point P and the end point of each continuous side of the convex polygon is calculated, and if all these have the same sign, it is determined that the point P is inside the convex polygon. If there is a different sign, it can be determined that the point P is outside the convex polygon.

To determine whether or not the assumed polygon is a convex polygon, the outer product of vectors corresponding to two sides sandwiching each vertex is calculated. If there is a square or a different sign, it can be determined to be a non-convex polygon.

The calculation of the cross product required for these determinations is a simple addition / subtraction calculation, and does not require calculation of a transcendental function such as an inverse trigonometric function.

Therefore, in the present embodiment, the following interior point determination method is performed.

First, a polygon formed by a set of oriented vertices on a plane and a point P on the plane corresponding to two sides sandwiching the ordered vertices of the polygon with respect to a point P
Checks whether the outer products of two vectors have the same sign. Thereby, it is determined whether or not the polygon is a convex polygon.

(A) In the case of a convex polygon, the outer product of two vectors starting from the continuous point of the ordered vertices of the convex polygon and ending with the point P has the same sign. By checking whether or not there is, it is determined whether or not the point P exists inside the convex polygon.

(B) If the polygon is not a convex polygon, two consecutive points of the ordered vertices of the polygon are defined as the end point and the point P
By examining the sum of the angles formed by the two vectors starting from, it is determined whether or not the point P exists inside the polygon.

Hereinafter, a specific description will be given with reference to FIGS.

FIG. 1 is a flow chart showing a method for determining an inner point of a graphic according to the first embodiment of the present invention. FIG. 2 is a flowchart showing the algorithm A in FIG. 1, and FIG.
3 is a flowchart showing Algorithm B in FIG. FIG. 4 is a diagram showing the polygon A used in the first embodiment.

As shown in FIG. 4, it is assumed that a polygon A oriented in a certain plane C exists. To be oriented means that a one-way arrow can be naturally defined with respect to a side constituting the polygon A. Naturally, there are two degrees of freedom in the direction of the arrow, that is, clockwise and counterclockwise. Orienting is equivalent to choosing one of these two degrees of freedom.

By starting from a certain vertex of the polygon and making a round in the direction given in this way, a point sequence of the vertices of the polygon is formed as shown in FIG. The sequence of points is (X
j, Yj) (j = 0,..., N). However, (X0,
Y0) = (XN, YN) At this time, it is assumed that another point P (X, Y) exists on the same plane C as shown in FIG. An object of the present invention is to determine whether or not this point P exists within a polygon A.

First, a description will be given of an interior point determination algorithm B according to the residue theorem described in the prior art (which requires a lot of calculation time).

The following quantities are defined:

Si = (Xi−X) (Yi + 1−Y) − (Yi−Y) (Xi + 1−X) Ci = (Xi−X) (Xi + 1−X) + (Yi−Y) (Yi + 1−Y) φi = atan2 (Si, Ci) where i = 0,..., N−1 and atan2 is FOR
It represents an arctangent function available in the TRAN language or the C language. That is, θ = atan2 (Y, X) is

[0043]

(Equation 2) Satisfy the relationship. If Si = Ci = 0, it means that P coincides with the vertex of the polygon, so it can be determined that it is not an interior point.

Thus,

[0045]

(Equation 3) Is ± 2π if the point P is inside the polygon A, and is zero if it is an outside point. This algorithm is referred to herein as algorithm B (steps S11 to S11 in FIG. 3).
Step S20). An algorithm similar to the algorithm B is described, for example, in the above-described prior art.

However, since this calculation uses atan2 and the ぃ ぅ transcendental function, the calculation takes time. Therefore,
Consider the following facts.

Consider the special case where a given polygon is a convex polygon. A convex polygon has all its interior angles π
It is the following polygon. In this case, the outer product Si of the vector connecting the point P and the end point of each side of the convex polygon has the same sign if it is an inner point, and includes a different sign if it is an outer point.

Therefore, in the case of a convex polygon, the interior point can be determined by performing the following calculation. First, select the vertices of the polygon appropriately. The Si is calculated from the selected vertex along the direction of the polygon, and the product with the next Si is calculated. If this is positive, it means that Si and Si + 1 have the same sign. That is, it is confirmed that Si · Si + 1> 0. If there is a point where Si · Si + 1 becomes 0 or negative, it means that the given point P is not an inner point.

The algorithm for determining the interior point that satisfies this fact is called algorithm A (steps S21 to S28 in FIG. 2). In this case, since it is possible to determine whether or not an inner point is obtained by a simple multiplication and subtraction calculation without using a transcendental function such as atan2, the processing speed is much higher than that of Algorithm B.

To determine whether or not the assumed polygon is a convex polygon, an outer product of vectors corresponding to two sides sandwiching each vertex is calculated, and if all these have the same sign, the convex polygon is determined. If there is a different sign, it can be determined that the object is a non-convex polygon.

Therefore, in the present embodiment, the flowchart shown in FIG. 1 (steps S21 to S28)
Provides an interior point determination algorithm according to

That is, first, it is determined whether or not the polygon A and the point P as shown in FIG. 4 are convex polygons (steps S31 to S37). If so, the algorithm A is applied (step S38). If it is a non-convex polygon, the interior point is determined by applying the above algorithm B (step S39).

According to this method, when the polygon is a non-convex polygon, the calculation of the transcendental function is required. However, when the polygon is a convex polygon, the interior point is determined only by simple addition, subtraction, multiplication and division. And the calculation time can be reduced in many cases.

[Second Embodiment] In the first embodiment, if it is known in advance that a given polygon is an oriented convex polygon, the interior point is determined only by the algorithm A. It can be carried out.

That is, given a point P on a plane formed as data on a memory in a computer and an oriented convex polygon having no intersection in the same plane formed as an array on the memory. The cross product formed by the obtained point P and the end point of each continuous side of the convex polygon is calculated, and if these are all the same sign, it can be determined that the point P is inside the convex polygon, If there is a different sign, the point P is determined to be outside the convex polygon.

[Third Embodiment] If the given polygon is a triangle, the obtained figure will be an oriented convex polygon regardless of the order in which the vertices are selected. Therefore,
The following interior point determination algorithm can be employed.

That is, in FIG. 5 showing a polygon (triangle) used in the present embodiment, points P on a plane formed as data on a memory in a computer are identical to points P formed on the memory as an array. For the triangular DEF in the plane, the outer product of the vectors PD × PE, PE × PF, PF × P
D is calculated, and if they are all the same sign, it is determined that the point P is inside the triangle DEF, and if there is a different sign, it is determined that the point P is outside the triangle DEF.

In short, for the triangle DEF on the plane and the point P on the plane, the outer products PD × PE, PE × P
By checking whether F and PF × PD have the same sign, it can be determined whether or not the point P exists inside the triangle ABC.

By storing a program according to the flowcharts shown in FIGS. 1 to 3 in a storage device of a computer and operating the computer, various programs using the computer, for example, a graphic simulation as a basic tool group for numerical simulation and graphics. By applying the present invention to a drawing apparatus, it is possible to realize the above-described method of determining the inside point of a figure. The program is stored in, for example, an external storage device (FD, CD-ROM, ROM, magnetic tape, etc.) connected to the computer, and the program is stored in a storage device (memory) by a reading device in the computer. May be.

[0060]

As described above in detail, according to the method for determining the interior point of a graphic according to the present invention, the amount of calculation required for determining the interior point of a graphic is reduced in many cases. Makes it possible to determine the interior point of the figure.

[Brief description of the drawings]

FIG. 1 is a flowchart illustrating a method of determining an inner point of a graphic according to a first embodiment of the present invention.

FIG. 2 is a flowchart showing an algorithm A in FIG.

FIG. 3 is a flowchart showing an algorithm B in FIG. 1;

FIG. 4 is a diagram showing a polygon A used in the first embodiment.

FIG. 5 is a diagram showing a polygon A used in a third embodiment.

FIG. 6 is a flowchart showing a conventional interior point determination algorithm.

FIG. 7 is an explanatory diagram showing a conventional interior point determination algorithm.

FIG. 8 is an explanatory diagram showing a conventional interior point determination algorithm.

[Explanation of symbols]

 A: Given polygon P: Given point Q1 to Q5 Vertex of polygon D, E, F Vertex of triangle

Claims (8)

[Claims] 1. A polygon formed by a set of oriented vertices on a plane and a point P on a plane correspond to two sides sandwiching the ordered vertices of the polygon.
It is determined whether or not the outer products of the two vectors have the same sign, and it is determined whether or not the polygon is a convex polygon. If the polygon is a convex polygon, the convex polygon is ordered. 2 with two consecutive vertices as end points and the point P as a start point
A method for determining whether or not the point P exists inside a convex polygon by checking whether or not all outer products of two vectors have the same sign. 2. When it is determined that the polygon is not a convex polygon in determining whether or not the polygon is a convex polygon, two consecutive vertices of the ordered vertices of the convex polygon are defined as end points. 2. The graphic according to claim 1, wherein a sum of angles formed by the two vectors starting from the point P is checked to determine whether the point P exists inside the convex polygon. Interior point determination method. 3. For a convex polygon formed by a set of oriented vertices on a plane and a point P on the plane,
It is checked whether or not the outer products of two vectors having the consecutive points of the ordered vertices of the convex polygon as an end point and the point P as a start point are all the same sign, and the point P is a convex polygon. A method for determining the inside point of a figure for determining whether or not it exists inside. 4. A triangle DEF on a plane and a point P on a plane
And the vector cross product PD × PE, PE × PF,
A method of judging an inside point of a figure, which checks whether or not PF × PD all have the same sign and judges whether or not the point P exists inside the triangle ABC. 5. A polygon corresponding to a polygon formed by a set of oriented vertices on a plane and a point P on the plane corresponding to two sides sandwiching the ordered vertices of the polygon.
A first check is made to determine whether or not the outer products of the two vectors have the same sign, and to determine whether or not the polygon is a convex polygon.
If the first determining means determines that the convex polygon is a convex polygon, two consecutive points of the ordered vertices of the convex polygon are the end point and the point P is the starting point. A second determining means for determining whether or not the outer products of the two vectors have the same sign, and determining whether or not the point P exists inside the convex polygon. A machine-readable storage medium. 6. If the first determining means determines that the convex polygon is not a convex polygon, two consecutive points of the ordered vertices of the convex polygon are defined as an end point, and the point P is defined as a start point. A program having third determining means for determining whether or not the point P exists inside a convex polygon by examining a sum of angles formed by the two vectors. 6. The storage medium according to claim 5, which is readable by a machine. 7. For a convex polygon formed by a set of oriented vertices on a plane and a point P on a plane,
It is checked whether or not the outer products of two vectors having the consecutive points of the ordered vertices of the convex polygon as an end point and the point P as a start point are all the same sign, and the point P is a convex polygon. A machine-readable storage medium storing a program having means for determining whether or not it exists inside. 8. A triangle DEF on a plane and a point P on a plane
And the vector cross product PD × PE, PE × PF,
A program having means for checking whether or not all of PF × PD have the same sign to determine whether or not the point P exists inside the triangle ABC; Storage medium.