KR-102963979-B1 - Method for randomly distributing 3D points with even density inside the regular octahedron

Abstract

The present invention relates to a method for randomly distributing a plurality of three-dimensional points with uniform density inside a regular octahedron. The method according to the present invention utilizes a random number Ux, which is uniformly distributed along the X-axis in an a-octahedron in XYZ three-dimensional space, to obtain the inverse function of the X-axis cumulative distribution function. Generate (=x), and for the x-cross-section corresponding to x, use the uniformly distributed random number Uy to obtain the inverse function of the Y-axis cumulative distribution function for the x-cross-section. Generate (=y), and for the x-cross-section corresponding to the above x,y, a uniformly distributed random number Uz is the inverse function of the Z-axis cumulative distribution function for the above x-cross-section. Generate (=z) to distribute points p(x,y,z) in the above a-octahedral space, and repeat the above steps until there are N points p.

Inventors

• 김창화

Assignees

• 강원대학교산학협력단

Dates

Publication Date

20260513

Application Date

20240627

Claims (5)

1. Step of establishing an a-octahedron in XYZ three-dimensional space; A step of generating a uniformly distributed random number Ux along the X-axis of the above a-octahedron; Using the Ux generated above, the inverse function of the X-axis cumulative distribution function Step to generate (=x); A step of generating a uniformly distributed random number Uy for an x-section corresponding to the above x; Using the above-generated Uy, the inverse function of the Y-axis cumulative distribution function for the above x-section Step to generate (=y); A step of generating a uniformly distributed random number Uz for an x-cross-section corresponding to the above x,y; Using the above-generated Uz, the inverse function of the Z-axis cumulative distribution function for the above x-section Step to generate (=z); and A method comprising the step of distributing point p(x,y,z) in the a-octahedral space based on the generated x,y,z.
2. In claim 1, The step of generating the above random number Uz is a method of generating the Uz at the position of the x-section and the y-line segment forming the x-section.
3. In claim 1, A method in which the above Ux, Uy, and Uz each follow a uniform distribution between 0 and 1.
4. In claim 1, The above , and A method in which each forms a uniform distribution.
5. In claim 1, A method of sequentially repeating the above steps until the number of the above points p becomes a preset N.

Description

Method for randomly distributing 3D points with even density inside the regular octahedron The present invention relates to a method for randomly distributing a plurality of three-dimensional points with an even density inside a regular octahedron. Due to recent advancements in communication technology, research on communication technologies for regions and environments that are traditionally difficult for humans to access is actively underway. This research includes various technologies such as Wireless Sensor Networks (WSN) including Smart Dust, underwater communication, and Intelligent Building Systems (IBS). Since establishing actual environments and conducting experiments for performance testing of these technologies entails significant costs, time, and risks, software-based simulations are mostly utilized. Among them, in the case of underwater acoustic communication capable of long-distance communication, the underwater environment itself is three-dimensional, so the distance between nodes can vary significantly depending on the depth of the node location, and furthermore, since propagation delay time and message delivery time vary significantly depending on the distance, simulation through the three-dimensional spatial arrangement of nodes is required to predict communication and network performance more accurately, including communication synchronization. Registered patents No. 10-1327325 and No. 10-1086450 disclose a technique for arranging nodes in a three-dimensional space. In addition, the inventor's prior art paper presents a method for arranging random points within a three-dimensional octahedron range that can be used in a node arrangement method. In conventional research, there was a problem in that when placing nodes (points) within a three-dimensional virtual network, the communication link distance could not be guaranteed and the nodes could not be created in a densely populated area of the network. In particular, while the aforementioned prior art guarantees the generation of point locations within the octahedron range, the probability of point generation for each location within the octahedron is evenly distributed, so there is a problem in that points are not evenly distributed between the cross-sections of the octahedron, nor even within the cross-sections, but are concentrated in certain areas. FIG. 1 is a block diagram of a device for processing a method of randomly distributing three-dimensional points with an even density inside a regular octahedron according to an embodiment of the present invention. FIG. 2 is a drawing showing a regular octahedron (a-octahedron) of height a in a three-dimensional space of an XYZ coordinate system according to an embodiment of the present invention. FIG. 3 is a drawing showing the YZ cross-section of the a-octahedron shown in FIG. 2 at 0≤x≤a/2 along the X-axis. Figure 4 is a YZ cross-sectional view shown in Figure 3. FIG. 5 is a drawing for showing the YZ cross-section of the a-octahedron shown in FIG. 2 at the X-axis where a/2 < x ≤ a. Fig. 6 is a YZ cross-sectional view shown in Fig. 5. FIG. 7 is an example diagram of x1-section and x2-section according to continuous variable x in an a-octahedron according to an embodiment of the present invention. FIG. 8 is an x-sectional view of an a-octahedron formed in the YZ plane with respect to the X-axis coordinate x according to an embodiment of the present invention. FIG. 9 is a drawing showing the associated coordinates and length, and the triangular region indicated by the shadow formed by y, when the range of the Y-axis coordinate y in the x-section according to the present invention is -x+(a/2)≤y≤a/2. FIG. 10 is a drawing showing the coordinates of a related point, the length of a line segment, a line segment formed by y, and a partial area of an x-section indicated by a shadow area from the left end of the x-section to this line segment, when the range of the Y-axis coordinate y in the x-section according to an embodiment of the present invention is a/2≤y≤x+(a/2). FIG. 11 is a drawing showing the area of the x-section formed between the y1-segment and the y2-segment according to an embodiment of the present invention. FIG. 12 is a diagram illustrating a method for evenly distributing three-dimensional points p(x,y,z) in an a-octahedral distribution according to an embodiment of the present invention. FIG. 13 is a diagram comparing the result of a method for randomly distributing three-dimensional points with even density inside an a-octahedron according to an embodiment of the present invention with the result of a conventional method. FIG. 14 is a diagram showing the distribution of points in various cross-sections in the result of the method according to an embodiment of the present invention. Hereinafter, some embodiments of the present invention will be described in detail with reference to exemplary drawings. It should be noted that in assigning reference numerals to the components of each d