JP2005149245A - Cad system, curved surface analysis device, curved surface reproducing device, and method and program therefor - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a CAD system which remarkably improves the use value of a CG model or a CAD model and realizes efficient design/production processes, and a curved surface analysis device, a curved surface reproducing device, and a method and a program therefor. <P>SOLUTION: A computer maps a curved surface on an actual space to a parameter space and calculates a primary base quantity and a secondary base quantity. Next, the computer calculates the curvature line of the curved surface and actual space coordinates where the curvature line passes, and calculates actual space coordinates of an intersection on the parameter space, between a line passing an inverse mapping target point on the parameter space and the curvature line mapped on the parameter space. Then the computer calculates a curve on the actual space, which passes the calculated actual space coordinates of the intersection and calculates a actual space coordinate value of the inverse mapping target point on the curve and thus reproduces curved surface data on the actual space. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a CAD system, a curved surface analysis device, a curved surface reproducing device, a method thereof, and a program thereof that handle a curved surface in real space on a parameter space defined by a tangent vector forming a tangent plane.

  Today, in order to meet consumer demand, it is desired to shorten the process from planning to design and production. In order to improve the efficiency of design and production processes, CG (Computer Graphics) and CAD (Computer Aided Design) systems are actively used. Conventionally, the following processing methods exist in order to represent a shape having a complicated curve or curved surface shape such as an automobile or home appliance on a computer.

The first is a solid model. A simple shape called a primitive is held in a computer, and an operation of combining the shapes is repeated to express a complicated shape. The primitives are, for example, a cylinder, a cube, a rectangular parallelepiped, a torus, a sphere, and the like. In the solid model, shape representation is performed by a set operation of these primitives. Therefore, in order to create a complex shape, many steps are required and strict calculation is required.

The second is a surface model, which uses operations such as bezier, b-spline, rational bezier, NURBS (Non-Uniform Relational b-spline) to perform operations such as cutting and connecting lines and surfaces. By repeating this process, a complex free curve / curved surface is expressed.

However, even if the above solid model or surface model has no problem in terms of expression, a problem may occur when used in a downstream application such as CAM or CAE. This is due to differences in support elements supported by the created CG and support elements supported by other CG, CAD, and downstream applications, differences in shape definitions, etc., and through applications such as translators that correct these differences The model is corrected (see Patent Document 3).
JP 2001-250130 A JP-A-11-65628 JP-A-10-69506 JP-A-4-134571 Japanese Patent Laid-Open No. 4-117572 JP-A-1-65628

However, the above-described correction work is extremely inefficient in order to shorten the design / production process. The reason why correction is necessary varies depending on the individual case, but the point that is particularly problematic in the production process is that in conventional CG and CAD systems, all curved and curved surface expressions are approximated by Euclidean geometry. That is.
For example, when a bowl-shaped tabsill surface is generated by a sweep operation, there are a long line at the bottom of the heel and a short line at the center of the heel. Therefore, this sweep operation is a deformation accompanied by expansion / contraction of the figure so as to maintain the continuity of the generated curved surface.
However, this expansion and contraction is not considered in the conventional CG or CAD system, and the internal representation is approximated as a cylindrical shape.
For this reason, when a CG model or a CAD model that is approximately expressed in Euclidean geometry is actually passed to the CAE, an error generated here becomes a problem in production.

Hereinafter, this problem will be described in detail. FIG. 14 is an explanatory diagram showing the appearance of a reproduction curved surface before and after converting CAD information expressed in NURBS into CAM information.
Now, in the curved surface (see the left figure) in which the CAD information expressed in NURBS is reproduced as it is, there is no curved surface portion which causes a problem in the curved surface reproduction. However, when this is converted into CAM information and reproduced. In the reproduced curved surface (see the right figure), wrinkles are generated in the non-Euclidean curved surface portion at the center of the curved surface (see FIGS. 14 and 15).
This is a manifestation of the influence of an error generated by approximating a curved surface expressed in non-Euclidean geometry by Euclidean geometry in NURBS interpolation.
That is, since the curved surface conversion and the data conversion are performed on the curved surface which is originally contracted without performing an appropriate interpolation process, so-called unusable CAD data and CG data are generated.

The present invention has been made in view of such circumstances, and a first object is to greatly increase the utility value of the CG model or the CAD model and to improve the efficiency of the design / production process. CAD system, curved surface analysis device for extracting point sequence information, primary basic quantity, secondary basic quantity from curved surface, curved surface playback apparatus for reproducing curved surface using point sequence information, primary basic quantity, secondary basic quantity, It is to provide a method and a program thereof.

  The present invention has been made to solve the above-described problems. The present invention provides a curved surface analysis apparatus that maps a curved surface in real space onto a parameter space defined by a tangent vector that forms a tangent plane, and the curved surface. A CAD system comprising a curved surface reproduction device that reproduces a curved surface in the real space from parameter space coordinates mapped by an analysis device, wherein the curved surface analysis device is generated on a real space generated by a predetermined graphic expression. A curved surface mapping means for mapping the curved surface to a parameter space of a plane, a cylindrical surface or a spherical surface defined by a tangent vector forming the tangent plane of the curved surface and a normal vector to the tangent plane, and the tangent vector A standard quantity calculating means for calculating a primary basic quantity and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface. The curved surface reproduction device calculates a curvature line of the curved surface and a real space coordinate through which the curvature line passes based on a primary basic amount and a secondary basic amount of the curved surface in the real space; The curvature line in the real space passes through the real space coordinates of the intersection point on the parameter space between the line passing the inverse mapping target point on the parameter space and the curvature line mapped on the parameter space. First real space coordinate value calculating means for calculating based on real space coordinates and a ratio of the distance from the start point of the curvature line to the intersection point on the parameter space and the total length of the curvature line; and the actual value of the calculated intersection point Based on the space coordinates, a predetermined graphic expression is used to calculate a curve in the real space that passes through the real space coordinates of the calculated intersection, and the target point from the start point of the line that passes through the inverse mapping target point in the parameter space Based on distance to Second real space coordinate value calculating means for calculating a real space coordinate value of the inverse mapping target point on the calculated curve, and reproducing the curved surface data in the real space based on the calculated real space coordinate value And a curved surface reproducing means.

Further, the present invention provides a primary basic quantity defined by a tangent vector that forms a tangent plane of a curved surface, a tangent vector, and a normal vector of the curved surface, from a curved surface in real space generated by a predetermined graphic representation. A curved surface analysis device for calculating a secondary basic quantity defined by: a differential equation parameter representing a curved surface in the real space based on the primary basic quantity and the secondary basic quantity, and the differential equation And a curved surface reproduction device for reproducing the curved surface data by calculating a real space coordinate value by the numerical calculation process.

  The present invention also provides a curved surface in real space generated by a predetermined graphic representation of a plane, cylindrical surface, or spherical surface defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector for the tangent plane. A curved surface mapping means for mapping to a parameter space; a standard quantity calculating means for calculating a primary basic quantity defined by the tangent vector; and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface; It is characterized by comprising.

  The present invention also provides a curved surface reproducing apparatus for reproducing a curved surface in the real space from a parameter space coordinate in which the curved surface in the real space is mapped onto the parameter space defined by the tangent vector forming the tangent plane of the curved surface. A curvature line calculating means for calculating a curvature line of the curved surface and real space coordinates through which the curvature line passes based on a primary basic quantity and a secondary basic quantity of the curved surface in the real space; and the parameter space The real space coordinates through which the curvature line in the real space passes are the real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point and the curvature line mapped onto the parameter space. First real space coordinate value calculating means for calculating based on the ratio of the distance from the starting point of the curvature line to the intersection point on the parameter space and the total length of the curvature line, and the real space coordinates of the calculated intersection point Based on the given By the shape representation, a curve on the real space passing through the real space coordinates of the calculated intersection is calculated, and based on the distance from the start point of the line passing the inverse mapping target point on the parameter space to the target point, A second real space coordinate value calculating means for calculating a real space coordinate value of the inverse mapping target point on the calculated curve, and a curved surface for reproducing curved surface data on the real space based on the calculated real space coordinate value; And a reproducing means.

  In addition, the present invention provides a computer in which a curved surface in real space generated by a predetermined graphic representation is defined as a plane defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector to the tangent plane, a cylindrical surface Alternatively, a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface are calculated by mapping to a spherical parameter space.

  In the present invention, the computer reproduces the curved surface in the real space from the parameter space coordinates in which the curved surface in the real space is mapped onto the parameter space defined by the tangent vector forming the tangent plane of the curved surface. A curved surface reproduction method, wherein a curvature line of the curved surface and a real space coordinate through which the curvature line passes are calculated based on a primary basic quantity and a secondary basic quantity of the curved surface in the real space, The real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point and the curvature line mapped on the parameter space, the real space coordinates through which the curvature line on the real space passes, and Calculated based on the ratio of the distance from the start point of the curvature line on the parameter space to the intersection point and the total length of the curvature line, and based on the real space coordinates of the calculated intersection point, the calculated intersection point using a predetermined graphic expression The fruit A curve in the real space passing through the inter-coordinates is calculated, and based on the distance from the starting point of the line passing through the inverse mapping target point in the parameter space to the target point, the calculated inverse mapping target point on the curve A real space coordinate value is calculated, and curved surface data is reproduced on the real space based on the calculated real space coordinate value.

  The present invention also provides a curved surface in real space generated by a predetermined graphic representation of a plane, cylindrical surface, or spherical surface defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector for the tangent plane. A process of mapping to a parameter space; a process of calculating a primary basic quantity defined by the tangent vector; a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface; Processing for calculating a curvature line of the curved surface and real space coordinates through which the curvature line passes based on a primary basic quantity and a secondary basic quantity of the curved surface, and a line passing through the inverse mapping target point on the parameter space; , The real space coordinates of the intersection point on the parameter space with the curvature line mapped on the parameter space, the real space coordinate through which the curvature line on the real space passes, and the parameter space Processing based on the ratio of the distance from the starting point of the curvature line to the intersection and the total length of the curvature line, and based on the real space coordinates of the calculated intersection, a predetermined graphic representation is used to calculate the actual value of the calculated intersection. A curve in the real space passing through the space coordinates is calculated, and based on the distance from the start point of the line passing through the reverse mapping target point on the parameter space to the target point, the calculated reverse mapping target point on the curve A CAD program for causing a computer to execute a process of calculating a real space coordinate value and a process of reproducing curved surface data in the real space based on the calculated real space coordinate value.

  The present invention also provides a curved surface in real space generated by a predetermined graphic representation of a plane, cylindrical surface, or spherical surface defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector for the tangent plane. Causes a computer to execute a process of mapping to a parameter space, a process of calculating a primary basic quantity defined by the tangent vector, and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface Is a curved surface analysis program.

  In addition, the present invention provides a computer that performs processing for reproducing a curved surface in the real space from the parameter space coordinates mapped on the parameter space defined by the tangent vector that forms the tangent plane of the curved surface. Processing for calculating the curvature line of the curved surface and the real space coordinates through which the curvature line passes, based on the primary and secondary basic quantities of the curved surface in the real space, The curvature line in the real space passes through the real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point in the parameter space and the curvature line mapped on the parameter space. Processing based on the real space coordinates and the ratio of the distance from the start point of the curvature line on the parameter space to the intersection and the total length of the curvature line, and a predetermined value based on the real space coordinates of the calculated intersection By calculating a curve in the real space that passes through the real space coordinates of the calculated intersection by the graphic representation, based on the distance from the start point of the line passing through the inverse mapping target point on the parameter space to the target point, Processing for causing a computer to execute processing for calculating a real space coordinate value of a reverse mapping target point on a calculated curve and processing for reproducing curved surface data in the real space based on the calculated real space coordinate value This is a curved surface reproduction program.

As described above, according to the present invention, a computer maps a curved surface in real space generated by a predetermined graphic expression to a parameter space defined by a tangent vector that forms a tangent plane of the curved surface. Calculate the primary basic quantity defined and the secondary basic quantity defined by the tangent vector and the normal vector of the curved surface, and based on the primary and secondary basic quantities of the curved surface in real space, Calculate the real space coordinates through which the curvature line and the curvature line pass, and the real space coordinates of the intersection in the parameter space between the line passing through the inverse mapping target point in the parameter space and the curvature line mapped in the parameter space. Calculated based on the real space coordinates through which the curvature line in the real space passes, and the ratio of the distance from the start point of the curvature line to the intersection in the parameter space and the total length of the curvature line, and the real space coordinates of the calculated intersection Based on the distance from the start point of the line passing through the inverse mapping target point on the parameter space to the target point, by calculating a curve in the real space that passes through the real space coordinates of the calculated intersection with a predetermined graphic representation Thus, the real space coordinate value of the inverse mapping target point on the calculated curve is calculated, and the curved surface data is reproduced in the real space based on the calculated real space coordinate value. Therefore, the utility value of the CG model or the CAD model is increased. In addition to being able to greatly increase the efficiency of the design / production process, the effect can be obtained.

  Hereinafter, the best mode for carrying out the present invention will be described.

First, the basic concept of the present invention will be described. FIG. 1 is a flowchart showing the flow of CAD data in the CAD system of the present invention.
A plurality of CAD data CAD-A, B, and C described by different curved surface description algorithms are input to CAD-A curved surface analysis programs to CAD-C curved surface analysis programs corresponding to CAD-A to C, respectively. In the data platform that is analyzed and realized by the CAD system, it is stored in a data format of predetermined point sequence data and curved surface feature lines.
Then, the point sequence data and the characteristic line of the curved surface are input to the CAD-A curved surface generation program to CAD-C curved surface generation program corresponding to CAD-A to C, respectively, and the curved surface based on CAD-A to C is generated. Is done.

Hereinafter, an embodiment of a CAD system of the present invention will be described with reference to the drawings. FIG. 2 is a configuration diagram showing the configuration of the CAD system of the present embodiment. The CAD system according to the present embodiment includes a curved surface analysis device and a curved surface reproduction device.
Depending on the implementation, the curved surface analysis device and the curved surface reproduction device may be constructed as stand-alone computers, respectively. However, in this embodiment, this is constructed in the same system. An example of the case will be described. Therefore, as will be described later, hardware that can be shared between the curved surface analysis device and the curved surface reproduction device will be described with the overlapping portions omitted.

The curved surface analysis apparatus is a computer that performs curved surface analysis processing for mapping a curved surface in a real space onto a parameter space defined by a tangent vector (described later) that forms a tangent plane. A central processing unit (not shown), a storage memory (not shown) such as a ROM or a RAM, and connected via a bus and shared with the curved surface reproduction device 10 and an image display processing unit 11, a display unit 12, an output unit 13, and a communication unit (not shown).
The CPU reads the curved surface analysis program 1 stored in the ROM and executes a series of processes related to free curved surface analysis. The RAM is a semiconductor memory for the CPU to temporarily store data.

The analysis program 1 is a graphic data represented by a surface model such as a solid model, bezier, b-spline, rational bezier, NURBS, etc. ), A point sequence information table 30, a primary basic quantity table 31, and a secondary basic quantity table 32 are created and stored in the database 10 by the CPU.
That is, the CPU reads the analysis program 1 to implement a curved surface analysis processing unit. The curved surface analysis processing unit includes a curved surface mapping processing unit and a standard amount calculation processing unit.
The curved surface mapping processing unit maps a curved surface in a real space generated by using a predetermined graphic expression algorithm adopted by other CAD data or the like to a parameter space defined by a tangent vector that forms a tangent plane of the curved surface. Process.
The standard quantity calculation processing unit performs a process of calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface.

のパラメータ形式で表示される曲面上の点列情報（ｕ、ｖ）からなる。
例えば、ｕ＝０、１／ｍ、２／ｍ、・・・ｍ−１／ｍ（ｍは自然数）であり、ｖ＝０、１／ｎ、２／ｎ、・・・ｎ−１／ｎ（ｎは自然数）とすると、図２に示す曲面はｍ×ｎのメッシュに分割される。この場合、点列情報（ｕ、ｖ）は、メッシュＩＤ１〜ＩＤｍｎまでのｍｎ個のデータ列となる。 Specifically, the point sequence information table 30 is a plane defined by a tangent vector forming a tangent plane of a curved surface and a normal vector with respect to the tangent plane of the curved surface in a real space generated by a predetermined graphic expression, A table comprising point sequence information obtained by mapping to a cylindrical or spherical parameter space, in this embodiment, as shown in FIG.

Point sequence information (u, v) on the curved surface displayed in the parameter format.
For example, u = 0, 1 / m, 2 / m, ... m-1 / m (m is a natural number), and v = 0, 1 / n, 2 / n, ... n-1 / n. If n is a natural number, the curved surface shown in FIG. 2 is divided into m × n meshes. In this case, the point sequence information (u, v) is mn data sequences from mesh ID1 to IDmn.

で表される。ここでｄｓの絶対値の二乗は

で表され，曲面の基本ベクトルより、上述の一次基本量が次式で定義される。

上述の一次基本量Ｅ，Ｆ，Ｇは、このように各メッシュに一意に定まり、一次基本量テーブル３１は、メッシュＩＤ１〜ＩＤｍｎそれぞれに対する値を格納する。
また上記式３及び式４をまとめると、

となる。 The primary basic quantity table 31 is composed of primary basic quantities E, F, and G derived by the following equations. When u and v have a functional relationship, s (u, v) represents a curve on the curved surface, partial derivative ∂s / ∂u = Su represents tangent vector of u = constant curve, The derivative ∂s / ∂v = Sv represents the tangent vector of v = constant curve.
At this time, the basic vectors Su and Sv form a curved tangent plane. A vector ds connecting two points s (u, v) on the curved surface to s (u + du, v + dv) is

It is represented by Where the square of the absolute value of ds is

The above-mentioned primary basic quantity is defined by the following equation from the basic vector of the curved surface.

The primary basic quantities E, F, and G described above are uniquely determined for each mesh in this way, and the primary basic quantity table 31 stores values for the meshes ID1 to IDmn.
Moreover, the above formulas 3 and 4 are summarized as follows:

It becomes.

この算出値Ｈを用いて、曲面上の単位法線ベクトルｎは以下の式で表される。

また、図４に示すように、曲面上の点Ｐにおける接線ベクトルの線束はこの接平面内に存在し、単位接線ベクトルｔの１つは、以下の式で表される。

図４に示すように、このｔとｎで定まる平面を法平面という。
この法断面上の点Ｐにおける曲率κを法曲率といい、ｔを法断面の弧長ｓで微分すると、

となる。両辺に法線ベクトルを掛けて、以下に示す二次基本量

を導入すると、

となる。
上述の二次基本量Ｌ，Ｍ，Ｎは、このように各メッシュに一意に定まり、二次基本量テーブル３２は、メッシュＩＤ１〜ＩＤｍｎそれぞれに対する値を格納する。
なお、式５を式１２に代入すると、以下の式が得られる。

以上によって一次基本量及び二次基本量から法曲率が算出される。 The secondary basic quantity table 32 includes secondary basic quantities L, M, and N derived by the following equations. Assuming that the angle formed by the basic vectors Su and Su is ω, the inner product F and the absolute value H of the vector product of the basic vectors are expressed as follows using a primary basic quantity.

Using this calculated value H, the unit normal vector n on the curved surface is expressed by the following equation.

Also, as shown in FIG. 4, the tangent vector line bundle at the point P on the curved surface exists in this tangent plane, and one of the unit tangent vectors t is expressed by the following equation.

As shown in FIG. 4, this plane determined by t and n is called a normal plane.
The curvature κ at the point P on the normal section is called the normal curvature, and t is differentiated by the arc length s of the normal section.

It becomes. Multiplying the normal vector on both sides, the secondary basic quantity shown below

Introduced

It becomes.
The secondary basic quantities L, M, and N described above are uniquely determined for each mesh as described above, and the secondary basic quantity table 32 stores values for the meshes ID1 to IDmn.
Note that the following formula is obtained by substituting Formula 5 into Formula 12.

Thus, the normal curvature is calculated from the primary basic quantity and the secondary basic quantity.

The curved surface reproducing device is a computer that reproduces a curved surface by performing inverse mapping processing from the coordinate value in the parameter space mapped by the curved surface analyzing device to the coordinate value in the real space, and more specifically, the curved surface analyzing device Similarly to the above, a central processing unit (not shown) such as a CPU and a storage memory (not shown) such as a ROM and a RAM are connected via a bus and shared with a curved surface analysis device 10, an image display processing unit 11, a display unit 12, an output unit 13, and a communication unit (not shown).
The CPU reads the curved surface conversion program 2 and the curved surface reproduction program 3 stored in the ROM, and executes a series of processes relating to curved surface conversion and curved surface reproduction. The RAM is a semiconductor memory for the CPU to temporarily store data.

The curved surface conversion program 2 reads information necessary for a free curved surface from the point sequence information table 30, the primary basic quantity table 31, and the secondary basic quantity table 32, generates free curved surface data, and can be interpreted by other CAD applications. This is a program that causes a computer to execute processing that transforms into.
Similar to the curved surface conversion program 2, the curved surface reproduction program 3 reads information necessary for reproducing a free curved surface from the point sequence information table 30, the primary basic quantity table 31, and the secondary basic quantity table 32, and generates free curved surface data. This is a program for causing a computer to execute processing to be output to the image display processing unit 11.

That is, the CPU reads the curved surface conversion program 2 and the curved surface reproduction program 3 to form a curved surface conversion processing unit and a curved surface reproduction processing unit.
The curved surface conversion processing unit and the curved surface reproduction processing unit include a curvature line calculation processing unit, a real space coordinate value calculation processing unit, and a curved surface reproduction processing unit.
The curvature line calculation processing unit calculates the curvature line of the curved surface and the real space coordinates through which the curvature line passes based on the primary and secondary basic quantities of the curved surface in the real space.
The real space coordinate value calculation processing unit includes a line passing through a reverse mapping target point (described later) on the parameter space (which may be either a straight line or a curve, but is a line represented by a known function parameter) , The real space coordinates of the intersection on the parameter space with the curvature line mapped on the parameter space, the real space coordinates through which the curvature line on the real space passes, and from the start point to the intersection of the curvature line on the parameter space And the ratio of the total length of the curvature line.
In addition, the real space coordinate value calculation processing unit calculates a curve on the real space that passes through the real space coordinates of the calculated intersection by a predetermined graphic expression based on the calculated real space coordinates of the intersection. Based on the distance from the start point to the target point of the line passing through the reverse mapping target point above (as described above, it may be either a straight line or a curve, but is a line expressed by a known function parameter) The real space coordinate value of the inverse mapping target point on the calculated curve is calculated.
The curved surface reproduction processing unit reproduces curved surface data in the real space based on the calculated real space coordinate value.

The database 10 stores the above-described point sequence information table 30, the primary basic quantity table 31, and the secondary basic quantity table 32, and the output result of the analysis program 1 is written in association with a mesh ID described later.
The image display processing unit 11 performs image display processing of output results from the curved surface reproduction program 3 and other CAD applications.
The display unit 12 displays the output result of the image display processing unit 11.
The output unit 13 outputs the output result of the image display processing unit 11 to a communication unit or another recording medium.
The communication unit transmits / receives data such as point sequence information, primary basic quantities, and secondary basic quantities stored in the database 1 to other servers and clients via a network such as a LAN or the Internet.

Next, the operation of the CAD system of this embodiment will be described.
FIG. 5 is a flowchart showing the process of curved surface analysis processing and curved surface reproduction processing by the CAD system of this embodiment.
In response to an analysis command of the actual measurement data 20 or other CAD format data 21 by the user's operation, the CPU reads the analysis program 1 from the ROM and executes a free-form surface analysis process.
First, the CPU performs a process of extracting a plurality of point sequences on a curved surface such as a two-dimensional NURBS surface or a bicubic surface held in the actual measurement value data 20 or other CAD format data 21 (step S1 in FIG. 5).
Then, a curved surface is generated from this point sequence using another CAD system, and the curved surface is divided into a predetermined number of meshes as shown in FIG. 3, and each mesh portion is normalized with basic vectors Su and Sv. In this specification, the three-dimensional space coordinate system before normalization with the basic vectors Su and Sv is defined as real space coordinates, and the two-dimensional space coordinate system normalized at this time is defined as parameter space coordinates. Each is associated and stored.
That is, the point sequence information (u, v) on the parameter space generated at the time of normalization is written in association with the mesh ID and the point sequence information on the real space in the point sequence information table 30 held in the database 10.
As described above, in this embodiment, the parameter space coordinates are expressed as a two-dimensional space coordinate system. However, the parameter space can be formed by a developable surface such as a plane or a cylindrical surface, or a spherical surface. The present invention is not necessarily limited to the case of using parameter space coordinates developed on a plane. That is, the mapping from the real space coordinates to the parameter space coordinates (Gaussian mapping) is not a mapping from only the tangent plane, but maps (or transforms) all the tangent vectors and normal vectors (orthogonal to each other) in the same direction. And mapping onto a spherical surface (or a part of the spherical surface) is also possible.

Next, the CPU executes differential geometric analysis processing. That is, a process of calculating primary basic quantities E, F, and G defined by basic vectors Su and Sv that form a tangent plane of the mesh is performed. The calculated primary basic quantities E, F, and G are written in association with the mesh ID in the primary basic quantity table 31 held in the database 10 as in the point sequence information.
Further, the CPU performs processing for calculating secondary basic quantities L, M, and N defined by the basic vectors Su and Sv and the unit normal vector n of the mesh. The calculated secondary basic quantities L, M, and N are written in association with the mesh ID in the secondary basic quantity table 32 held in the database 10 in the same manner as the primary basic quantities E, F, and G.
This first basic quantity defines how the parameter space (uv plane) is stretched and contracted to create a curved surface, determines the length of the u and v curve groups that make up the curved surface, and the angle between the curves, The second basic quantity is a quadratic expression that approximates the height of the curved surface with respect to the tangent plane, and determines various curvatures of the curved surface, that is, the degree of bending of the curved surface.

そして、ｅ１、ｅ２より単位法線ベクトルｎが次のように計算できる。

このようにして、3本のベクトル｛ｅ１、ｅ２、ｎ｝が曲面上の各点において定義される。 In addition, the CPU performs a process of calculating a condition for the differential equations representing the meshes described above to be continuous at each mesh boundary, in other words, an integrable condition that is a condition for the differential equations to have a unique solution. .
Now, the above-mentioned curved surface coordinates (u, v) are replaced with (u1, u2), and this point is defined as p (u1, u2). A curve formed by fixing u2 and moving u1 is called a u1 curve, a curve formed by fixing u1 and moving u2 is called a u2 curve, and a point p (u1, u2) on the curved surface is the starting point, and u1 The tangent vector along the curve, u2 curve can be calculated as follows.

The unit normal vector n can be calculated from e1 and e2 as follows.

In this way, three vectors {e1, e2, n} are defined at each point on the curved surface.

そして、第１基本テンソル（ｇｉｊ、ｉ,ｊ＝１，２）を以下のように定義する。

また、４個の数の組ｇｉｊ、ｉ,ｊ＝１，２を次のように定義する。

さらに、各点において二次基本量Ｌ、Ｍ、Ｎを以下のように定義する。

そして、さらに第２基本テンソル（ｈｉｊ、ｉ,ｊ＝１，２）を以下のように定義する。

At each point, the primary basic quantities E, F, and G are defined as follows.

The first basic tensor (gij, i, j = 1, 2) is defined as follows.

Further, a set of four numbers gij, i, j = 1, 2 is defined as follows.

Further, secondary basic quantities L, M, and N are defined as follows at each point.

Further, the second basic tensor (hij, i, j = 1, 2) is defined as follows.

ただし、式２３はクリストッフェルの記号を示す。
この構造方程式２１、２２の積分可能条件は次の２式（式２４のガウスの方程式及び式２５のマイナル・コダッツィの方程式）で示される。

ただし、式２６はリーマン・クリストッフェルの曲率テンソルを示す。 Now, when the moving frame {e1, e2, n} is differentiated with respect to the curved surface coordinates (u1, u2), the structure of the curved surface represented by the following two formulas (Gauss formula of formula 21 and Weingarten formula of formula 22): Get the equation.

However, Formula 23 shows Christoffel's symbol.
Integrable conditions of the structural equations 21 and 22 are expressed by the following two equations (Gaussian equation of Equation 24 and Minor Kodazzi equation of Equation 25).

However, Formula 26 shows the curvature tensor of Riemann Christoffel.

A first basic tensor (gij, i, j = 1, 2) and a second basic tensor (hij, i, j = 1, 2) are given as functions of the surface coordinates (u1, u2), and these are the Gaussian mentioned above. Since the shape of the curved surface having such gij and hij is uniquely determined (see Bonnet's basic theorem), each mesh is C2 continuous.
The CPU performs these arithmetic processes and calculates the above-described integration possible condition.

とおき、さらにκをγの関数κ（γ）と書き直すと、

となる。このγの２次式よりｄκ（γ）／ｄγ＝０において、κ（γ）は極値を取る。そして、この極値条件のもとで、式１５を微分し、κとγを（κ〜）と（γ〜）と書き換えると、

を得る。そして、数１６に代入すると、

を得る。これらの式より以下の関係式が得られる。

式１８を変形すると、

が得られる。κ〜２の係数は、式７より正であり、この根をκ１、κ２とすると、図６に示すように、この値が主曲率となる。 Next, the CPU executes a curvature line analysis process, a characteristic line analysis process, and a curvature / garth length conversion process. First, the main curvatures κ1 and κ2 in the mesh are calculated based on the primary basic quantities E, F, and G and the secondary basic quantities L, M, and N by the curvature line analysis process.
That is, first, the extreme value of the curvature κ is calculated. The shape of the normal section, which is the intersection of the normal plane and the curved surface shown in FIG. 3, changes with the tangential direction, and the normal curvature also changes accordingly. This shape returns to the original state after half rotation of the normal plane.
Γ now

And rewriting κ as a function κ (γ) of γ,

It becomes. From the quadratic expression of γ, κ (γ) takes an extreme value when dκ (γ) / dγ = 0. Then, under this extreme value condition, when Equation 15 is differentiated and κ and γ are rewritten as (κ˜) and (γ˜),

Get. And when substituting into Equation 16,

Get. From these equations, the following relational expressions are obtained.

By transforming Equation 18,

Is obtained. The coefficients of κ˜2 are positive from Equation 7, and when the roots are κ1 and κ2, this value is the main curvature as shown in FIG.

が表現される。ここで、Ｋｍ、Ｋｇはそれぞれ平均曲率及びガウス曲率である。Ｋｇ＝０となるのは、曲面が可展面となる場合であり、曲面上の曲率線は直線になる。本実施形態においては、このガウス曲率が０となる点を後述する変形の基準点とする。
この点以外に変形の基準点として適当な点として、例えば、曲率線、境界線（稜線）、等傾斜直交線、主曲率極値線，傾斜極値線、臍点を選択してもよい。
これらは、曲面の特徴を示す特徴量である主曲率，主方向，ガウス曲率，平均曲率，曲率線のうち、１または2以上の特徴量の変化パターンによって規定される変形の基準点または基準線となる点または線であり、一次基本量及び二次基本量に基づいて算出可能である。 Next, a Gaussian curvature or an average curvature is calculated based on the main curvature. That is, from the relationship between the root of the quadratic equation and the coefficient,

Is expressed. Here, Km and Kg are an average curvature and a Gaussian curvature, respectively. Kg = 0 is when the curved surface becomes a developable surface, and the curvature line on the curved surface becomes a straight line. In the present embodiment, a point at which the Gaussian curvature becomes 0 is set as a reference point for deformation described later.
In addition to this point, for example, a curvature line, a boundary line (ridge line), an equal inclined orthogonal line, a main curvature extreme value line, an inclined extreme value line, or an umbilical point may be selected as an appropriate point as a reference point for deformation.
These are reference points or reference lines of deformation defined by a change pattern of one or more feature values among main curvature, main direction, Gaussian curvature, average curvature, and curvature line, which are feature quantities indicating the characteristics of the curved surface. And can be calculated based on the primary basic quantity and the secondary basic quantity.

または、

を得る。これら２式はともに、曲率線の式であり、２次方程式であるので、γ１、γ２は以下の関係がある。

曲面上の点において、γ１、γ２で決まる方向において、曲率は極値を取る。曲面上の接線ベクトルは、（Ｓｕｄｕ＋Ｓｖｄｖ）であり、γ１、γ２に対応する２つの接線ベクトルの内積は、

となり、この{ }内を変換すると、

はゼロとなる。すなわち、主曲率の法断面の２つの接線方向は、直交している事が分かる。この方向は主方向と呼ばれ、この主方向と曲面上の接線が一致する場合、これが曲率線となる。
以上により、メッシュの主方向を示す曲率線の算出処理が行われる。 Also, a curvature line indicating the main direction of the mesh is calculated based on the main curvature. That is, if κ˜ is eliminated from Equation 19,

Or

Get. Both of these two equations are equations of curvature and are quadratic equations, so that γ1 and γ2 have the following relationship.

At a point on the curved surface, the curvature takes an extreme value in a direction determined by γ1 and γ2. The tangent vector on the curved surface is (Sudu + Svdv), and the inner product of the two tangent vectors corresponding to γ1 and γ2 is

And if you convert within this {},

Becomes zero. That is, it can be seen that the two tangent directions of the normal curvature normal section are orthogonal. This direction is called the main direction, and when this main direction and the tangent on the curved surface coincide, this becomes a curvature line.
Thus, the calculation process of the curvature line indicating the main direction of the mesh is performed.

Next, curvature / garth length conversion processing is executed. That is, the CPU calculates the girth length based on the curvature calculated based on the primary basic quantities E, F, G and the secondary basic quantities L, M, N. A curvature radius r is calculated from the curvature (1 / r) along the curvature line calculated by the above-described curvature line calculation process, and the girth length of the curvature line is expanded or contracted for each calculation section.
Thus, the analysis process is performed (step S2).

  Therefore, according to the curved surface analysis apparatus of the present embodiment, as shown in FIG. 7, only the point sequence information, the primary basic quantity, and the secondary basic quantity are extracted from the curved surface expressed using an arbitrary curved surface expression algorithm. In addition, it is possible to eliminate the error at the time of the maximum curved surface, and to obtain the effect of greatly improving the efficiency of data to be transferred.

Next, the CPU performs curved surface data transfer processing in response to collection of the created and extracted point sequence information, primary basic quantity, and secondary basic quantity (step S3).
On the other hand, if these pieces of information are not available, database evaluation processing is performed.
That is, the shape reproduced based on the calculated main direction, reference position (point, line, etc.) and deformation amount is compared with the shape reproduced based on the point sequence information, primary basic amount, and secondary basic amount. If they match, curved surface data transfer processing is performed.
If they do not match, approximate complement accuracy improvement processing is performed.
That is, the original curved surface is approximated and supplemented so that second order differentiation is possible, and the above-described processing is repeated again from step S1. Then, when the comparative evaluations coincide, the process proceeds to curved surface data transfer processing.

Curved surface data consisting of point sequence information (real space coordinates, parameter space coordinates) and curved surface feature information (primary basic quantity and secondary basic quantity) is transferred from the curved surface analysis apparatus to the curved surface reproducing apparatus.
In the curved surface reproduction apparatus, the curved surface data is input to the curved surface conversion program 2 or the curved surface reproduction program 3 shown in FIG. 2 (step S4).

When the CPU receives the conversion command and the playback command, the CPU executes the curved surface conversion program 2 or the curved surface playback program 3 to calculate the curvature line and the real space coordinates through which the curvature line passes from the primary basic quantity and the secondary basic quantity. Perform the calculation process.
FIG. 8 is a flowchart showing the process from calculation of the curvature line in the real space from the primary basic quantity and the secondary basic quantity to the mapping to the parameter space and the spatial coordinate conversion.
The CPU calculates the main curvature from the primary basic quantity and the secondary basic quantity as in the above-described curvature line analysis process, and calculates a curvature line based on the main curvature.
That is, the CPU first expands and deforms the curvature line by the length of the girth in the direction of the curvature line with the Gaussian curvature point selected as the feature point or feature line as a reference for deformation, and in the real space, the curvature line and Real space coordinates through which the curvature line passes are obtained (step S5).

Next, the CPU maps the obtained curvature line and the real space coordinates through which the curvature line passes on the parameter space by the basic vectors Su and Sv forming the tangent plane described above (step S6). The two-dimensional parameter space coordinate system shown in FIG. 8 shows the result of mapping the obtained curvature line and real space coordinates through which the curvature line passes.
Then, the CPU performs reverse mapping on the real space as shown in FIG. 10 for an arbitrary reverse mapping target point on the parameter space shown in FIG. Although the inverse mapping target point can be arbitrarily set, in this embodiment, in order to make the inverse mapping target point uniform to some extent, the parameter space is set to a predetermined value as shown in the flowchart of FIG. It divides in the u direction and the v direction by the number of parameter space coordinate divisions, and this division intersection is set as a reverse mapping target point (step S10 in FIG. 11).
Specifically, if the number of parameter space coordinate divisions in the u direction and v direction is m and n (m and n are real numbers), m × n inverse mapping target points are set.

Next, the CPU sets a line passing through the inverse mapping target point on the set parameter space. As described above, the line passing through the reverse mapping target point may be either a straight line or a curved line, but is a line expressed by a known function parameter. In this embodiment, for simplicity, a case where u = constant (const_u in FIG. 9) is described.
Next, the CPU passes through this inverse mapping target point, and the real space of the intersection point (white circle point shown in FIG. 9) on the parameter space between the constant line u = constant and the curvature line mapped on the parameter space. Coordinates (white circle points shown in FIG. 10) are calculated.
Specifically, the CPU intersects the actual space coordinates through which the input curvature line in the real space passes and the starting point of the curvature line in the parameter space (the intersection of v = 0 and the curvature line) (see FIG. 9). The ratio of the actual length distance along the curvature line up to the point of the white circle) and the total length of the curvature line,
(Distance from the start point to the intersection point) / (Total length of curvature line)
The real space coordinates of the intersection (the white circle shown in FIG. 10) are calculated using a predetermined graphic expression algorithm (for example, NURBS) based on the parameters indicated by (Step S11).

Next, based on the calculated real space coordinates of the intersection, the CPU calculates a curve in the real space that passes through the calculated real space coordinates of the intersection using a predetermined graphic expression algorithm. The CPU then calculates the inverse mapping target on the curve calculated based on the distance from the starting point of u = constant straight line (u = constant, v = 0) passing through the inverse mapping target point in the parameter space to the target point. The real space coordinate value of the point is calculated (Step S7 in FIG. 5, Step S12 in FIG. 11).
Through the above processing, the CPU calculates the real space coordinates of the intersection point on the parameter space between the line passing through the inverse mapping target point on the parameter space and the curvature line mapped on the parameter space.
Then, the CPU reproduces the curved surface data (mesh) in the real space based on the calculated real space coordinate value.

When the curved surface conversion program is executed, a plurality of point sequences on the curved surface are extracted from the reproduced mesh or curved surface, and the point sequences are converted according to a graphic expression algorithm in another CAD system.
The converted graphic data is reproduced by another CAD application 22 and then output to the image display processing unit 11. The image display processing unit 11 performs display processing of data output from the CAD application 22 and outputs this to the display unit 12. The display unit 12 receives display data and displays it.

When the curved surface conversion program is executed, the reproduced curved surface data is output to the image display processing unit 11. The image display processing unit 11 performs display processing of data output from the CAD application 22 and outputs this to the display unit 12. The display unit 12 receives display data and displays it.
That is, using a point at which the Gaussian curvature is 0 as a reference for deformation, the curvature line is stretched and deformed by the length of the girth in the curvature line direction, and the mesh or curved surface is reproduced. Then, the reproduced graphic data is output to the image display processing unit 11 and displayed on the display unit 12 after the display processing.

As described above, according to the CAD system of the present embodiment, an effect of performing analysis, conversion, and reproduction of a free-form surface while maintaining C2 continuity can be obtained. Therefore, it is possible to greatly increase the utility value of the CAD model and to increase the efficiency of the design / production process.

The applicants conducted a verification test as shown in FIG. 12 as verification of the effectiveness of the curved surface analysis and the reproduction algorithm. In this verification test, for the torus defined by the radius R of the torus surface and the radius r of the ring surface,
Torus surface direction: 0 ° ≦ v ₀ ≦ 45 °
However, 0 ° is a predetermined annular surface.
Ring surface direction: 135 ° ≦ u ₀ ≦ 180 °
However, the outer diameter of the torus surface is 0 °.
The trimmed surface that was trimmed only on the curved surface defined by, and the reproduction error were evaluated. Specifically, the point sequence coordinates on the curved surface are reproduced from the curvature line information obtained by the curved surface analysis by the CAD system described above, and the coordinate value indicated by the reproduction result is compared with the coordinate value calculated from the mathematical expression. The error was calculated.
FIG. 13 is obtained by normalizing the above v ₀ and u ₀ in a form corresponding to u and v in the above-described CAD system, and applying a curved surface analysis and a reproduction algorithm (reproduced coordinate values) and a mathematical expression. It is a figure which shows the original coordinate value and its deviation.
The deviation is
(Playback coordinates-original coordinates) / (original coordinate maximum value-original coordinate minimum value)
Defined in
As a result, when the curved surface analysis and reproduction algorithm of the present invention were applied, the maximum deviation was 10 to the minus fifth power or less, and the PDQ (Product Data Quality) level could be satisfied.

  Further, FIGS. 16 and 17 show a comparison between the influence of the NURBS interpolation error pointed out as a problem of the prior art and the influence of the interpolation error in the parameter space of the present invention. In the rendering model of FIG. 14 and the mesh model of FIG. 15, unexpected wrinkles occur as an effect of data conversion. However, in FIGS. 16 and 17, the data conversion error is suppressed only to the influence of the rounding error of the computer. Therefore, the same wrinkles are not generated.

  Therefore, according to the CAD system of the present embodiment, it has been verified that the utility value of the CAD model can be greatly increased and the effect of improving the efficiency of the design / production process can be obtained.

In the CAD system of the present embodiment, a series of processes related to free-form surface analysis, conversion, and reproduction in a CAD model has been described. However, the CAD system of the present invention is not limited to this, and the above-described CG system and computer The present invention can be applied to a system and a program for performing image expression using the.
Further, in the CAD system of the present embodiment, as a suitable example, the curved surface is divided into meshes as shown in FIG. 2 and then normalized with the basic vectors Su and Sv, and u using point sequence information (u, v) is used. However, the CAD system of the present invention is not limited to this, and coordinate values based on (x, y, z) coordinate parameters may be used.

が主曲率となり、ガウス曲率Ｋの定義から

となる。 Next, a second embodiment of the CAD system of the present invention will be described.
The CAD system of this embodiment is different from the first embodiment in a curved surface reproduction algorithm in the curved surface reproduction apparatus. That is, in the above embodiment, the curved surface reproduction device calculates a curvature line from the primary basic quantity and secondary basic quantity input from the situation analysis apparatus, and twists obtained by mapping the point sequence information to the parameter space of (u, v). Interpolation between point sequences was executed in a state where no occurrence occurred, and a curved surface was generated by inverse mapping. This is a curved surface reproduction algorithm that employs geometric processing using vectors (tensors). On the other hand, in this embodiment, a curved surface is reproduced by obtaining a direct solution of a general curved surface by partial differential equations (simultaneous equations, determinants) using point sequence information, primary basic quantities, and secondary basic quantities.
In general, when solving a partial differential equation in a curved surface coordinate system, there are many differential coefficient variables, and the convergence calculation is complicated. Therefore, it is necessary to select an appropriate coordinate system so that the differential coefficient variable is reduced. In the playback apparatus of this embodiment, the main curvature line coordinates (u, v) of the main curvatures λ, μ (= κ1, κ2) are selected as the coordinate system.
In the main curvature line, the two main directions (maximum curvature and minimum curvature) are orthogonal to each other, and therefore, in equation (4), the primary basic amount F = 0. Further, in equation (13), when du = cos θ and dv = sin θ, and θ = 0 and π / 2 are substituted,

Is the main curvature, and from the definition of Gaussian curvature K

It becomes.

また、第二基本形式は、

と表される。ここで、式（４４）にL=λE、N＝μGを代入すると、

と表すことができる。 Since M must be 0, the primary basic quantity F = secondary basic quantity M = 0 holds. That is, the first basic form of the curved surface p (u; v) is

The second basic format is

It is expressed. Here, substituting L = λE and N = μG into equation (44),

It can be expressed as.

これにより、ガウスの公式（ｐの２階微分）が

（但し、νは曲面の単位法線ベクトル）と表される。また、単位法線ベクトルの微分（ワインガルテンの公式）は、

である。 Hereafter, the differential equation for determining the surface p under these coordinates is written. In general, in order to show a Gaussian equation (an equation that expresses a Gaussian curvature by a primary basic quantity), first, if a geodesic differential equation is used with primary basic quantities E, F, and G, the following Christoffel symbols are defined: ,

As a result, Gauss's formula (the second derivative of p) becomes

(Where ν is a unit normal vector of a curved surface). Also, the derivative of the unit normal vector (Winegarten formula) is

It is.

となり、また、単位法線ベクトルの微分（ワインガルテンの公式）は、

となるから、それぞれを式（４７）に代入すると、

を得る。これを（曲率座標系での）ガウス・ワインガルテンの公式またはフルネ方程式³とよぶ． Here, by substituting F = 0, L = λE, M = 0, N = μG,

And the derivative of the unit normal vector (Winegarten formula) is

Therefore, substituting each into equation (47),

Get. This is called the Gauss-Weingarten formula (in the curvature coordinate system) or the Fournet equation ³ .

となるが、これをフレームの方程式で表すと、

となる。特にＦは、正則（非特異）行列、つまり、Ｆの各項は、全て０になることはない。するとフルネ方程式は、フレームの方程式

を満たす。とくに、式（４５）の第1式をｕで微分したものと、第2式をｖで微分したものは、等しくなる。これを整理すると次式のようになる。

Considering pu, pv, ν as a cubic column vector, the cubic square matrix is

However, when this is expressed by the frame equation,

It becomes. In particular, F is a regular (non-singular) matrix, that is, each term of F is not all zero. Then the Fournet equation is the frame equation

Meet. In particular, the one obtained by differentiating the first expression of the expression (45) by u is equal to the one obtained by differentiating the second expression by v. This can be organized as follows:

As described above, on the curvature line coordinates, F = 0, M = 0, L = λE, and N = μG. Therefore, the above-mentioned curved surface structural equation Frunet equation (Gauss-Weingarten formula) The Gaussian equation and the Kodatch equation, which are integral conditions of the structural equation of the curved surface, can be expressed by primary fundamental quantities E, G and primary fundamental quantities E, G, first and second derivatives and principal curvatures λ, μ.
That is, conversely, when the primary basic quantities E and G and the principal curvatures λ and μ are input to the curved surface reproducing device, the Fournet equation is set by the parameter expression as described above, and any point ( Since it is known that there is only one solution having an arbitrary matrix as an initial value by fixing u ₀ , v ₀ ), a coordinate value in real space is calculated by a numerical solution using this. .
Therefore, according to the CAD system of the present embodiment, as in the above-described embodiment, it is possible to obtain an effect of performing analysis, conversion, and reproduction of a free-form surface while maintaining C2 continuity. Therefore, it is possible to greatly increase the utility value of the CAD model and to increase the efficiency of the design / production process.

The above-described CAD system, curved surface analysis apparatus, and curved surface reproduction apparatus have a computer system therein.
A series of processes related to the above-described curved surface analysis processing and curved surface reproduction processing are stored in a computer-readable recording medium in the form of a program, and the above processing is performed by the computer reading and executing the program. Done.
That is, each processing means and processing unit in the above-described CAD system, curved surface analysis device, and curved surface reproduction device is a central processing unit such as a CPU that reads the above program into a main storage device such as a ROM or RAM, and processes information. It is realized by executing arithmetic processing.
Here, the computer-readable recording medium means a magnetic disk, a magneto-optical disk, a CD-ROM, a DVD-ROM, a semiconductor memory, or the like. Alternatively, the computer program may be distributed to the computer via a communication line, and the computer that has received the distribution may execute the program.

It is a flowchart which shows the flow of CAD data in the CAD system of this invention. It is a block diagram which shows the structure of the CAD system of this embodiment. It is explanatory drawing which shows a mode that a curved surface is divided | segmented into a mxn mesh and basic vectors _Su and _Sv are defined. It is explanatory drawing which shows the surface which unit tangent vector t and unit normal vector n extend. It is a flowchart which shows the process of the curved surface analysis process by the CAD system of this embodiment, and a curved surface reproduction | regeneration process. It is explanatory drawing which shows the mode of a curvature change. It is explanatory drawing which shows the outline of a curved surface analysis process. It is explanatory drawing which shows the outline of curved surface reproduction | regeneration processing. It is explanatory drawing which shows the relationship between the intersection of a curvature line in a parameter space, a reverse mapping target point, a reverse mapping target point, and a curvature line. It is explanatory drawing which shows the relationship between the intersection of a curvature line in real space, a reverse mapping target point, a reverse mapping target point, and a curvature line. It is a flowchart which shows the flow of the process which reversely maps a reverse mapping target point on real space. It is explanatory drawing which shows the outline of this verification test. It is explanatory drawing which shows the test result of this verification test. It is explanatory drawing which shows the rendering model of the reproduction | regeneration curved surface before and behind converting CAD information expressed by NURBS into CAM information. It is explanatory drawing which shows the mesh model of the reproduction | regeneration curved surface before and behind converting CAD information expressed by NURBS into CAM information. It is explanatory drawing which shows the rendering model of the reproduction | regeneration curved surface before and behind converting CAD information expressed by NURBS into CAM information by the CAD system of this embodiment. It is explanatory drawing which shows the mesh model of the reproduction | regeneration curved surface before and behind converting CAD information expressed by NURBS into CAM information by the CAD system of this embodiment.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Analysis program 2 ... Conversion program 3 ... Reproduction program 10 ... Database 11 ... Image display process part 12 ... Display part 13 ... Output part

Claims (9)

A curved surface analysis device for mapping a curved surface in real space onto a parameter space defined by a tangent vector forming a tangent plane, and reproducing the curved surface in the real space from the parameter space coordinates mapped by the curved surface analysis device A CAD system composed of a curved surface playback device,
The curved surface analysis apparatus is
A curved surface that maps a curved surface in real space generated by a predetermined graphic representation to a parameter space of a plane, a cylindrical surface, or a sphere defined by a tangent vector that forms the tangent plane of the curved surface and a normal vector to the tangent plane Mapping means;
A standard quantity calculating means for calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface;
The curved surface reproducing apparatus is
A curvature line calculating means for calculating a curvature line of the curved surface and real space coordinates through which the curvature line passes based on a primary basic quantity and a secondary basic quantity of the curved surface in the real space;
The real space coordinates of the intersection in the parameter space between the line passing through the reverse mapping target point in the parameter space and the curvature line mapped in the parameter space are the actual space through which the curvature line in the real space passes. First real space coordinate value calculating means for calculating based on the space coordinates and the ratio of the distance from the start point of the curvature line on the parameter space to the intersection and the total length of the curvature line;
Based on the calculated real space coordinates of the intersection, a curve in the real space that passes through the real space coordinates of the calculated intersection is calculated by a predetermined graphic expression, and passes through the inverse mapping target point in the parameter space. Second real space coordinate value calculating means for calculating a real space coordinate value of the inverse mapping target point on the calculated curve based on a distance from the start point of the line to the target point;
A CAD system comprising: a curved surface reproduction means for reproducing curved surface data in the real space based on the calculated real space coordinate value. From a curved surface in real space generated by a predetermined graphic representation, a primary basic quantity defined by a tangent vector forming a tangent plane of the curved surface, and a second defined by the tangent vector and the normal vector of the curved surface. A curved surface analysis device for calculating a next basic quantity;
Based on the primary basic quantity and the secondary basic quantity, parameters of a differential equation representing a curved surface in the real space are set, real space coordinate values are calculated by numerical calculation processing of the differential equation, and curved surface data is reproduced. A CAD system comprising a curved surface reproducing device. A curved surface that maps a curved surface in real space generated by a predetermined graphic representation to a parameter space of a plane, a cylindrical surface, or a sphere defined by a tangent vector that forms the tangent plane of the curved surface and a normal vector to the tangent plane Mapping means;
A curved surface analysis comprising: a standard quantity calculating means for calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and a normal vector of the curved surface. apparatus. A curved surface reproducing apparatus for reproducing a curved surface in the real space from a parameter space coordinate mapped on a parameter space defined by a tangent vector that forms a tangent plane of the curved surface in the real space,
A curvature line calculating means for calculating a curvature line of the curved surface and real space coordinates through which the curvature line passes based on a primary basic quantity and a secondary basic quantity of the curved surface in the real space;
The curvature line in the real space passes through the real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point in the parameter space and the curvature line mapped on the parameter space. First real space coordinate value calculating means for calculating based on real space coordinates and a ratio of the distance from the start point of the curvature line on the parameter space to the intersection and the total length of the curvature line;
Based on the calculated real space coordinates of the intersection, a curve in the real space that passes through the real space coordinates of the calculated intersection is calculated by a predetermined graphic expression, and passes through the inverse mapping target point in the parameter space. Second real space coordinate value calculating means for calculating a real space coordinate value of the inverse mapping target point on the calculated curve based on a distance from the start point of the line to the target point;
A curved surface reproducing apparatus comprising: curved surface reproducing means for reproducing curved surface data in the real space based on the calculated real space coordinate value. Computer
A curved surface in real space generated by a predetermined graphic representation is mapped to a parameter space of a plane, a cylindrical surface or a sphere defined by a tangent vector forming a tangent plane of the curved surface and a normal vector to the tangent plane;
A curved surface analysis method, comprising: calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and a normal vector of the curved surface. A curved surface reproduction method in which a computer reproduces a curved surface in real space from parameter space coordinates in which the curved surface in real space is mapped onto a parameter space defined by a tangent vector that forms a tangent plane of the curved surface. ,
Based on the primary and secondary basic quantities of the curved surface in the real space, the curvature line of the curved surface and the real space coordinates through which the curvature line passes are calculated,
The real space coordinates of the intersection in the parameter space between the line passing through the reverse mapping target point in the parameter space and the curvature line mapped in the parameter space are the actual space through which the curvature line in the real space passes. Calculate based on spatial coordinates and the ratio of the distance from the start point of the curvature line on the parameter space to the intersection and the total length of the curvature line,
Based on the calculated real space coordinates of the intersection, a curve in the real space that passes through the real space coordinates of the calculated intersection is calculated by a predetermined graphic expression, and passes through the inverse mapping target point in the parameter space. Based on the distance from the starting point of the line to the target point, calculate the real space coordinate value of the inverse mapping target point on the calculated curve,
A curved surface reproduction method, wherein curved surface data is reproduced in the real space based on the calculated real space coordinate value. A process of mapping a curved surface in real space generated by a predetermined graphic representation to a parameter space of a plane, a cylindrical surface, or a sphere defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector to the tangent plane When,
Processing for calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and the normal vector of the curved surface;
Processing for calculating the curvature line of the curved surface and the real space coordinates through which the curvature line passes based on the primary and secondary basic quantities of the curved surface in the real space;
The curvature line in the real space passes through the real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point in the parameter space and the curvature line mapped on the parameter space. Processing based on real space coordinates and the ratio of the distance from the starting point of the curvature line on the parameter space to the intersection and the total length of the curvature line;
Based on the calculated real space coordinates of the intersection, a curve in the real space that passes through the real space coordinates of the calculated intersection is calculated by a predetermined graphic expression, and passes through the inverse mapping target point in the parameter space. Based on the distance from the starting point of the line to the target point, a process of calculating the real space coordinate value of the inverse mapping target point on the calculated curve;
A CAD program for causing a computer to execute a process of reproducing curved surface data in the real space based on the calculated real space coordinate value. A process of mapping a curved surface in real space generated by a predetermined graphic representation to a parameter space of a plane, a cylindrical surface, or a sphere defined by a tangent vector that forms a tangent plane of the curved surface and a normal vector to the tangent plane When,
A curved surface analysis program for causing a computer to execute a process of calculating a primary basic quantity defined by the tangent vector and a secondary basic quantity defined by the tangent vector and a normal vector of the curved surface. A program for causing a computer to execute processing for reproducing a curved surface in the real space from a parameter space coordinate in which the curved surface in the real space is mapped onto the parameter space defined by the tangent vector forming the tangent plane of the curved surface Because
Processing for calculating the curvature line of the curved surface and the real space coordinates through which the curvature line passes based on the primary and secondary basic quantities of the curved surface in the real space;
The curvature line in the real space passes through the real space coordinates of the intersection on the parameter space between the line passing through the inverse mapping target point in the parameter space and the curvature line mapped on the parameter space. Processing based on real space coordinates and the ratio of the distance from the starting point of the curvature line on the parameter space to the intersection and the total length of the curvature line;
Based on the calculated real space coordinates of the intersection, a curve in the real space that passes through the real space coordinates of the calculated intersection is calculated by a predetermined graphic expression, and passes through the inverse mapping target point in the parameter space. Based on the distance from the starting point of the line to the target point, a process of calculating the real space coordinate value of the inverse mapping target point on the calculated curve;
A curved surface reproduction program for causing a computer to execute a process of reproducing curved surface data in the real space based on the calculated real space coordinate value.

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