1
DESCRIPTION
CURVED-SURFACE GENERATION METHOD, PROGRAM, AND 3-DIMENSIONAL
SHAPE PROCESSING DEVICE
Technical Field
[0001]
The present invention relates to a curved-surface
generating method for generating a curved surface from a group
of points formed of a plurality of measurement points obtained
as the results of shape measurement of an object.
Background Art
[0002]
Conventionally, when reproducing, in parameter space, a
three-dimensional shape model for use in CAD (Computer Aided
Design) or CAM (Computer Aided Manufacturing) from the results
of shape measurement of an object, a known technique is to
create polygonal surfaces, such as triangles etc., by
connecting each of a group of points with straight lines.
For example, Japanese Unexamined Patent Application,
Publication No. 2003-346182 discloses a three-dimensional
polygon data creating method which can obtain polygon data for
an object surface with superior performance and stability from
three-dimensional group-of-points data which are the results

2
of shape measurement of an object, using a matching triangles
method.
Patent Document 1:
Japanese Unexamined Patent Application, Publication No.
2003-346182 (pages 2 to 8, and Figs. 1 and 4).
Disclosure of Invention
[0003]
However, in the invention of Patent Document 1 described
above, because the shape is defined with polygonal
approximations, there are some approximation errors for
undevelopable curved surfaces among curved surfaces.
Conventionally, to reduce these approximation errors, a
technique for creating polygons as finely as possible to
achieve precise approximation is used. However, with this
technique, the amount of data becomes enormous, resulting in
the problem that, in practice, the amount of required computer
memory increases.
Furthermore, with the conventional technique, because
surface normals are determined from polygons, it is not
possible to obtain accurate normals according to the curvature
of the curved surface. Therefore, there is a problem in that
an error is included in the reproduced curved surface, and it
is not possible to generate a curved surface with good
precision.

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[0004]
The present invention has been conceived to solve the
problems described above, and an object thereof is to provide
a curved-surface generating method and program and a three-
dimensional shape processing apparatus which can reduce the
amount of data and generate a curved surface with high
precision.
[0005]
In order to solve the problems described above, the
present invention employs the following solutions.
A first aspect of the present invention is a curved-
surface generating method for generating a curved surface from
a group of points formed of a plurality of measurement points
obtained as results of shape measurement of an object, the
curved-surface generating method including a representative-
point selection step of selecting a plurality of
representative points from the group of points; a principal-
curvature calculating step of calculating respective principal
curvatures at each of the representative points on the basis
of positional relationships between each representative point
and a plurality of the measurement points existing around this
representative point; a line-of-curvature creating step of
creating a line of curvature on the basis of the principal
curvatures at each representative point; and a curved-surface
generating step of generating a curved surface by using the

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line of curvature.
[0006]
With this curved-surface generating method, after
selecting the plurality of representative points from the
group of points, the principal curvature at each
representative point is calculated on the basis of the
positional relationships between each representative point and
the points existing around it, and the line of curvature is
created from these principal curvatures. Then, by using this
line of curvature, the curved surface is reproduced with a
curve-surface reproduction technique or the like. In this
way, the plurality of points around each representative point
are taken as auxiliary points for obtaining the principal
curvatures. Therefore, the reproduced curved surface can be
represented using only the representative points.
Accordingly, although the curved surface can be reproduced
with an extremely small group of points, because the curved
surface is reproduced using information about all points, it
is possible to reproduce the curved surface with high
precision.
[0007]
In the curved-surface generating method described above,
the principal-curvature calculating step may include a step of
setting a normal at each representative point; a step of
generating element vectors by joining each representative

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point and each points existing around this representative
point; and a step of determining the principal curvatures on
the basis of relationships between the normal and the element
vectors.
[0008]
With this method, a normal is set at each representative
point and the same number of element vectors as the number of
points existing around the representative point are generated
by joining each representative point and the points existing
around this representative point. Therefore, it is possible
to obtain the principal curvatures with a simple technique on
the basis of the relationships between the normal of the
representative point and each element vector generated around
the representative point.
[0009]
In the curved-surface generating method described above,
the principal-curvature calculating step may include a step of
setting a normal at each representative point; a tangential-
vector setting step of setting a tangential vector that is
orthogonal to the normal; a step of generating element vectors
by joining each representative point and each point existing
around this representative point; a step of calculating angles
formed by the tangential vector and the element vectors around
the normal; a step of calculating angles formed by a
tangential plane including the tangential vector and planes

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including the element vectors and defining the angles as
curvatures; a step of creating angle-curvature tables by
plotting the calculation results on a horizontal axis in
angle-curvature tables; and a step of obtaining the principal
curvatures on the basis of the curvature tables.
[0010]
In this way, normal is set at the representative point
and tangential vector which is orthogonal to the normal is
set. In addition, by joining the representative point and
each point existing around the representative point, the same
number of element vectors as the number of points existing
around the representative point is generated. Then, the
angles that the tangential vector and the element vectors form
around the normal are calculated, the angles formed by the
tangential plane including the tangential vector and the
planes including the element vectors are calculated for each
element vector, and these calculation results are plotted in
angle-curvature tables in which angle is shown on the
horizontal axis and curvature is shown on the vertical axis to
create angle-curvature tables. Thus, by obtaining the maximum
curvature and the minimum curvature in these angle-curvature
tables, it is possible to easily obtain the principal
curvatures.
[0011]
In the curved-surface generating method described above,

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the principal-curvature calculating step may include a step of
setting a normal at each representative point; a tangential-
vector setting step of setting a tangential vector that is
orthogonal to the normal; a step of generating element vectors
by joining each representative point and each point existing
around this representative point; a step of calculating angles
formed by the tangential vector and the element vectors around
the normal; a step of calculating angles formed by a
tangential plane including the tangential vector and planes
including the element vectors and defining the angles as
curvatures; a step of creating angle-curvature tables by
plotting the calculation results on a horizontal axis in
angle-curvature tables; an extraction step of extracting only
a fundamental frequency from the curvature tables; and a
principal-curvature obtaining step of obtaining the principal
curvatures from the angle-curvature table on the basis of the
fundamental frequency.
[0012]
In this way, a normal is set at each representative point
and a tangential vector which is orthogonal to the normal is
set. In addition, by joining each representative point and
each point existing around this representative point, the same
number of element vectors as the number of points existing
around the representative point is generated. Then, at each
representative point, the angles formed by the tangential

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vector and the element vectors around the normal are
calculated, the angles formed by the tangential plane
including the tangential vector and the planes including the
element vectors are calculated for each element vector, and
these calculation results are plotted in angle-curvature
tables in which angle is shown on the horizontal axis and
curvature is shown on the vertical axis to create angle-
curvature table. Then, only the fundamental frequency is
extracted from these angle-curvature tables. Extraction of
this fundamental frequency obtains the fundamental frequency
by, for example, regarding the angle-curvature table as a
time-series table and performing, for example, a fast Fourier
transform on this table. Thereafter, by performing, for
example, an inverse fast Fourier transform on this fundamental
frequency, it is possible to obtain an angle-curvature table
in which only the fundamental frequency is reflected. Because
this angle-curvature table is a high-precision table from
which noise is eliminated and accurate values are reflected,
it is possible to obtain the principal curvatures with
extremely high precision.
[0013]
In the curved-surface generating method described above,
the principal-curvature calculating step may include a step of
generating element vectors by joining each representative
point and each point existing around this representative

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point; a step of determining a group of normal vectors at each
representative point by calculating vector products of the
element vectors; and a step of determining an average vector
of the group of normal vectors and defining the average vector
as a normal vector at each of the representative points.
[0014]
With this method, by obtaining the vector products of the
plurality of element vectors generated by joining each
representative point and each of the points existing around
this representative point, the group of normal vectors at each
representative point is obtained, and the average vector of
this group of normal vectors is defined as the normal vector
of that representative point. Therefore, it is possible to
set a normal vector with higher reliability by using
information about a plurality of points. Accordingly, it is
possible to improve the precision of the curved surface
reproduction.
[0015]
A second aspect of the present invention is curved-
surface generating program for executing, on a computer
system, curved-surface generating processing for generating a
curved surface from a group of points formed of a plurality of
measurement points obtained as results of shape measurement of
an object, the curved-surface generating program including a
representative-point selection step of selecting a plurality

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of representative points from the group of points; a
principal-curvature calculation step of calculating respective
principal curvatures at each representative point on the basis
of positional relationships between each representative point
a plurality of the measurement points existing around this
representative point; a line-of-curvature creating step of
creating a line of curvature on the basis of the principal
curvature at each representative point; and a curved-surface
generating step of generating a curved surface by using the
line of curvature.
[0016]
A third aspect of the present invention is a three-
dimensional shape processing apparatus which is provided with
a curved-surface generating program and which generates a
curved surface from a group of points formed of a plurality of
measurement points obtained as results of shape measurement of
an object by executing the curved-surface generating program,
wherein the curved-surface generating program includes a
representative-point selection step of selecting a plurality
of representative points from the group of points; a
principal-curvature calculating step of calculating respective
principal curvatures at each representative point on the basis
of positional relationships between each representative point
and a plurality of the measurement points existing around this
representative point/ a line-of-curvature creating step of

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creating a line of curvature on the basis of the principal
curvature at each representative point; and a curved-surface
generating step of generating a curved surface by using the
line of curvature.
[0017]
The present invention affords an advantage in that it is
possible to generate a curved surface with high precision,
while reducing the amount of data.
Brief Description of Drawings
[0018]
[FIG. 1] Fig. 1 is a block diagram showing, in outline,
the configuration of a three-dimensional shape processing
apparatus according to a first embodiment of the present
invention.
[FIG. 2] Fig. 2 is a flowchart showing the procedure of
a curved-surface generating method according to the first
embodiment of the present invention.
[FIG. 3] Fig. 3 is a diagram for explaining
representative points.
[FIG. 4] Fig. 4 is a diagram for explaining a principal-
curvature calculation step.
[FIG. 5] Fig. 5 is a flowchart showing the procedure of
the principal-curvature calculation step according to the
first embodiment of the present invention.

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[FIG. 6] Fig. 6 is a diagram for explaining the
principal-curvature calculation step.
[FIG. 7] Fig. 7 is a diagram showing an example of an
angle-curvature table according to the first embodiment of the
present invention.
[FIG. 8] Fig. 8 is a diagram showing a fast Fourier
transformed table according to a second embodiment of the
present invention.
[FIG. 9] Fig. 9 is a diagram showing an example of an
angle-curvature table according to the second embodiment of
the present invention, in which only a fundamental frequency
is reflected.
[FIG. 10] Fig. 10 is a diagram showing an example of an
angle-curvature table depicting a line of curvature obtained
from measurement values and a line of curvature obtained on
the basis of Euler's formula.
Best Mode for Carrying Out the Invention
[0019]
An embodiment of a three-dimensional processing apparatus
for realizing a curved-surface generating method according to
the present invention will be described below with reference
to the drawings.
First Embodiment
Fig. 1 is a block diagram showing, in outline, the

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configuration of a three-dimensional shape processing
apparatus according to a first embodiment of the present
invention. As shown in Fig. 1, the three-dimensional shape
processing apparatus according to this embodiment, which is a
computer system such as a CAD (Computer Aided Design) or CAM
(Computer Aided Manufacturing) system, is formed of a CPU
(central processing unit) 1, a main storage device 2 such as a
RAM (Random Access Memory), a secondary storage device 3 such
as an HDD (Hard Disk Drive), an input device 4 such as a
keyboard or mouse, an output device such as a monitor or
printer, and so forth.
Various programs are stored in the secondary storage
device 3, and the CPU 1 realizes various types of processing
by reading programs from the secondary storage device 3 into
the main storage device 2 such as the RAM and executing them.
[0020]
Next, curved-surface generating processing (curved-
surface generating method) for generating a curved surface
from a group of points, in the three-dimensional shape
processing apparatus having the above-described configuration,
will be described with reference to the drawings. The
processing shown below is realized by, for example, the CPU 1
reading a curved-surface generating program stored in the
secondary storage device 3 into the main storage device 2 such
as the RAM and executing it.

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First, the CPU 1 acquires group-of-points data from a
plurality of measurement points obtained as a result of shape
measurement of an object. This group-of-points data may be
stored in advance in a memory, such as the secondary storage
device 3, installed in the three-dimensional shape processing
apparatus, or it may be obtained online from another external
device. In the present invention, the way in which this
group-of-points data is acquired is not particularly limited.
[0021]
When the group-of-points as described above (hereinafter
referred to as "group of points") is acquired, a plurality of
representative points are selected from the group of points
(step SA1 in Fig. 2: representative points selection step).
For example, in a group of points such as that shown in Fig.
3, a plurality of points are selected as representative points
P0. Then, based on the positional relationship between each
representative point P0 selected in step SA1 and a plurality
of points existing around this representative point P0, the
respective principal curvatures at each of the representative
points P0 are calculated (step AS2: principal-curvature
calculation step).
Details of the principal-curvature calculation step are
described below by giving as an example a group of points
belonging to an arbitrarily selected area Q from among the
group of points shown in Fig. 3.

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[0022]
First, as shown in Fig. 4, element vectors L01, L02, L03,
and L04 are generated by joining the representative point PO
to the points P1, P2, P3, and P4, respectively, which exist
around PO (step SB1 in Fig. 5).
Then, calculating the vector products of all combinations
of the element vectors L01, L02, L03, and L04 yields a group
of normal vectors (not shown in the drawing) at the
representative point PO (step SB2 in Fig. 5). Then, an
average vector of the group of normal vectors is determined,
and this average vector is defined as a normal vector n at the
representative points PO (step SB3 in Fig. 5).
[0023]
Next, principal curvatures are determined on the basis of
the respective relationships between this normal vector n and
the element vectors L01, L02, L03, and L04. Specifically, a
tangential vector t that is orthogonal to the normal n is set,
the angles which this tangential vector t and each of the
element vectors L01, L02, L03, and L04 form around the normal
n are calculated, and the angles formed by a tangential plane
including the tangential vector and planes including each of
the element vectors L01, L02, and L04 are calculated as the
curvatures (step SB4 in Fig. 5).
For example, when the element vector L02 shown in Fig. 4
has the relationship shown in Fig. 6 with respect to the

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representative point PO, an angle 02 formed by the tangential
vector t and the vector component L02(XY) in the tangential
plane of the element vector L02 is calculated, and the angle
formed by the plane including the element vector L02 and the
tangential plane including the tangential vector t is obtained
as a curvature K2.
In a similar fashion, angles 9 and curvatures K are also
calculated for point P1, P3, and P4 shown in Fig. 4.
[0024]
Once such calculations for the points existing around the
representative point PO have been completed, angle-curvature
tables are created by plotting the calculation results in
angle-curvature tables showing angle 9 on the horizontal axis
and curvature K on the vertical axis and fitting these points
using Euler's law (step SB5 in Fig. 5). As a result, an
angle-curvature table such as that shown in Fig. 7, for
example, is obtained. Points may be plotted in the angle-
curvature table in parallel with the calculation.
Then, a maximum curvature Kmax and a minimum curvature
Kmin in this angle-curvature table are obtained as the
principal curvature (step SB6 in Fig. 5).
Then, by performing the respective principal-curvature
calculation processes described above for each of the
representative points PO set in the group of points shown in
Fig. 3, the maximum curvature Kmax and the minimum curvature

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Kmin at each of the representative points PO are obtained.
[0025]
When the principal curvatures at each of the
representative points PO are obtained, a line of curvature is
created by connecting these principal curvatures (step SA3 in
Fig. 2: line-of-curvature creation step).
Then, a curved surface is generated by a curved-surface
reproduction technique using this line of curvature (step SA4
in Fig. 2: curved-surface generating step). For example,
Gaussian mapping / inverse mapping are performed to generate a
curved surface on the basis of the line of curvature.
Specifically, the curved surface is generated by interpolation
of the curved surface after performing a coordinate
transformation to a parameter space in which Euclidean
geometry is established.
[0026]
As described above, with the curved-surface generating
method according to this embodiment, after selecting a
plurality of representative points PO from the group of
points, the principal curvatures are calculated at the
representative points PO on the basis of the positional
relationships between each representative point PO and the
points existing around it, and a line of curvature is created
from these principal curvatures. Then, by using this line of
curvature, a curved surface is generated by a curved-surface

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reproduction technique or the like. In this way, a plurality
of points around each representative point PO are taken as
auxiliary points for obtaining the principal curvatures, and a
curved surface is reproduced. Therefore, the reproduced
curved surface can be represented using only the
representative points PO. Accordingly, although the curved
surface can be represented with an extremely small group of
points, because the curved surface is generated using
information about all points, it is possible to generate the
curved surface with high precision. As a result, it is
possible to reduce the amount of data, thus making it possible
to speed up the processing.
For example, if a group of points consisting of a
plurality of measurement points obtained as the results of
shape measurement of an object contains 3 million points,
when using the curved-surface generating method according to
this embodiment, it is possible to represent a curved surface
by using several thousand points.
[0027]
Second Embodiment
Next, a curved-surface generating method according to a
second embodiment of the present invention will be described
with reference to the drawings. The difference between the
curved-surface generating method according to this embodiment
and the curved-surface generating method according to the

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first embodiment is the addition of the following step in the
principal-curvature calculation step (step SA2 in Fig. 2)
according to the first embodiment described above in order to
further improve the precision of the curved-surface
generation.
That is, in the curved-surface generating method of this
embodiment, a step of extracting only a fundamental frequency
from the angle-curvature table after creating the angle-
curvature table (step SB5) and a step of recreating the angle-
curvature table for this fundamental frequency are added to
the detailed procedure of the principal-curvature calculating
step according to the first embodiment shown in Fig. 4, and
the principal curvature is obtained from this recreated angle-
curvature table.
[0028]
The step of extracting only the fundamental frequency
from the angle-curvature table can be realized by the
following technique, for example.
Frequency components (spectrum) are obtained by fast
Fourier transforming (FFT) the angle-curvature table, and in
the fast Fourier transformed table, the frequency at which the
spectrum shows a maximum value is extracted as the fundamental
frequency. An example of the fast Fourier transformed table
is shown in Fig. 8. In this fast Fourier transformed table,
the horizontal axis is frequency and the vertical axis is the

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amplitude spectrum.
Then, by inverse fast Fourier transforming (IFFT) the
fundamental frequency component extracted in this way, it is
possible to obtain an angle-curvature table in which only the
fundamental frequency component is reflected. An example of
an angle-frequency table in which only the fundamental
frequency component is reflected is shown in Fig. 9. As is
clear from this figure, noise is removed from the angle-
curvature table according to this embodiment, and it has
higher precision compared to the angle-curvature table
according to the first embodiment shown in Fig. 7.
Then, by obtaining the maximum curvature Kmax and the
minimum curvature Kmin from this angle-curvature table, it is
possible to obtain the principal curvature with an extremely
small error.
[0029]
As described above, with the curved-surface generating
method according to this embodiment, because the principal
curvature is obtained from the angle-curvature table with
extremely small error, it is possible to generate a curved
surface with extremely high precision. Accordingly, a curved
surface generated from a group of points can be assumed to be
a smooth surface.
In the embodiment described above, extraction of the
fundamental frequency is achieved using a fast Fourier

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transform (FFT), but it is not limited to this method. For
example, it is also possible to use the MEM (Maximum Entropy
Method), the BT method (Blackman-Tukey Method), a wavelet
method, and so on.
[0030]
In the embodiment described above, in step SB3 in Fig. 5,
the average vector of the group of normal vectors is defined
as the normal vector. As described below, however, this
average vector may be further corrected and the corrected
average vector may be defined as the normal vector.
First, in the embodiment described above, the tangential
vector which is orthogonal to the average vector is set, and
the angle-curvature table as shown in Fig. 7 is created from
the respective positional relationships between this
tangential vector and each of the element vectors L01, L02,
L03, and L04. Here, when the average vector described above
is not actually a tangential vector, the curve depicted in the
angle-curvature table is not an exact cosine wave based on
Euler's formula, as shown by the dotted line is Fig. 10, nor a
cosine wave but is a curve such as that shown by the solid
line in Fig. 10, in other words, a curve whose phase is
shifted from the curve based on Euler's formula.
Thus, while finely swinging the average vector to an
arbitrary angle to eliminate this error, more concretely, by
changing the orientation or direction angle of the average

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vector shown in Fig. 6 by a small amount at a time, the curve
depicted in the angle-curvature table (the solid line in Fig.
10) is corrected, and the average vector when this curve is
substantially aligned with the curve based on Euler's formula
(the dotted line in Fig. 10) is defined as the normal vector.
With this approach, because it is possible to determine
the normal vector based on Euler's formula, it is possible to
increase the precision.
[0031]
Although the embodiments of the present invention have
been described above with reference to the drawings, the
actual configuration is not limited to these embodiments.
Various modifications are possible so long as they do not
depart from the spirit of the invention.

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CLAIMS
1. A curved surface generating method for generating a curved
surface from a group of points formed of a plurality of
measurement points obtained as results of shape measurement of
an object, comprising:
a representative-point selection step of selecting a
plurality of representative points from the group of points;
a principal-curvature calculating step of calculating
respective principal curvatures at each of the representative
points on the basis of positional relationships between each
representative point and a plurality of the measurement points
existing around this representative point;
a line-of-curvature creating step of creating a line of
curvature on the basis of the principal curvatures at each
representative point; and
a curved-surface generating step of generating a curved
surface by using the line of curvature.
2. A curved-surface generating method according to Claim 1,
wherein the principal-curvature calculating step includes
a step of setting a normal at each representative point;
a step of generating element vectors by joining each
representative point and each points existing around this
representative point; and

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a step of determining the principal curvatures on the
basis of relationships between the normal and the element
vectors.
3. A curved-surface generating method according to Claim 1,
wherein the principal-curvature calculating step includes
a step of setting a normal at each representative point;
a tangential-vector setting step of setting a tangential
vector that is orthogonal to the normal;
a step of generating element vectors by joining each
representative point and each point existing around this
representative point;
a step of calculating angles formed by the tangential
vector and the element vectors around the normal;
a step of calculating angles formed by a tangential plane
including the tangential vector and planes including the
element vectors and defining the angles as curvatures;
a step of creating angle-curvature tables by plotting the
calculation results on a horizontal axis in angle-curvature
tables; and
a step of obtaining the principal curvatures on the basis
of the curvature tables.
4. A curved-surface generating method according to Claim 1,
wherein the principal-curvature calculating step includes

25
a step of setting a normal at each representative point;
a tangential-vector setting step of setting a tangential
vector that is orthogonal to the normal;
a step of generating element vectors by joining each
representative point and each point existing around this
representative point;
a step of calculating angles formed by the tangential
vector and the element vectors around the normal;
a step of calculating angles formed by a tangential plane
including the tangential vector and planes including the
element vectors and defining the angles as curvatures;
a step of creating angle-curvature tables by plotting the
calculation results on a horizontal axis in angle-curvature
tables;
an extraction step of extracting only a fundamental
frequency from the curvature tables; and
a principal-curvature obtaining step of obtaining the
principal curvatures from the angle-curvature table on the
basis of the fundamental frequency.
5. A curved-surface generating method according to Claim 1,
wherein the principal-curvature calculating step includes
a step of generating element vectors by joining each
representative point and each point existing around this
representative point;

26
a step of determining a group of normal vectors at each
representative point by calculating vector products of the
element vectors; and
a step of determining an average vector of the group of
normal vectors and defining the average vector as a normal
vector at each of the representative points.
6. A curved-surface generating program for executing, on a
computer system, curved-surface generating processing for
generating a curved surface from a group of points formed of a
plurality of measurement points obtained as results of shape
measurement of an object, the curved-surface generating
program comprising:
a representative-point selection step of selecting a
plurality of representative points from the group of points;
a principal-curvature calculation step of calculating
respective principal curvatures at each representative point
on the basis of positional relationships between each
representative point a plurality of the measurement points
existing around this representative point;
a line-of-curvature creating step of creating a line of
curvature on the basis of the principal curvature at each
representative point; and
a curved-surface generating step of generating a curved
surface by using the line of curvature.

27
7. A three-dimensional shape processing apparatus which is
provided with a curved-surface generating program and which
generates a curved surface from a group of points formed of a
plurality of measurement points obtained as results of shape
measurement of an object by executing the curved-surface
generating program, wherein the curved-surface generating
program includes
a representative-point selection step of selecting a
plurality of representative points from the group of points;
a principal-curvature calculating step of calculating
respective principal curvatures at each representative point
on the basis of positional relationships between each
representative point and a plurality of the measurement points
existing around this representative point;
a line-of-curvature creating step of creating a line of
curvature on the basis of the principal curvature at each
representative point; and
a curved-surface generating step of generating a curved
surface by using the line of curvature.

A plurality of representative points are selected from a
group of points obtained as results of shape measurement of an
object, respective principal curvatures are calculated at the
representative points on the basis of positional relationships
between the representative point and a plurality of points
existing around the representative point, a line of curvature
is created on the basis of the principal curvatures of the
representative points and a curved surface is generated using
this line of curvature.