WO2004053741A1 - 三角形と線分の交点計算方法とそのプログラム - Google Patents
三角形と線分の交点計算方法とそのプログラム Download PDFInfo
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- WO2004053741A1 WO2004053741A1 PCT/JP2003/015586 JP0315586W WO2004053741A1 WO 2004053741 A1 WO2004053741 A1 WO 2004053741A1 JP 0315586 W JP0315586 W JP 0315586W WO 2004053741 A1 WO2004053741 A1 WO 2004053741A1
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- intersection
- coordinate
- triangle
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T19/00—Manipulating 3D models or images for computer graphics
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T15/00—3D [Three Dimensional] image rendering
- G06T15/06—Ray-tracing
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T2210/00—Indexing scheme for image generation or computer graphics
- G06T2210/21—Collision detection, intersection
Definitions
- the present invention relates to a method of calculating an intersection of a triangle and a line segment and a program therefor. Description of related technology
- Entity data that integrates shape and physical properties can be stored with a small storage capacity, which enables the integrated management of the shape, structure, physical property information, and history of an object, from design to processing, assembly, testing, and evaluation.
- Patent Literature 1 discloses a method of storing substantial data in which data relating to a series of processes can be managed by the same data, and CAD and simulation can be unified.
- the “patent document 1” “method of storing entity data integrating shape and physical properties” includes an external data input step (A), an octree splitting step (B), and a cell data storage step.
- the external data input step (A) the external data 12 comprising the boundary data of the object acquired in the external data acquisition step S1 is input to a computer or the like storing the method of the present invention.
- the split tree division step (B) the external data 1 and 2 are divided by octree division into rectangular parallelepiped cells whose boundary planes are orthogonal.
- Celde overnight storage step (C) various physical property values are set for each cell. It is to remember.
- Patent Document 1 divides an external data consisting of shape data of an object into rectangular parallelepiped cells whose boundary planes are orthogonal by an octree division, and assigns various physical property values to each cell. It is something to memorize.
- Each divided cell consists of an internal cell located inside the object and a boundary cell including a boundary surface.
- the internal cell has one attribute
- the boundary cell has two physical properties, inside and outside the object.
- V-CAD data data obtained by this method
- V-CAD data designs and simulations using this data
- V-CAD virtualized computer
- FIG. 2 is a diagram showing a procedure for creating V-CAD data from external data based on [Patent Document 1].
- a poxel space is defined by STE P-1, and the intersection of the geometrical shape and the edge of each cell is calculated by STE P-2 ( Step 3) to create an outer loop on the inner surface of the cell from the cutting point information for each cell, and to STEP 3 refer to the original geometric shape to divide the inside of the outer loop into triangles.
- V-CAD data is created by setting the medium value of the cell with STEP-5.
- Non-Patent Literature 1] to [Non-Patent Literature 3] are conventionally known as a method of calculating an intersection between a triangle and a line segment.
- ⁇ ,) 3 is calculated as shown in FIG. 4, when projected on the Xy plane, P i (x 15 yi »zi), P 2 (2.y 2.z 2 ), P a (x 3 , y 3, z 3 ) and P rnt (x, y, z) of the linear transformation equation (Eq. (4) in [Equation 3]) omitting the Z coordinate of each vertex You just need to create an inverse transform
- Equation 4 In FIG. 6A , V Nrml 2 , V Nrm23 , and V Nrm31 point in the same direction, and in FIG. 6B, V Nrm31 points in the opposite direction.
- the purpose of the present invention is to provide a method for calculating the intersection between a triangle and a line segment, which can reduce the calculation time and increase the speed when finding the intersection of the same triangle with a plurality of different line segments.
- P 2 a unit length of the second axis (V axis), a coordinate system setting step of setting a coordinate system R for a unit vector PcP IXPcPs the third axis (N axis) (A), a normal coordinate system A transformation matrix calculation step (B) that computes a transformation matrix M that transforms the coordinates of the points in the coordinate system R into coordinate values, and a line segment that calculates the u, V, and n coordinate values of both ends A and B of the line segment End coordinate calculation step (C); intersection determination step (D) for determining whether there is an intersection with a triangle based on the u and v.
- P. P is a program for calculating the intersection of a triangle with three vertices P 2 and a line segment connecting A and B.
- the origin, P. P i is the unit length of the first axis (U axis), P.
- an intersection determination step (D) for determining whether there is an intersection with the triangle based on the u, v, and n coordinate values of both ends A and B, and an intersection point for calculating u and V coordinate values of the intersection when the intersection occurs.
- a program for executing a marker operation step (E) and an intersection determination step (F) for determining the presence or absence of an intersection in a triangle from the u and V coordinate values of the intersection.
- the transformation matrix M is an Affine transformation matrix.
- a transformation matrix M can be easily calculated for a triangle in a short time.
- Equation (8) makes it possible to easily calculate the u, V, and n coordinate values of both ends A and B of a line segment for a plurality of different line segments in a short time.
- intersection determination step (D) (1) it is determined that there is no intersection if the n coordinate of one point is equal to or greater than the length of the line segment, and / or (2) the n coordinate of the two points has the same sign. If there is no intersection, it is judged that there is an intersection if it has a different sign.
- intersection determination step (D) (1) the u coordinate of both points is 0 or less or 1 or more, (2) the V coordinate of both points is 0 or less, or (3) the u coordinate + v coordinate of both points If it is 1 or more, the intersection is judged to be outside the triangle.
- intersection coordinate calculation step (E) a point internally divided by the n-coordinate values of both ends A and B is set as an intersection, and the u and V coordinate values are calculated.
- the u and V coordinate values of the intersection can be calculated easily and in a short time.
- the intersection determination step (F) when the u and V coordinate values of the intersection satisfy Expression (13) in [Equation 1], the intersection is determined to be inside the triangle.
- FIG. 1 is a flowchart of a method for storing entity data of a prior application.
- FIG. 2 is another flowchart of a method of storing the entity data of the prior application.
- FIG. 3 is a schematic diagram showing a conventional method.
- FIG. 4 is another schematic diagram showing a conventional method.
- FIG. 5A and FIG. 5B are other schematic diagrams showing the conventional method.
- FIG. 6A and FIG. 6B are other schematic views showing the conventional method.
- FIG. 7 is a configuration diagram of an apparatus for performing the method of the present invention.
- FIG. 8 is a flowchart of the method of the present invention and its program.
- FIG. 9 is a schematic diagram showing a coordinate system R according to the present invention.
- FIG. 10 is a flowchart specifically showing the processing method according to the present invention.
- FIG. 11 is a flowchart showing the first embodiment of the present invention.
- FIG. 12 is an image showing the display of the input triangle.
- FIG. 13 is an image showing V-CAD data composed of cells and cell inner surfaces.
- Figure 14 is an image showing the cell cutting points generated by the calculation of the intersection between the input data and the cell ridge line.
- FIG. 15 is a flowchart showing a second embodiment of the present invention.
- FIG. 16 is an image showing the VCAD data after the shape conversion processing. DESCRIPTION OF THE PREFERRED EMBODIMENTS
- FIG. 7 is an apparatus configuration diagram for executing the method of the present invention.
- the device 10 includes an input device 2, an external storage device 3, an internal storage device 4, a central processing unit 5, and an output device 6.
- the input device 2 is, for example, a keyboard, and inputs external data 12 composed of shape data of the object 1.
- the external storage device 3 is a hard disk, a floppy disk, a magnetic tape, a compact disk, or the like, and stores an entity data in which a shape and a physical quantity are integrated and a storage operation program thereof.
- the internal storage device 4 is, for example, a RAM, a ROM, or the like, and stores calculation information.
- the central processing unit 5 (CPU) intensively processes operations, input / output, and the like, and executes a storage program together with the internal storage device 4.
- the output device 6 is, for example, a display device and a printer, and outputs the stored entity data and the execution result of the storage program.
- FIG. 8 is a flowchart of an intersection calculation method and a program thereof according to the present invention. As shown in this figure, the intersection calculation method and the program according to the present invention include a coordinate system setting step.
- the intersection calculation method of the present invention is a method of calculating an intersection of a triangle having P 0 , P i, and P 2 as three vertices and a line connecting A and B.
- the program of the present invention This is a program for calculating the intersection of a triangle with three vertices P 0 and P lt P 2 and a line connecting A and B.
- P in the coordinate system setting step (A). Is the origin, P. P i Unit length of primary axis (U axis), P. Set the coordinate system R where P 2 is the unit length of the second axis (V axis) and the unit vector of PoP t XPoP z is the third axis (N axis).
- a transformation matrix M for transforming the coordinates of the point in the ordinary coordinate system into the coordinate values of the coordinate system R is calculated.
- the line segment end coordinate calculation step (C) the u, v, and n coordinate values of both ends A and B of the line segment are calculated.
- intersection determination step (D) the presence or absence of intersection with the triangle is determined from the u, V, and n coordinate values of both ends A and B.
- intersection coordinate calculation step (E) when they intersect, the u and V coordinate values of the intersection are calculated.
- intersection determination step (F) the presence or absence of an intersection in the triangle is determined from the u and V coordinate values of the intersection.
- volume data such as VCAD (data representing the entire three-dimensional spread, CT and MRI, data for volume rendering, etc.)
- VCAD volume data representing the entire three-dimensional spread, CT and MRI, data for volume rendering, etc.
- the technique of the present invention is indispensable.
- the intersection calculation and the inside / outside judgment can be performed at high speed and stably by converting the input data into an Affine transformation matrix generally used for shape display on a computer or the like.
- the intersection calculation is repeatedly performed on the same triangle, the calculation cost can be significantly reduced by reusing the previously created matrix unique to the triangle.
- many planes consist of planes, and many triangles used as input surface shapes are large.
- the intersection of the same triangle and the cell edge is calculated many times, so that the advantage of the present invention becomes remarkable.
- the two sides of the triangle are the first axis, the second axis, the normal vector is the third axis, and the vertex common to the first and second axes is the origin.
- the coordinates of both ends of the line segment for which the intersection is calculated are converted into the spatial coordinate system R by the above matrix, and from these values, the presence or absence of the intersection, the inside / outside judgment of the triangle at the intersection, and the coordinate calculation of the intersection are performed.
- the result of the coordinate transformation includes the signed distance, it can be used to calculate the distance from the original curved surface as attribute data of Voumeume data and to confirm the front and back directions of the surface.
- FIG. 10 is a flowchart specifically showing the processing method according to the present invention.
- three-dimensional coordinate data of an input triangle having P 0 and PLP 2 as three vertices is input.
- P. Is the origin PoP i is the unit length of the first axis (U axis)
- P Q P 2 is the unit length of the second axis (V axis)
- the unit vector of PoPiX PoPz is the third axis (N axis).
- the coordinate system R to be set.
- S1 a matrix corresponding to the coordinate system R corresponding to the triangle is calculated.
- This step corresponds to the above-described transformation matrix calculation step (B).
- S2 the uv coordinate and the signed distance of both ends A and B of the line segment to be subjected to the intersection calculation are obtained by A ffine conversion operation.
- This step corresponds to the line end coordinate calculation step (C) described above.
- PoP i is the unit length of the first axis (U axis)
- P is the unit length of the second axis (V axis)
- the unit vector of PoPiX PaPz is the third axis (N axis).
- the coordinate value converted by the formula is calculated as u, V, and n coordinates of the UV value on the coordinate surface determined by the triangle and the signed distance from the triangle, respectively.
- this transformation matrix includes scaling, rotation, transformation of coordinate axes, and translation, it uses a transformation matrix called Affine transformation, which is usually used for displaying 4x4 CG.
- the scale in each axis direction is XS cale, yS cale, Z sca 1 e
- the translation vector is (xMove, yMove, zMove).
- Equation (1 1) ( ⁇ ⁇ , vy n, ⁇ ⁇ ⁇ ) can be expressed as Equation (1 1) in [Equation 4]. These are integrated to calculate equation (12) in [Equation 5].
- 4.1.5 Determine the uv coordinates by using the points internally divided by the n-coordinate values at both ends as intersections. 4.1.6 Determine whether the u and V coordinates of the intersection are inside or outside the triangle based on the following criteria.
- Equation (14) in [Equation 7] comes out twice, so they are calculated as vectors P and Q, respectively. It is calculated by equation (15) in [7].
- the calculation cost of vector P is 6 multiplications
- the calculation cost of vector Q is 6 multiplications
- the calculation cost until t is calculated is the inner product calculation, reciprocal calculation, and reciprocal calculation of one vector and another vector, respectively. Multiplication results in 19 multiplications and one division.
- Figure 11 shows the flow from acquisition of external data to creation of V-CAD data.
- a process of mapping a geometric shape (S-CAD data, a triangle patch) to each cell was performed before performing the above-mentioned intersection calculation.
- S-CAD data geometric shape
- a triangle patch a geometric shape
- candidates for intersection target line segments edges of cells
- the mapping process is not performed, all the line segments must be searched, which takes a long calculation time. Therefore, it is preferable to perform the mapping process before the intersection calculation as in the present embodiment.
- FIGS. 12 to 14 show display screens when the processing of FIG. 11 is performed. That is, FIG. 12 is a display screen of a triangle input in STEP (T 2), and FIG. 13 is a display image of V-CAD data created in STEP (T 13).
- FIG. 14 is an enlarged view of a part of FIG. 13 and shows a cell cutting point generated by calculating an intersection of input data and a cell edge.
- the benchmark achieved a maximum speed-up of about 30%.
- FIG. 15 is a flow chart when the method of the present invention is applied to an interference check at the time of shape deformation processing.
- STEP (U4) as a pre-process before performing the series of processes in Fig. 11, the intersection calculation (from STEP (T 5) to (T l 1) in Fig. 11) is performed, and the tick of self-interference is calculated. Was done. Thereafter, by performing a series of processes in FIG. 11 (corresponding to S TEP (U5)), the shape after the deformation can be displayed.
- Figure 16 shows the displayed image.
- the method for calculating the intersection of a triangle and a line segment according to the present invention and the program therefor reduce the calculation time and increase the speed when finding the intersection of a same triangle with a plurality of different line segments. It has excellent effects such as being able to do.
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Priority Applications (3)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
EP03777281A EP1569143A4 (en) | 2002-12-06 | 2003-12-05 | METHOD FOR CALCULATING THE INTERFACES BETWEEN A TRIANGLE AND A LINE SEGMENT AND CORRESPONDING PROGRAM |
US10/537,160 US7187390B2 (en) | 2002-12-06 | 2003-12-05 | Method and program for determing intersection point of triangle with line segment |
JP2004558419A JP4526121B2 (ja) | 2002-12-06 | 2003-12-05 | 三角形と線分の交点計算方法とそのプログラム |
Applications Claiming Priority (2)
Application Number | Priority Date | Filing Date | Title |
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JP2002354594 | 2002-12-06 | ||
JP2002-354594 | 2002-12-06 |
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WO2004053741A1 true WO2004053741A1 (ja) | 2004-06-24 |
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PCT/JP2003/015586 WO2004053741A1 (ja) | 2002-12-06 | 2003-12-05 | 三角形と線分の交点計算方法とそのプログラム |
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US (1) | US7187390B2 (ja) |
EP (1) | EP1569143A4 (ja) |
JP (1) | JP4526121B2 (ja) |
CN (2) | CN102156773A (ja) |
WO (1) | WO2004053741A1 (ja) |
Cited By (3)
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WO2009004676A1 (ja) * | 2007-06-29 | 2009-01-08 | Fujitsu Limited | 判定プログラム、判定装置、判定方法 |
CN101872470A (zh) * | 2010-05-30 | 2010-10-27 | 李滨 | 一种ct或mri图像的矫正和靶点定位方法 |
JP2014229254A (ja) * | 2013-05-27 | 2014-12-08 | 株式会社ア−キテック | Cadシステム |
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EP1574988B1 (en) * | 2004-03-08 | 2014-06-18 | Siemens Product Lifecycle Management Software Inc. | Determining and using geometric feature data |
JP2006107093A (ja) * | 2004-10-05 | 2006-04-20 | Konica Minolta Medical & Graphic Inc | 画像処理装置、およびプログラム |
US7555163B2 (en) * | 2004-12-16 | 2009-06-30 | Sony Corporation | Systems and methods for representing signed distance functions |
KR101705072B1 (ko) * | 2010-09-28 | 2017-02-09 | 삼성전자주식회사 | 영상 처리 장치 및 방법 |
WO2019126869A1 (en) * | 2017-12-29 | 2019-07-04 | Bombardier Inc. | Method and system for operating a configuration platform |
CN114101825B (zh) * | 2021-12-08 | 2023-02-10 | 通用技术集团机床工程研究院有限公司 | 一种基于电火花成形机的多边形工件找中方法 |
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- 2003-12-05 JP JP2004558419A patent/JP4526121B2/ja not_active Expired - Fee Related
- 2003-12-05 EP EP03777281A patent/EP1569143A4/en not_active Withdrawn
- 2003-12-05 CN CN2011100648843A patent/CN102156773A/zh active Pending
- 2003-12-05 WO PCT/JP2003/015586 patent/WO2004053741A1/ja active Application Filing
- 2003-12-05 US US10/537,160 patent/US7187390B2/en not_active Expired - Fee Related
- 2003-12-05 CN CNA2003801044914A patent/CN1717688A/zh active Pending
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Cited By (3)
Publication number | Priority date | Publication date | Assignee | Title |
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WO2009004676A1 (ja) * | 2007-06-29 | 2009-01-08 | Fujitsu Limited | 判定プログラム、判定装置、判定方法 |
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JP2014229254A (ja) * | 2013-05-27 | 2014-12-08 | 株式会社ア−キテック | Cadシステム |
Also Published As
Publication number | Publication date |
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EP1569143A4 (en) | 2009-08-12 |
CN1717688A (zh) | 2006-01-04 |
EP1569143A1 (en) | 2005-08-31 |
JP4526121B2 (ja) | 2010-08-18 |
CN102156773A (zh) | 2011-08-17 |
US7187390B2 (en) | 2007-03-06 |
JPWO2004053741A1 (ja) | 2006-04-13 |
US20050283513A1 (en) | 2005-12-22 |
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