JP2897245B2 - Free curve creation method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention will be described in the following order.

A Industrial application field B Outline of the invention C Conventional technology (FIG. 8) D Problems to be solved by the invention E Means for solving the problems F Function (FIGS. 1 and 3) G Implementation Example (FIGS. 1 to 7) (G1) Principle of Free Curve (FIG. 2) (G2) Embodiment of Free Curve Creation Method (FIGS. 1 to 7) Effect of Invention H Use in Industry A FIELD The present invention relates to a method for creating a free curve, for example, a CAD (compu
ter aided design) or CAM (computer aided manufa)
cturing).

B. Summary of the Invention The present invention relates to a method for creating a free curve, in which a first plane in a three-dimensional space and a midpoint between two nodes of the first plane are orthogonal to a straight line connecting the two nodes. And the intersection between the intersection line and a spherical surface having a predetermined radius generated with one of the two nodes as the center point, and using the arc as a center point, By generating a free curve, an arc-shaped free curve having a predetermined radius while passing through two nodes on a first plane in a three-dimensional space can be generated.

C Prior Art For example, when designing the shape of an object having a free-form surface using a CAD method (geometric modeling), a designer generally uses a plurality of points in a three-dimensional space through which the surface passes (this is called a node). Is designated, and a boundary curve network connecting the designated nodes is calculated by a desired vector function, thereby creating a curved surface represented by a so-called wire frame.

Thus, a large number of framework spaces surrounded by boundary curves can be formed (this process is called a framework process).

The boundary curve network formed by such framework processing is
It itself represents the rough shape that the designer intends to design, and if the surface that can be expressed by a predetermined vector function can be interpolated using the boundary curves surrounding each framework space, the designer as a whole designed A free-form surface (which cannot be defined by a quadratic function) can be generated. Here, the curved surface formed in each frame space forms a basic element constituting the entire curved surface, and this is called a patch.

By the way, in order to give a more natural external shape to the generated free-form surface as a whole, a shared space is established between two frame spaces adjacent to each other with the shared boundary therebetween so as to satisfy the condition of continuation of the tangent plane at the shared boundary. There has been proposed a free-form surface creation method in which control edge vectors around a boundary are reset (Japanese Patent Application No. 60-277448).

This free-form surface creation method uses a patch that is set in a quadrilateral frame space as shown in FIG. Is a vector function consisting of a cubic Bezier equation Represented by two patches Nodes are given by the framework processing to connect Based on adjacent patches Control edge vector such that the condition of tangent plane continuity is satisfied at the shared boundary COM of Are set, and control points are set by these control edge vectors. The principle is to set again.

If such a method is applied to other shared boundaries as well, Can be smoothly connected to adjacent patches according to the condition of continuation of the tangential plane.

Where the vector function is a cubic Bezier equation Is Is expressed using parameters u and v in the u and v directions, shift operators E and F, and the control point For the following equation 0 ≦ u ≦ 1 (4) 0 ≦ v ≦ 1 (5)

Further, the tangent plane means a plane formed by tangent vectors in the u and v directions at each point of the sharing boundary. For example, for the sharing boundary COM shown in FIG. When the tangent planes are the same, the condition of continuation of the tangent plane holds.

According to this method, it is possible to easily design even an object shape in which the curved surface shape changes smoothly as a whole, which is actually difficult to design with the conventional design method, as intended by the designer.

D Problems to be Solved by the Invention Generally, when designing the outer shape of a product on a two-dimensional flat upper surface in this way, the rough outer shape of the product is represented by a straight line, and then the outer shape is determined as necessary. A so-called squaring process is performed in which corners where lines intersect are replaced with an arc having a predetermined radius. Even when a large number of framework spaces are formed by boundary curves as described above, a three-dimensional space is used. It is thought that if such a cornering process can be realized, the usability of the designer can be remarkably improved.

For this purpose, first, a free curve expressed by a third-order Pezier equation consisting of an arc having a predetermined radius on a predetermined plane including the two nodes while passing through two predetermined nodes in the three-dimensional space is used. It is necessary to stretch.

The present invention has been made in view of the above points, and aims to propose a free curve creating method capable of forming an arc-shaped free curve having a predetermined radius on a predetermined plane including two nodes. is there.

Means for Solving E Problem In order to solve such a problem, in the present invention, two nodes specified in a three-dimensional space are used. On a given first plane π ₁ containing the two nodes In the free curve forming method for generating an arc-shaped free curve having a predetermined radius r passing through a central processing unit of a free curve forming apparatus using a computer, the central processing unit includes two nodes. And two nodes A processing step SP7 for calculating a _second plane π ₂ orthogonal to a straight line connecting the two nodes, One of the nodes Centered on a spherical surface having a predetermined radius r And a first and second plane π _1.
And π ₂ Two intersections with Step SP9 for calculating One of the intersections And two nodes on the first plane π ₁ by arc approximation Free circular arc of a given radius r passing through And a processing step SP11 for generating the program.

F action A first plane π ₁ in a three-dimensional space and this first plane π ₁
Upper two nodes And two nodes Intersecting line with the second plane π ₂ orthogonal to the straight line connecting between And this intersection line And two nodes One of the nodes Sphere of a given radius r generated with Two intersections with One of the intersections The center point As the free curve is generated by arc approximation, two nodes in the three-dimensional space An arc-shaped free curve having a predetermined radius r on a first plane π ₁ including Can be generated.

G Example Hereinafter, an example of the present invention will be described in detail with reference to the drawings.

(G1) Principle of Free Curve In this embodiment, the free curve is composed of nodes set at arbitrary intervals as shown in FIG. K _SG (K _SG1 , K _SG2 , K
_SG3 ) is connected.

This curve segment K _SG is the third-order bezier
Using the following equation, Parametric space curve represented by Is represented by

Where t is one of the nodes From the other node in the direction along the curve segment K _SG Is a parameter that changes from value 0 to value 1 as represented by the following equation: 0 ≦ t ≦ 1 (7)

Thus, the curve segment K _SG represented by the cubic Bezier equation is converted to a node by the shift operator E. Two control points between By specifying, each point on the curve segment K _SG is Position vector from the origin O of xyz space by the expansion formula of It is expressed as

Where the shift operator E is the control point on the curve segment K _SG For i = 0, 1, 2,... (10)

Therefore, if equation (6) is expanded and the relation of equation (9) is substituted, the following equation is obtained. , And as a result, equation (8) is obtained.

Thus, each curve segment K _SG1 , K _SG2 ,
K _SG3 has two nodes and control points based on equation (8). Can be represented by Of the specified node Control point in between By setting each curve segment
_Nodes and control points representing _KSG1 , _KSG2 , and _KSG3 Can be used to express a free curve having a desired curve shape.

(G2) Embodiment of Free Curve Creation Method In this embodiment, the free curve creation device includes display means such as a display, input means such as a mouse and a keyboard, and a central processing unit (CPU). As shown in the figure, two nodes specified and input in a three-dimensional space On the plane π ₁ containing , And an arc-shaped free curve with a specified and input radius r Create

That is, the CPU of the free curve creation device moves from step SP1 to step SP2 of the free curve creation processing program shown in FIG. 1 according to the instruction of the designer, and executes two nodes from the designer. Wait for the specified input.

Here, as shown in FIG. 4, the designer uses a mouse or the like to display two nodes in the display. Move the cursor to each of the set positions, click the mouse, and select the two nodes. When you input the coordinate data of, for example, from a keyboard or the like, the CPU Is displayed on the display, and this node Captures coordinate data for.

Subsequently, the CPU proceeds to the next step SP3, where an arc-shaped free curve to be created is created. Wait for the input of the radius r, and here the design is a node When a value of 1/2 or more of the distance between them is entered as a radius r from a keyboard or the like, the CPU moves to the next step SP4,
Node to specify plane π ₁ Wait for the specified input.

In this state, the designer sets the nodes in the same way as in step SP2. If you specify and input Captures coordinate data for, three nodes Using these coordinate data, these three nodes Calculating a plane [pi ₁ comprising, display it on Deisupurei, proceeds to the following step SP5.

In this step SP5, the CPU executes the above-mentioned step SP4.
Normal vector of the specified plane π _{1 found} at And calculate this as a node And display it on the display.

Subsequently, the CPU proceeds to the next step SP6, where the two nodes specified and input are set. Using these coordinate data, these two nodes Midpoint of Is calculated, displayed on the display, and proceeds to the next step SP7.

In this step SP7, the CPU Unit vector of the straight line going And calculate this as a node On the display, and the midpoint determined in step SP6 above. And this unit vector Is calculated as a normal vector, and this is displayed on the display as a plane π ₂ as shown in FIG.

Thereafter, the CPU executes the following equation in the next step SP8. Normal vector based on And unit vector Cross product vector Is calculated.

The outer product vector calculated in this way Is the intersection of the specified and input plane π ₁ and the plane π ₂ calculated in step SP7 described above. The direction of is shown.

Subsequently, in the next step SP9, the CPU obtains the following equation. Of the planes π ₁ and π ₂ Next formula For example, a node represented by the vector equation of Any point separated by radius r specified and input from Consists of Is calculated and displayed on the display as shown in FIG.

In practice, equation (14) uses the inner product to Can be represented by Consists of Is Needs to be satisfied. Therefore, substituting equation (13) into equation (15), It can be seen that the solution of the vector equation represented by Therefore, the left side of equation (17) is (17) can be replaced by Can be represented by a quadratic equation for the scalar quantity S, and the left side is If the equation is replaced as follows, then the solution of equation (21) is Can be calculated as represented by
But a node By selecting a value that is at least 1/2 times the distance between, it is possible to always obtain one or more real solutions, and thus, You can ask.

Subsequently, the CPU calculated as described above, Arc free curve to find Waits for the designer to enter the setting.

Here the designer moves the cursor with the mouse, For example, If you click the mouse on the above, the CPU will go to the next step SP11
Then, as shown in FIG. As the other intersection using the arc approximation technique Arc-shaped free curve bulging to the side Create

In the case of this embodiment, the CPU first specifies and inputs in the above-described step SP10. First unit vector of a straight line going to On the basis of, Unit vector of the first tangent of the direction , And The second unit vector of the line going to A node based on Intersection at Unit vector of the second tangent of the direction Is calculated.

Then the CPU A straight line r ₁ toward the Arcuate free curves by connexion formed in a linear r ₂ toward the Using the fan angle θ of The first and second tangent vectors are obtained using the scalar quantity m calculated by , Which gives two nodes Two control points between Set.

In this way, as described above with respect to the expressions (6) to (11), the CPU Two control points set as described above And the two nodes If a free curve is set between them, it is designated and input in a three-dimensional space.
Two nodes On the plane π ₁ containing , And an arc-shaped free curve with a specified radius r Is displayed on the display, and the process proceeds to the next step SP12 to terminate the free-form surface creation processing program SP1.

Thus, in the designer, two nodes specified in three-dimensional space On the plane π ₁ containing Arc free curve that passes through and has the specified radius r Can be visually checked to see whether or not the free curve imaged by the user has been created.

According to the above method, the first plane π _{1 in} the three-dimensional space
And two nodes on this first plane π ₁ And two nodes Intersecting line with the second plane π ₂ orthogonal to the straight line connecting between And this intersection line And two nodes A spherical surface with a predetermined radius r generated with any of Two intersections with The center point of either , A free curve is generated by arc approximation, so that two nodes in a three-dimensional space And its two nodes On a first plane π ₁ containing a circular arc of a given radius r Can be realized.

H Effects of the Invention As described above, according to the present invention, an arc-shaped free curve passing through the two nodes and having the specified radius on a plane including the two nodes specified and input in the three-dimensional space. Can be realized, and thus the efficiency of the designer's design work can be significantly improved.

[Brief description of the drawings]

FIG. 1 is a flowchart showing a processing procedure of a free curve creating method according to an embodiment of the present invention, FIG. 2 is a schematic diagram used to explain the principle of the free curve, and FIG. 3 is a schematic diagram showing an arc-shaped free curve. 4 to 7 are schematic diagrams for explaining the processing procedure of FIG. 1, and FIG. 8 is a schematic diagram for explaining a free-form surface.

Claims (1)

(57) [Claims] An arc-shaped free curve having a predetermined radius passing through said two nodes on a first plane including two nodes specified in a three-dimensional space is formed by a computer. In the method for creating a free curve generated by a processing device, the central processing device calculates a second plane including a midpoint between the two nodes and orthogonal to a straight line connecting the two nodes. A processing step of generating a spherical surface having the predetermined radius with one of the two nodes as a center point; and two intersections of an intersection line of the first and second planes and the spherical surface. And calculating an arc-shaped free curve of the predetermined radius passing through the two nodes on the first plane by arc approximation with one of the two intersections as a center point. Processing steps and Free curve creation method which is characterized in that run.