JP2001022959A - Method and device for modeling - Google Patents

Abstract

PROBLEM TO BE SOLVED: To detect and express a branching point and a coupling point in the case of branching or coupling at points more than two by finding phase information on the basis of a change in the number of contour lines of an object provided by cutting the object with a plane vertical to a specified height direction at various positions along with the height direction. SOLUTION: Polygon data PD are read and a Morse function is determined. Next, contour lines are found and corresponding to a change in the number of contour lines, a top point, a branching point, a coupling point and a bottom point are found. The coordinate transformation of the polygon data PD is conducted. In this coordinate transformation, the coordinate values of the polygon data are transformed so that a determined height direction HT coincides with the axial direction of an object Q. In this case, the height direction HD is matched with a z-axis direction. Thus, the specified direction concerning the object Q is defined as height direction HT, and the phase information is found on the basis of a change in the number of contour lines of the object provided by cutting the object with the plane vertical to the height direction HT at various positions along the height direction HT.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to a method and an apparatus for modeling a three-dimensional shape of an object.

[0002]

2. Description of the Related Art Conventionally, a non-contact type three-dimensional input device has been used to input a three-dimensional shape (three-dimensional data) of an object. The three-dimensional shape obtained by the three-dimensional input device is usually represented by a polygon.

When a three-dimensional shape is described by polygons, if the shape is to be described in detail, the data amount becomes enormous. As a result, the memory capacity increases and the processing speed decreases, so that handling is inconvenient.

As a modeling technique with a reduced data amount, Japanese Patent Application Laid-Open Nos.
There is a modeling method disclosed in Japanese Patent No. 31474.
According to this method, modeling of a three-dimensional shape of an object is performed using topological information and geometric information on the object, that is, by expressing topological data. In a typical model based on this, an object is represented by a geometric singular point (such as a bifurcation point) when the object is scanned in a certain direction and a contour by a cross section at each scanning position. This topological data is called homotopy model data.

That is, according to the homotopy model data, since an object is represented by information on a singular point in the object and data of a contour at a predetermined position, the data amount is significantly reduced as compared with the case where polygons are used.

FIG. 20 is a flowchart showing a conventional procedure for converting polygon data into homotopy model data. In FIG. 20, first, polygon data is read (# 801). Determine Morse function (# 8
02). This will generally determine the height direction. Next, a singular point is detected (# 803).

A singular point is to look at a side (edge) connected to a vertex of each polygon clockwise (or clockwise) and determine whether the side is rising or falling from the vertex of the polygon. Is determined by Here, a rising case is represented by "+", and a falling case is represented by "-". Then, when all sides are "-", the vertices of the polygon are regarded as the local maximum points. All sides are "+"
In the case of, the vertex of the polygon is regarded as a minimum point. In addition, when the consecutive “+” or “−” side group is represented by one “+” or “1”, all the sides are represented by “− ++ −” or “+ − + −”. Since the vertex of the polygon is a saddle point, this is set as a singular point.

Further, in order to remove the topological uncertainty, connection information of a singular point is obtained (# 804). This is determined by tracing the sides and determining whether there is a side of the polygon connecting the singular points. After obtaining the phase information of the object in this way, the shape information is obtained. At the height position including the singular point, cross the polygon with a plane perpendicular to the height direction,
The shape of the contour of the object obtained by the cross section is obtained, and connection information between the contours is obtained by following the sides of the polygon (# 805).

The topology and the topology model to which the geometric information is added are described in "Basics of Morse Theory" (written by Yukio Matsumoto, Iwanami Lecture, Basic Mathematics 8, Iwanami Shoten, 19)
1997) and "Surface coding b
used on Morsetheory "(Y. Sh
inagawa, Y .; L. Kergosien and
T. L. Kunii, IEEE Computer
Graphics and Applications,
11 (5), pp68-78, September19
91).

[0010]

As described above, a method for converting an object already represented by a polygon into a topological model is disclosed in the official gazette. That is, in order to detect a branch point, an edge connected to a vertex of a polygon is traced, and a vertex in which an ascending edge group and a descending edge group alternately appear twice each is regarded as a singular point (branch point). is there.

In the flowchart shown in FIG. 20, in step # 803, a singular point is defined based on a topological concept, and the singular point is detected by a method of applying it to a discretized value.

However, according to the method described in the above publication, a branch point such as a three-pronged or four-pronged point cannot be detected. That is, if one part of the object branches into two parts, or if the two parts combine into one part,
Although they can be expressed as branch points or connection points, the case where they branch into three or more parts and the case where three or more parts are connected are not conventionally defined. Therefore, it is not possible to detect branches and joins of three or more parts due to the limitations of the algorithm, and there is no way to express them.

SUMMARY OF THE INVENTION The present invention has been made in view of the above-described problems, and has been made in consideration of the above-described problems in the case where an object is topologically modeled when it branches into three or more parts and when three or more parts are combined. It is an object of the present invention to be able to detect and represent even those branch points and connection points.

[0014]

According to a first aspect of the present invention, there is provided a method for modeling a three-dimensional shape of an object using topological information and geometrical information on the object.
The specific direction for the object is a height direction, and a cross section is taken on a plane perpendicular to the height direction at different positions along the height direction, based on a change in the number of contours of the object obtained by the cross section. The phase information is obtained.

In the method according to the second aspect of the present invention, when a position where the number of the contour lines increases is detected, the position is set as a branch point. In the method according to the third aspect of the present invention, when a position where the number of the contour lines decreases is detected, the position is set as a connection point.

According to a fourth aspect of the present invention, when a cross section is taken at a position in a certain height direction, the distance in the height direction among the vertices of the polygon patch connected to the contour line obtained by the cross section. Let the position of the farthest vertex be the position in the height direction for the next section.

According to a fifth aspect of the present invention, there is provided an apparatus for modeling a three-dimensional shape of an object by using phase information and geometric information of the object, wherein a specific direction of the object is set in a height direction. Means for determining as, and a means for cross-section at a plane perpendicular to the height direction at different positions along the height direction, and for obtaining the phase information based on a change in the number of contours of the object obtained by the cross-section , Has.

[0018]

FIG. 1 is a block diagram showing a configuration of a three-dimensional data processing apparatus 1 for implementing a method according to the present invention.

The three-dimensional data processing device 1 includes a CPU 10 and the like for executing arithmetic processing, a hard disk device 11 for storing data and programs, a memory 12 for temporarily storing data for arithmetic operations, A display device 13 for displaying the input data and calculation results, a mouse 14 used for designating and inputting a position on the display surface HG of the display device 13, and other auxiliary devices 15. Such a three-dimensional data processing device 1 is realized by, for example, a personal computer or a workstation.

The three-dimensional data processing device 1 receives polygon data PD as three-dimensional data from outside. If the polygon data PD is stored in a storage medium (recording medium) such as a floppy disk, the attached device 15
The reading is performed by a storage medium drive such as a floppy (registered trademark) disk drive.
The read data is temporarily stored in the memory 12 or stored in the hard disk device 11. Also,
When the polygon data PD is stored in an external device other than the three-dimensional data processing device 1, the polygon data PD is connected to the external device via an external device connecting device, which is one of the attached devices 15, and is appropriately connected therefrom. The data is read out and stored in the memory 12
Is temporarily stored in the hard disk drive 11.

The polygon data PD input to the three-dimensional data processing device 1 is converted into homotopy model data YD and YDA, which are three-dimensional shape-added phase data, and output from the three-dimensional data processing device 1 to the outside. The program for such conversion processing is stored in the hard disk drive 1
1, stored in a floppy disk, a magneto-optical disk, or another recording medium, loaded into the memory 12 as appropriate, and executed by the processing device 10. [Homotopy Model Data] Next, the homotopy model data YD will be described.

The homotopy model data YD expresses a three-dimensional shape of an object by adding geometric information such as a cross-section contour line to three-dimensional phase information. The homotopy model data YD itself is described in the document “Surface coding” described in the section of the related art.
based on Morse theory ", and an application example as a modeling system is shown in Japanese Patent Application Laid-Open No. Hei 10-31472. The basic part of the data handled in this embodiment is the same as the data format shown in these known documents.

In the data format of the homotopy model data YD, a three-dimensional surface is regarded as a two-dimensional manifold, and an object is represented by a singular point of a mapping (referred to as a Morse function) from a three-dimensional point to a certain height and its connection relation. I do. In the case of a two-dimensional manifold, a point on a curved surface is represented by a two-dimensional local coordinate system, and a point at which the differential coefficient of the mapping at that point becomes zero in the two-dimensional direction is called a singular point.

Here, the local coordinate system around an arbitrary point P0 is (s, t), and the height function is f (s, t).
In this case, the necessary and sufficient condition that the point P0 is a singular point is expressed by the following equation (1).

[0025]

(Equation 1)

Is satisfied, and from the stability of the singularity,
The following equation (2) which is a Hesse matrix:

[0027]

(Equation 2)

That is, the matrix value becomes as shown in the following equation (3).

[0029]

(Equation 3)

With respect to a singular point, the eigenvalue of the Hessian matrix is determined, and when the number of negative eigenvalues (called an exponent) is 2, the singular point becomes a local maximum. When the index is 1, the singular point is a saddle point, and when the index is 0, the singular point is a minimum value. When a three-dimensional object is scanned from top to bottom in the height direction, the maximum value is the top point of the object portion, the saddle point is the branch point of the object portion or the junction point of the two object portions, and the minimum value is the It will be the valley bottom point.

By this, it is possible to represent the "structure" of the object, but this alone does not represent the shape. Therefore, the three-dimensional shape of the object is expressed by specifying the contour of the cross section at each height position of the object. further,
Since the shape is not specified between the contour lines, specify multiple curves called auxiliary lines between adjacent contour lines,
Thus, the shape between the contour lines is determined.

Summarizing these, the data necessary to represent the three-dimensional shape is as follows. Height direction that is a Morse function Singular point in height direction Connection information of singular point Contour line of cross section at each position in height direction Plural auxiliary lines connecting adjacent contour lines See FIGS. 2 to 6 below. This will be specifically described.

FIG. 2 is a diagram showing an example of the height direction of a "pot" which is an example of a three-dimensional object, and FIG. 3 is a diagram showing singular points n1 to n6 of the pot and their connection relations by edges e1 to e6. FIG. 4 shows cut outlines c1 to c9 of the pot.
FIG. 5 is a diagram showing auxiliary lines s1 to s7 for pots, and FIG. 6 is a diagram showing the relationship between conventional polygon data PD and homotopy model data YD.

For the object Q shown in FIG. 2, the direction indicated by the arrow is the height direction HT. In this case, the height direction HT
Coincides with the height direction of the object Q along the vertical line when the object Q is placed along its normal usage.

As shown in FIG. 3, each singular point n1 of the object Q
To n6 are the top point [or the start point (n1, n2)], the branch point (n3), and the connection point (n
4, n5) and the valley bottom point [or end point (n6)].

The edges e1 to e6 are connection information indicating how the singular points n1 to n6 are connected, and sequentially connect the edges (sides) of the polygons in the polygon data PD.

As shown in FIG. 4, the contour lines c1 to c9 are
It shows the contour of the cross section at each position in the height direction HT including the singular point. By the contour lines c1 to c9, the object Q
Is expressed. The contour lines c1 to c9 can be designated as needed, but the contour lines c5 and c5 at the height positions where the singular points n3 to n5 excluding the top point and the valley bottom point exist.
c2 and c6, c3 and c12 must always be specified. Note that singular points n1, n, which are the top point and the bottom point,
The contour line at 2, n6 is regarded as a point.

As shown in FIG. 5, the auxiliary lines s1 to s7 are
It is a curve connecting adjacent contour lines. The contour line and the auxiliary line may be represented in any manner, but N has a high degree of freedom in expression.
URBS (Non Uniformed Ratio)
al B-Spline) curve is suitable.

In this way, the three-dimensional object Q can be represented by the homotopy model data YD. However, depending on the direction of the height direction HT, the singular point and the contour Etc. are very different.

In the conventional method of converting to homotopy model data YD, a singular point is obtained topologically using the above-described equations (1) to (3), so that three or more singular points are obtained. A singular point could not be detected when branching to a part or when three or more parts were combined.

In the method according to the present embodiment described below,
In order to improve the conventional disadvantage, instead of using the singular point of the topological concept, it is changed to a top point, a branch point, a joint point, and a valley point of a part (object part) of the object Q defined below. Then, a similar model expression is performed. Where the top point is
The point where the object part first appears along the scanning direction,
A bifurcation point is a point where an object part branches into a plurality of parts,
A connection point is a point where a plurality of object parts are connected, and a valley bottom point is defined as a point at which no object part appears thereafter along the scanning direction.

In this manner, the phase information is changed from the singular point of the conventional topological concept to the top point, the branch point, the connection point, and the valley point in the present embodiment. When the homotopy model data YD in the present embodiment is particularly distinguished from the conventional homotopy model data YD, it is described as “homotopy model data YDA”.

Empirically, for a correctly connected polygon, the representation by the phase information of the homotopy model data YDA is the same as that of a conventional polygon except for branching into three or more object parts and coupling of three or more object parts. Has the same meaning as the singularity of the topological concept of [Homotopy model data YDA in the present embodiment]
When converting the polygon data PD into the homotopy model data YDA, the specific direction of the object Q is set to the height direction HT, and a plane perpendicular to the height direction HT (a height plane) at a different position along the height direction HT. ), And obtains phase information based on a change in the number of contour lines of the object Q obtained by the cross section. The phase information is the peak point, the branch point, the joint point, and the valley point as described above.

Next, a description will be given with reference to flowcharts shown in FIGS. 7 and 8 and FIGS. 9 to 19. 7 and 8 are flowcharts showing a procedure for converting polygon data to homotopy model data YDA according to the present embodiment, FIG. 9 is a diagram showing a polygon with respect to one contour line CLi and its relationship with vertices, and FIG. FIG. 11 is a diagram showing an example of a junction point at a fork; FIG. 11 is a diagram showing an example of a branch point;
Is a diagram showing an example of a top point, and FIG. 13 is a diagram showing an example of a contour line. FIG. 14 is a diagram showing an example of an object Q1 including a trifurcated branch point, and FIGS. 15 to 19 show respective height positions P of the object Q1.
It is a figure which shows the outline in 1-7.

In FIG. 7, first, the polygon data PD
Is read (# 11), and a Morse function is determined (# 1).
2). These are the same as the conventional processing. Next, a contour line is determined, and a top point, a branch point, a joint point, and a valley point are determined according to a change in the number of the contour lines (# 13). This processing is realized by repeatedly performing up to a height position where the object Q does not exist. In addition, the connection relation of the contour lines is obtained at the same time.

In FIG. 8, first, the polygon data PD
Is performed (# 101). The coordinate transformation here is
The coordinate values of the polygon data are converted such that the height direction HT determined in step # 12 coincides with a certain axis direction of the object Q. In the present embodiment, coordinate conversion is performed so that the height direction HT matches the z-axis direction. This facilitates later calculations.

Next, a range of heights to be considered for the object Q, that is, a maximum height and a minimum height at which data exists for the object Q is obtained (# 102). This is obtained by finding the maximum and minimum z values of the vertices of all the patches that make up the polygon.

Then, a contour line at the maximum height is obtained (# 103). The contour line can be obtained by obtaining the intersections between all the patches and the height plane. The contour at the maximum height is usually represented by a point.

Next, a height position for obtaining the next contour is calculated (# 104). Here, step # 103
The minimum height of the vertices of the patch that intersects with the height plane is obtained for each of the contour lines obtained in step (1), and the maximum value of the minimum height is obtained for all the contour lines. The next height position Hnext is expressed by the following equation (4).

[0050]

(Equation 4)

Where num-countour is the number of contours, num-countour
patch (i) is the number of patches including the intersection forming the i-th contour, num-edge (j) is the number of vertices of the j-th patch, ed
ge-z (k) indicates the Z coordinate value of the k-th vertex.

That is, the i-th contour line CL shown in FIG.
In i, first, the minimum value of the Z coordinate values of all vertices included in a certain patch is obtained. Next, the minimum value of the Z coordinate value in all the patches included in the i-th contour line CLi is obtained. And
All contour lines C including other contour lines CL not shown in FIG.
The maximum value among L1 to CLnum-countour is obtained. This is defined as the next height position Hnext.

By using the height position Hnext obtained in this way, it is possible to recognize how the contour line CL obtained this time and the contour line CL obtained in the next cutting (section) are connected. Is done.

Next, it is determined whether the next height position Hnext is the same as the current height position (# 105). If they are the same (yes in # 105), one of the contours becomes a valley bottom point, indicating that there is no patch below this for a series of contours connected to this contour. Therefore, a valley bottom point is set for the contour line giving the height position Hnext (# 106), and a series of contour lines connected to this contour line is regarded as completed, and is excluded from subsequent calculations (# 107). ). It is determined whether there is a contour group that has not been completed yet (# 108). When all the contour groups have been completed, the data is arranged to generate necessary data, and then the processing is terminated.

If there is an unfinished outline group, the height position Hnext for the next cutting is re-determined except for the completed outline group, and the calculation is repeated (# 104). By the way, when the height position Hnext of the next cutting is different from the current height position, the intersection of the height plane and the patch is actually obtained.
An outline is created (# 109). The contour created here is obtained as a partial segment of the contour. For example, when an outline as shown in FIG. 13 is obtained, partial segments SEG1, SEG2, and SEG3 are obtained.
Further, it is calculated in advance which contour line at the height position of the previous cut is connected to this partial segment SEG. If the vertices of the two contours are included in the same patch, the two contours are connected;
Two contours can be considered as not connected.

Further, using this partial segment SEG, it is determined whether or not the height plane is a singular plane (# 110). Here, the “singular plane” refers to a height plane at which branching or joining of contour lines occurs, and can be determined as follows. That is, it is calculated whether or not any of the end points of all the obtained partial segments of the contour line matches. If present, it can be considered as a critical time, so that branching or bonding occurs at this height plane. In the case of FIG. 13, the height plane giving this contour is a “singular plane”.

When the height plane under consideration is the "singular plane", the section is cut at a height plane that is raised and lowered by a small amount from the "singular plane", and the number of contour lines and the change in their connection relation are determined. , Branch or join. When the number of contour lines on the height plane that is only a small amount below the singular plane connected to the height line that is connected to the height plane that is a minute amount above the singular plane is increased, It is considered that the bifurcation occurred at the “singular plane”. Conversely, when the number of contour lines is reduced, it is considered that the binding has occurred in the “singular plane”.

In the examples shown in FIGS. 10 and 11, the height position of the previous cut is H0, the height position of the "singular plane" is H1, the height position slightly above the height position H1 is Ha, and the height position is H. Position H
The height position below a small amount from 1 is represented by Hb.

As described above, when a branch or a connection occurs, a branch point or a connection point is set on the "singular plane"(# 111). If the number of contour lines on the height plane slightly above the considered height position is greater than the number of contour lines at the height position of the previous cut, it is considered as the previous height position. It can be determined that the peak point exists between the current height position (# 112).

In this case, the height position at which the peak point exists can be determined by sequentially increasing the cutting height position from the considered height position and determining the changing height position. The height position of this section takes the height position where the vertex of the patch exists in order from the height position under consideration in the upward direction. A top point is set at the height position thus obtained (# 113).

FIG. 12 shows an example of the top point. The height of the previous cut is H0, the height position under consideration is H1, the height position slightly higher than the height position H1 is Ha, and the height is higher from the height position H1 in order. The height position where the number of lines has changed is defined as Hup. At this time, it is considered that the peak point exists at the height position Htop immediately before the height position Hup.

That is, when the height position of the cutting is gradually lowered, a point (contour line) of the object portion suddenly appears at the height position Htop. Since that point has no connection with the other contour lines that existed up to that point, it can be determined that the point is the top point.

After the branch point, the connection point, and the peak point are obtained as described above, the connection relationship between the obtained contour lines is calculated (# 114), and the processing at the next height position Hnext is performed. Is repeated (# 104).

The object Q1 shown in FIG. 14 is a three-legged table, and has a branch point to a fork near the height position P3.
As shown in FIGS. 16A and 16B, the number of contour lines increases from one to three between the height position P3 and the height position P4. In the meantime, by precisely obtaining the singular plane as described above, it is possible to accurately obtain the trifurcation branch point and its height position.

In FIGS. 15 to 19, the contour at the height position P1 is the “trunk” portion of the table, and the contour at the height position P2 is around the “trunk” from the “leg”. Is shown. At the height position P2, there is still one contour line. The height position P3 is the height position at the moment when the “leg” is divided into three parts. The outline of the height position P4 shows the state after the “leg” is divided into three parts. The height position P5 is
This is the height position at which a new outline has appeared due to the “toe” of the “leg”. The height position P6 is a height position at the moment when the outline of the “leg” and the outline of the “toe” are combined. In this case, each of the two outlines becomes one. The height position P7 indicates a contour after the “leg” and the “toe” are combined. Thereafter, no topological change occurs, and the contour ends at the “valley bottom point”.

As described above, by performing conversion processing to homotopy model data YDA, phase information equivalent to a singular point based on a topological concept can be extracted, and further, topological information can be extracted. Branching and joining that cannot be handled by the extraction method based on the general concept,
It can be accurately extracted from the polygon data PD.

In this embodiment, when extracting the phase information, first, a contour line is obtained at a certain height position. The contour is a point because it initially appears at the top point. One outline is connected at one point, and it branches into two or three. That point is the branch point. Conversely, a plurality of contour lines are connected at one point at a certain height position, and this is a complete one contour line. In that case, that point is the connection point.

In order to change the height position of the cut, it is moved by a very small amount where necessary, that is, at a place such as a branch point or a joint point. Where they are not needed, move them roughly. Even in such a case, it is necessary to obtain information as to which contour line at the previous height is connected to the contour line at a certain height position. To obtain this information, the edge of the polygon in the polygon data PD
Since it is determined whether or not the outlines are connected to each other, basically, a small amount is moved so that the connected state can be determined by the polygon. However, when there is only one contour line and it can be determined that the current contour line and the next contour line are connected, the height position can be largely changed.

As described above, according to the present embodiment, the object Q
In the case of topologically modeling, when branching into three or more parts and when three or more parts combine, it is also possible to detect and represent those branch points and connection points. . Further, by obtaining the height position Hnext of the next cut for obtaining the contour line as in the above-described equation (4), the information indicating the connection between the contour lines can be obtained, and as few contour lines as possible can be obtained. Can be represented by

[0070]

According to the present invention, when an object is topologically modeled, when the object is branched into three or more parts and when three or more parts are combined, the branch points and the joint points are determined. Can also be detected and expressed.

[Brief description of the drawings]

FIG. 1 is a block diagram showing a configuration of a three-dimensional data processing device for implementing a method according to the present invention.

FIG. 2 is a diagram illustrating an example of a height direction of a “pot” which is an example of a three-dimensional object.

FIG. 3 is a diagram showing singular points of a pot and their connection relationship by edges.

FIG. 4 is a view showing a cutting contour line for a pot.

FIG. 5 is a diagram showing an auxiliary line for a pot.

FIG. 6 is a diagram showing the relationship between conventional polygon data and homotopy model data.

FIG. 7 is a flowchart illustrating a procedure for converting polygon data to homotopy model data according to the embodiment.

FIG. 8 is a flowchart illustrating a procedure for converting polygon data to homotopy model data according to the present embodiment.

FIG. 9 is a diagram showing the relationship between a polygon and its vertices for one outline.

FIG. 10 is a diagram illustrating an example of a connection point at a fork.

FIG. 11 is a diagram illustrating an example of a branch point.

FIG. 12 is a diagram showing an example of a top point.

FIG. 13 is a diagram illustrating an example of a contour line.

FIG. 14 is a diagram illustrating an example of an object including a trifurcated branch point.

FIG. 15 is a diagram showing a contour line at each height position of the object.

FIG. 16 is a diagram showing a contour line at each height position of the object.

FIG. 17 is a diagram showing a contour line at each height position of an object.

FIG. 18 is a diagram showing a contour line at each height position of the object.

FIG. 19 is a diagram showing a contour line at each height position of the object.

FIG. 20 is a flowchart showing a conventional procedure for converting polygon data into homotopy model data.

[Explanation of symbols]

Reference Signs List 1 3D data input device (modeling device) 10 Processing device (means for determining, means for obtaining phase information) 11 Hard disk device 12 Memory Q, Q1 Object HT Height direction c1 to c9, CL Contour line

 ──────────────────────────────────────────────────続 き Continued on the front page F term (reference) 5B046 CA04 DA10 EA00 FA12 FA18 GA01 5B050 BA09 CA04 CA07 EA05 EA06 EA07 EA28 FA02 FA06 5B080 AA03 AA05 AA10 AA13 BA01 GA02

Claims (5)

[Claims] 1. A method for modeling a three-dimensional shape of an object using topological information and geometric information of the object, wherein a specific direction of the object is a height direction, and At different positions in a plane perpendicular to the height direction, and calculating the phase information based on a change in the number of contours of the object obtained by the cross section. 2. The modeling method according to claim 1, wherein when a position where the number of contour lines increases is detected, the position is set as a branch point. 3. The modeling method according to claim 1, wherein when a position where the number of contour lines decreases is detected, the position is set as a connection point. 4. When a cross section is taken at a position in a certain height direction, the position of the vertex having the longest distance in the height direction among the vertices of the polygon patch connected to the contour obtained by the cross section is set as the following. The modeling method according to claim 1, wherein the position is a position in a height direction for a cross section of the object. 5. An apparatus for modeling a three-dimensional shape of an object using phase information and geometric information about the object, wherein: a means for determining a specific direction about the object as a height direction; Means for cross-section in a plane perpendicular to the height direction at different positions along the height direction, and means for obtaining the phase information based on a change in the number of contour lines of the object obtained by the cross-section. Characteristic modeling equipment.