JP2570954B2 - Straight line shape judgment method at the time of three-dimensional spline dimension conversion - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

The present invention relates to a component type is performed when the dimension conversion for shifting three-dimensional the three-dimensional CAD system <br/> B- spline curve data in a two-dimensional CAD cis <br/> Te arm Regarding the determination method, in particular, determining the linear shape after the dimension change 3
The present invention relates to a method for determining a linear shape at the time of dimension conversion of a dimensional spline.

[0002]

2. Description of the Related Art As shown in FIG. 10, a B-spline curve has four control points P _0, P _1, P
_2, P ₃ and eight knot vectors (t _0, t _0, t _0, t _0, t ₁
t _1, t _1, t ₁ ) is the smallest constituent unit, and the curve definition formula of this section is expressed by spline functions N ₀ (t), N ₁ (t) _, N ₂ ( t) _,
It is expressed as shown in the following equation (1) using N ₃ (t).

P ₀ N ₀ (t) + P ₁ N ₁ (t) + P ₂ N ₂ (t) + P ₃ N ₃ (t) (1) where t） [t _0, t ₁ ]

[0004] When the line segment indicated by the B-spline curve data is a straight line, it may be necessary to determine that and to obtain the coordinate values of the start point and end point of the straight line. Conventionally, whether or not a line segment indicated by B-spline curve data is a straight line is determined by determining the cross product of adjacent vectors (P _{i + 1} −P _i ) _and (P _{i + 2} −P _{i + 1} ). Is calculated, and it is determined whether or not the line is a straight line based on whether or not the component in each axis direction is 0 (for example, see Japanese Patent Application Laid-Open No.
No. 211071). At this time, the coordinate values of the start point and the end point, which are the basic data of the straight line, are the coordinate values of the start point and the end point of the control point.

[0005] Also, three-dimensional B on a three-dimensional CAD system
After performing dimensional conversion in order to transfer the spline curve data to the two-dimensional CAD system, it is determined whether or not the line segment indicated by the two-dimensional B-spline curve data after the dimensional conversion is a straight line. As a method of obtaining the coordinate value of the end point, a method as shown in FIG. 7 is conceivable. Hereinafter, the method shown in FIG. 7 will be described in detail.

When an operator specifies a three-dimensional B-spline curve on a three-dimensional CAD system and inputs a projection method, in step S61, the basic data of the three-dimensional B-spline curve stored in the database (step S61). Control point information and a knot vector) and a conversion matrix to a projection plane are read.

Next, in S62, it is determined whether the projection method is parallel projection or perspective projection. If parallel projection is determined, S63
Then, the coordinate value of each control point is converted by a conversion matrix to obtain 2
Only (x, y) which is a part of the dimensional coordinates is extracted, and this and the knot vector input first are used as basic data of a new two-dimensional B-spline curve. If it is determined that the projection is a perspective projection, the coordinate value of the passing point is calculated based on the basic data in S64, and the coordinate value of the passing point is converted by the conversion matrix in S65 to obtain a two-dimensional coordinate part (x, y). ) And take it out as a new passing point. In step S66, a new two-dimensional B-spline curve is re-established with the new passing point, and the basic data is obtained.

Next, the shape of the new two-dimensional B-spline curve is determined based on the coordinate values of the control points as basic data. First, in S67, it is determined whether or not the two-dimensional B-spline curve becomes one point according to the flowchart of FIG.

In FIG. 8, first, in S81, basic data of a two-dimensional B-spline curve (control point information and knot vector)
Receive, at the next S82, the control points _{P 0, P 1, ...,} P i,
Find the center of gravity of the polygon whose vertex is.

[0010] Thereafter, in step S83, the variable i is set to 0, and the following processing is performed until the value of the variable i becomes equal to or more than (the number of control points -1) or it is determined in step S89 that the variable i is not a two-dimensional point.

First, at step S85, the distance between the control point P _i and the center of gravity is determined, and at step S86, it is checked whether the distance is within an error range. If it is within a certain error, S87
Then, 1 is added to the variable i to continue the processing, otherwise, in S89, information indicating that it is not a two-dimensional point is returned to the main processing, and it is determined whether or not the two-dimensional B-spline curve becomes one point. Finish the process. If the distances between all the control points and the center of gravity are within the error range (NO in S84), the two-dimensional point data (the coordinate values of the center of gravity) is returned to the main processing (S88). To end.

In the main processing of FIG. 7, the two-dimensional point data is returned, so that the two-dimensional point data is stored in a memory (not shown) in S68, and the processing is terminated. If information indicating that the point is not a two-dimensional point is returned, it is determined in S69 whether the point is a straight line.

The determination as to whether or not a straight line is made is carried out as follows in accordance with the flowchart shown in FIG.

At S91, basic data (control point information and knot vector) of the two-dimensional B-spline curve is received, and at S92, 0 is set to a variable i.

[0015] Then, in S93, a direction vector P ₀ for each control _{point, P 1, ..., P i} , ... to vector V _i and its size _{L i (i = 0,1,2, ...} , control Points-1) are defined as in the following equations (2) and (3), respectively.

V _i = P _{i + 1} −P _i (2) L _i = {V _i } (3)

Thereafter, the following processing is performed until the value of i becomes equal to or more than (the number of control points -1) (NO in S94) or until it is determined in S103 that the line is not a two-dimensional straight line.

At S95, the vector Vi _{+ 1} and its magnitude L
_{i + 1} is defined as shown in the following equations (4) and (5).

V _{i + 1} = P _{i + 2} −P _{i + 1} (4) L _{i + 1} = {V _{i + 1} } (5)

[0020] In S96, L _i and L _{i + 1} certain errors ε _l
Are classified as (6) _, (7) _and (8).

L _i <ε _l → Case 4 (6) L _i > ε _l ∩L _{i + 1} <ε _l → Case 5 (7) L _i > ε _l ∩L _{i + 1} > ε _l → Case 6 (7) 8)

In the case of Case 4, in step S97, V _i ← V
_{i + 1} , L _i ← L _{i + 1} , 1 is added to the variable i in S98,
When the value of the variable i is less than (the number of control points -1) (S94 is Y
In the case of ES, new V _{i + 1} and L _{i + 1} are defined in S95, and the same classification processing as described above is performed in S96.

[0023] For Case5 is, V _i, L _i is intact, 1 is added to the variable i in S98, if the value of the variable i is smaller than (the number of control points -1) (if S94 is YES), S9
5, new V _{i + 1} and L _{i + 1} are defined, and the same classification processing as described above is performed in S96.

In the case of Case6, first, the vector V _i at S99, following equation V _{i + 1 (9),} determining the unit vector n _i, n _{i + 1} is normalized as in (10).

[0025] _{n i = V i / L i} ... (9) n i + 1 = V i + 1 / L i + 1 ... (10)

Next, in S100, the inner product N = n _i · n
_{i + 1} and the outer product G = n _i × n _{i + 1} are calculated, and it is determined in S101 whether or not the following expression (11) is satisfied.

N = 1∩G = 0 (11)

In equation (11), the inner product N = 1 is a condition for determining whether or not the control points P _i , P _{i + 1} , and P _{i + 2} are arranged in a straight line in a permutation. , The outer product G = 0 is a condition for determining whether the control points P _i , P _{i + 1} , and P _{i + 2} are on a straight line.

When N = 1∩G = 0, that is, when it is determined that the control points P _i , P _{i + 1} , and P _{i + 2} are arranged in a straight line in a permutation, V _i is determined in S102. ← V _{i + 1,} L _i ← L _{i + 1}
In S98, 1 is added to the variable i. If the value of the variable i is less than (the number of control points −1) (YES in S94), V _{i + 1 and} L _{i + 1} are newly defined in S95, and in S96. The same classification processing as described above is performed.

If N = 1∩G = 0, then S1
At 03, it is determined that the line is not a two-dimensional line, the information is returned to the main process, and the line determination process ends.

In step S94, the value of the variable i is set to (the number of control points-
1) If it is determined that the above has been reached, the coordinate values of the start point and the end point of the control point are returned to the main processing in S104 as the coordinate values of the start point and the end point of the straight line, respectively, and the processing ends.

In the main processing of FIG. 7, when the two-dimensional straight line data (the coordinate values of the start point and end point of the straight line) is returned, S70 is executed.
To store it in memory. When the determination result that the line is not a two-dimensional straight line is returned, the two-dimensional B-spline curve data created in S63 or S66 is stored in the memory in S71.

[0033]

The method of determining a straight line by the method shown in FIG. 7 is a case where the control points are arranged in a permutation on one straight line as shown in FIG. The inner product = 1 in equation (11) is a condition for guaranteeing this permutation. In fact, in addition to this, FIG.
As shown in FIG. 2, the control points may not be arranged in a straight line in a permutation. In order to determine a straight line including this case, the part related to the inner product N = 1 in equation (11) is removed, and
It is sufficient to simply set G = 0, but if the control points are not arranged in a permutation on the straight line as shown in FIG. 12, the coordinate values of the start point and the end point which are the basic data of the straight line cannot be obtained. A two-dimensional B-spline curve was determined, and the basic data of the two-dimensional B-spline curve had to be returned.

In the method disclosed in Japanese Patent Application Laid-Open No. 63-211101, the coordinate values of the start point and the end point of the control point are used as the coordinate values of the start point and the end point of the straight line.
If it is determined whether the line segment indicated by the two-dimensional B-spline curve data obtained by dimensionally converting the spline curve data is a straight line, and the coordinate values of the start point and end point of the straight line are obtained, as shown in FIG. If the control points are not arranged in a permutation on a straight line, there is a problem that the coordinate values of the start point and the end point become erroneous.

An object of the present invention is to determine the coordinate values of the starting point and the ending point, which are basic data of a two-dimensional straight line, even for a two-dimensional straight line in which the control points are not arranged in a permutation. It is an object of the present invention to provide a method for determining a linear shape at the time of dimension conversion of a dimensional spline.

[0036]

[Means for Solving the Problems] On a three-dimensional CAD system
3D B-spline curve data of 2D CAD system
When converting to two-dimensional B-spline curve data on the system
In the element type determination method performed in the above, the three-dimensional CAD
Data of 3D B-spline curve data on the system
The display device is stored in the base and
The three-dimensional B-splice displayed on the screen
Of the three-dimensional B-spline curve corresponding to the
Three-dimensional B-spline designated by the designation means
The three-dimensional B-spline curve data corresponding to the curve is
Input from the database, 2-dimensional B- spline curve data after <br/> dimensional transform is the result row Tsu Na dimensional transform to the three-dimensional B- spline curve data the input is accepted and shows a straight line, If the control points are not arranged in a permutation on a straight line, a curve defining polynomial and its first derivative are derived for each control point section, and the maximum value of each control point section is calculated using them. , A minimum value, and a maximum value and a minimum value in all control point sections are calculated based on the maximum value and the minimum value in each control point section. Specify the start point and end point of the straight line based on the value ,
The coordinates of the start point and the end point are stored in a memory .

[0037]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Next, embodiments of the present invention will be described in detail with reference to the drawings.

FIG. 1 is an external view showing a hardware configuration of an embodiment of the present invention. An engineering workstation (EWS) 10 for practicing the present invention includes a calculation of shape determination at the time of dimension conversion of a three-dimensional B-spline curve,
CP that plays a role in notifying each device device of the calculation result
An arithmetic unit 1 incorporating a U, a memory, etc., and a CRT display 2 for displaying and outputting elements such as curves as CAD drawings
And a keyboard 3 used by the operator to input numbers and characters, and a tablet 4 and a stylus pen 5 used by the operator to specify curved elements on the screen.

Next, a procedure for executing the present invention using the arithmetic unit 10 will be described.

The essential point of the method for judging the shape of a straight line at the time of dimension conversion of a three-dimensional B-spline curve according to the present invention is that an operator uses a stylus pen 5 to generate a CR as shown in the flowchart of FIG.
When the user designates a three-dimensional B-spline curve on the screen of the T display 2 and inputs a projection method using the keyboard 3, in S1, basic data (control point information, knot vector) of the three-dimensional B-spline curve, Information necessary for conversion (projection method, conversion matrix) is input from the database, and in S2, two-dimensional B-
The dimension is converted to a spline curve to obtain its basic data (control point information, knot vector).

Further, in S3, the type of the shape of the two-dimensional B-spline curve is determined. If the shape is a two-dimensional point or two-dimensional B-spline curve, the data is stored in the memory of the arithmetic unit 1 in S8, The process ends. If it is a two-dimensional straight line, the process proceeds to S4.

In S4, the coefficients of the polynomial defining the curve in the section of each control point are determined. In the next S5, the maximum value and the minimum value of x (or y) in the section of each control point are obtained. Further, in S6, based on the maximum value and the minimum value of x (or y) in the section of each control point, the maximum value and the minimum value of x (or y) in the whole curve and the corresponding y (or x) are obtained. ) Is calculated. In the last step S7, two-dimensional straight line data in which the minimum value of x and the value of y corresponding thereto are set as the starting point coordinate values and the maximum value of x and the corresponding value of y are set as the end point coordinate values are stored in the memory in the arithmetic unit 1. Then, the processing ends.

FIGS. 3 to 6 are detailed flowcharts of the embodiment of the present invention.

When the operator designates a three-dimensional B-spline curve displayed on the screen of the CRT display 2 by using the stylus pen 5 and inputs other necessary information with the keyboard 3, the arithmetic unit 1 executes the processing shown in FIG. S1 of 3
In step 1, basic data such as control point coordinates and knot vectors of a designated three-dimensional B-spline curve and data necessary for conversion such as a conversion matrix are input from a database.
It is stored in a variable area reserved in advance.

Next, in S12, S62 to S62 in FIG.
The dimension conversion of the three-dimensional B-spline curve data is performed in the same procedure as in S66, and the data of the control point coordinates and the knot vector which are the basic data of the two-dimensional B-spline curve are obtained and stored.

Thereafter, S81 to S81 of FIG.
It is determined whether or not a two-dimensional point is obtained by the same procedure as in S89. If the two-dimensional point is obtained, the two-dimensional point data is stored in the memory of the arithmetic unit 1 in S14, and the process is terminated.

[0047] Further, to turn OFF the linear judgment flag at S15, sets 0 to the variable i in S16, the control point in the next _{S17 P 0, P 1, ...} , P i, ... and the vector V _i with respect to and its size L _i equation (12) is defined as shown in (13).

V _i = P _{i + 1} −P _i (12) L _i = {V _i } (13)

Thereafter, it is determined that the value of the variable i does not exceed (the number of control points -1) (until S18 in FIG. 4 becomes NO) or that the variable i is not a two-dimensional straight line in S25 (NO in S25). Until the following processing is performed.

At S19 in FIG. 4, the vector Vi _{+ 1} and its magnitude L _{i + 1} (i = 0, 1, ₁ ) are obtained for the direction vectors P _0, P _1, ... _, P _i ,. 2, ..., number of control points-1)
Is defined as shown in Expressions (14) and (15).

V _{i + 1} = P _{i + 2} −P _{i + 1} (14) L _{i + 1} = {V _{i + 1} } (15)

[0052] in S20 by comparing L _i and L _{i + 1} and certain errors epsilon _l following equation (16), (17), it is classified as (18).

L _i <ε _l → Case 1 (16) L _i > ε _l ∩L _{i + 1} <ε _l → Case 2 (17) L _i > ε _l ∩L _{i + 1} > ε _l → Case 3 (17) 18)

In the case of Case 1, in S21, V _i ← V
_{i + 1,} L _i ← L _{i + 1} , 1 is added to the variable i in S22,
When the value of the variable i is less than (the number of control points -1) (S18 is Y
In the case of ES), new V _{i + 1} and L _{i + 1} are defined in S19, and the same comparative classification is performed in S20 as described above.

[0055] For Case2 is, V _i, L _i is intact, 1 is added to the variable i in S22, if the value of the variable i is smaller than (the number of control points -1) (if S18 is YES), S19
Defines new V _{i + 1 and} L _{i + 1} , and performs the same comparative classification as described above in S20.

In the case of Case 3, first, in step S23, the vectors Vi _and Vi _{+ 1} are normalized using the following equations (19) and (20) to obtain unit vectors _{ni and ni + 1} .

N _i = V _i / L _i ... (19) _{ni + 1} = Vi _{+ 1} / _{Li + 1} ... (20)

Next, in S24, an inner product N = n _i · n _{i + 1} and an outer product G = n _i × n _{i + 1} are calculated, and in S25, it is determined whether or not the outer product G = 0, that is, two-dimensional. It is determined whether or not it is a straight line.

If it is determined that the outer product G is not 0, the basic data (control point information, knot vector) is stored in the memory as the two-dimensional spline curve data, and the process is terminated.

When it is determined that the outer product G = 0,
That is, if it is determined that the line becomes a two-dimensional straight line, V _i ← in S27.
New V _{i + 1,} L _{i + 1} are defined as V _{i + 1,} L _i ← L _{i + 1} .

Thereafter, it is determined in S28 whether or not the inner product N = 1. If the inner product N = 1, 1 is added to the variable i in S22, and if the value of the variable i is less than (the number of control points −1) (S1
8 is YES), new V _{i + 1 and} L _{i + 1} are defined in S19, and the same comparative classification as described above is performed in S20. If it is determined that the inner product N is not 1, the straight line determination flag is set to ON in S29, and then the process of S22 is performed.

The above processing is repeated, and when the value of the variable i becomes equal to or more than (the number of control points -1), the processing shown in FIG. 5 is performed. At this stage, it has already been distinguished that the result of the dimension conversion is a linear shape.

In S30 of FIG. 5, the straight line determination flag is evaluated. If the flag is OFF, the start point of the control point is determined in S31 _.
The coordinate values of _the end point are stored in the memory in the arithmetic device 1 as the coordinate values of the start point _{and the} end point of the straight line, the data is returned, and the process ends.

When the straight line determination flag is ON, the following processing is performed to calculate straight line data (the coordinate values of the start point and the end point).

First, in S32, it is checked from the start point and end point of the control point whether or not this linear shape is parallel to the y-axis. y
If not parallel to the axis, a variable i is set to 0 in S34,
If they are parallel, after turning on the y-axis parallel flag in S33,
In S34, 0 is set to a variable i.

Thereafter, the following processing is performed until the variable i becomes equal to or more than (the number of control points−3) (until S35 becomes YES).

[0067] First, S43 in the control point interval i in in spline N ⁱ ₀ in FIG. _{^{6 (t), N i 1}} (t), N i 2
(T) _and ^Ni ₃ (t) are obtained. In the next step S44, the spline function and the control point coordinate values P ⁱ , P ^{i + 1} , P ^{i + 2} and P ^{i + 3} are substituted into the following curve definition expressions (21) and (22).

[0068] ^{x i (t) = P i} x N i 0 (t) + P i + 1 x N i 1 (t) + P i + 2 x N i 2 (t) + P i + 3 x N i 3 (t ^{) ... (21) y i (} t) = P i y N i 0 (t) + P i + 1 y N i 1 (t) + P i + 2 y N i 2 (t) + P i + 3 y N i 3 (T) ... (22)

Next, in S45, the equations (21) and (22) are rearranged into polynomials as shown in the following equations (23) and (24), and the coefficients A ⁱ , B ⁱ , C ⁱ , and D ^{i of the} polynomials are obtained. Is calculated.

[0070] ^{x i (t) = A i} x t 3 + B i x t 2 + C i x t + D i x ... (23) y i (t) = A i y t 3 + B i y t 2 + C i y t + D i _y ... (24)

Thereafter, in S46, it is checked whether or not the y-axis parallel flag is ON. If it is OFF, the following processing is performed.

Since x ⁱ (t) is at most a cubic function, the first derivative represented by the following equation (25) can be easily obtained.

[0073] ^{dx i (t) / dt =} 3A i x t 2 + 2B i x t + C i x ... (25)

The equation of the first derivative = 0 is solved in S47 by using the formula of the solution of the quadratic equation, and the value of t is obtained.

In the next S48, the maximum value and the minimum value of x ⁱ (t) are obtained by substituting the value of t into the polynomial (23) obtained in S45. Further, by substituting the values of the knot vectors t _i and t _{i + 1} corresponding to the control point section i in S49 into the polynomial (23) obtained in S45, the control point section i
Find the value at both ends of.

Thereafter, in S50, the maximum value and the minimum value of x in the control point section i are obtained by comparing the values obtained in S48 and S49. When the processing of S50 ends,
In S51, the variable i is increased by 1, and the same processing is performed for the next control point section i.

By performing these processes for all control point sections, that is, for the entire curve, the result of determination in S35 in FIG.
At S, the process at S36 is performed.

At S36, the y-axis parallel flag is ON or OFF.
Check if. In this example, the y-axis parallel flag is OFF
Therefore, the process of S37 is performed, and the maximum value and the minimum value of x in the entire curve are obtained. Then, S38
Calculates the y value that forms a pair with the maximum value and the minimum value of x, and determines the minimum value of x and the y value that forms a pair with the starting point coordinate value, the maximum value of x, and the y that forms a pair in next S39 Is stored in the memory in the arithmetic unit 1, and then the processing is terminated.

Similarly, when the y-axis parallel flag is ON, the maximum value and the minimum value of y in each control point section are obtained in S52 to S55 in FIG. 6, and the straight line data is calculated in S40 to S42 in FIG. After the data is stored in the memory in the device 1, the process ends.

[0080]

As described above, the present invention provides three-dimensional
3D B-spline curve data on CAD system
It is stored in the database and displayed on the screen of the display device.
3D B-spline curve stored in database
Display the 3D B-spline curve corresponding to the data
The three-dimensional B-
When a spline curve is specified, an arithmetic unit is used.
And three-dimensional B-splines specified from the database
Input 3D B-spline curve data corresponding to the curve
Then, the input three-dimensional B-spline curve data
2D after dimension conversion which is the result of performing dimension conversion on
B- spline curve becomes indicates a straight, and if the control point reached that not arranged in permutation in a straight line, based on the curve defining polynomial of the control points for each segment and the equation of the first derivative Calculate the maximum value and minimum value in all control point sections, further specify the start point and end point of the line based on the maximum value and minimum value in all control point sections, and store the coordinate values in the memory. <br> Therefore, even when the control points are not arranged in a permutation on a straight line, the effect that the coordinate values of the start point and the end point which are the basic data of the straight line can be specified is obtained. is there.

[Brief description of the drawings]

FIG. 1 is an external view showing a hardware configuration of an embodiment of the present invention.

FIG. 2 is a flowchart for explaining the main points of the processing content of the present invention.

FIG. 3 is a flowchart showing detailed processing contents of the embodiment of the present invention.

FIG. 4 is a flowchart showing detailed processing contents of the embodiment of the present invention.

FIG. 5 is a flowchart showing a detailed processing content of the embodiment of the present invention.

FIG. 6 is a flowchart showing detailed processing contents of the embodiment of the present invention.

FIG. 7 is a flowchart showing processing contents of a conventionally conceivable method.

FIG. 8 is a flowchart showing a method for determining whether a point is a two-dimensional point.

FIG. 9 is a flowchart showing a method for determining whether or not a two-dimensional straight line exists;

FIG. 10 is a diagram showing an expression form of a B-spline curve.

FIG. 11 is a diagram illustrating an example of a case where a B-spline curve has a linear shape.

FIG. 12 is a diagram illustrating an example of a case where a B-spline curve has a linear shape.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 ... Calculator 2 ... CRT display 3 ... Keyboard 4 ... Tablet 5 ... Stylus pen 10 ... Engineering workstation (EWS)

Claims (2)

(57) [Claims] 1. A three-dimensional B-source on a three-dimensional CAD system.
Plein curve data can be converted to two-dimensional CAD system
Element type used when converting to B-spline curve data
In another determination method, a three-dimensional B-spline tune on the three-dimensional CAD system is used.
By storing the line data in a database and using an arithmetic device , the three-dimensional B-space displayed on the screen of the display device is displayed.
3D B-spline tune corresponding to the pline curve data
The three-dimensional B-sp
3D B-spline curve data corresponding to the line curve
The input from the database, three-dimensional and the input B- 2-dimensional B- spline curve data after dimensional transform is the result of dimensional transform lines Tsu name against the spline curve data is assumed to indicate a straight line, and the control When the points are not arranged in a permutation on a straight line, a curve defining polynomial and its first derivative are derived for each control point section, and the maximum value and the minimum value of each control point section are derived using them. Calculating the maximum value and the minimum value in all the control point sections based on the maximum value and the minimum value in each of the control point sections, and calculating the maximum value and the minimum value in the all control point sections. The start point and the end point of the straight line are specified based on the coordinate values of the start point and the end point.
A method for determining a linear shape at the time of dimension conversion of a three-dimensional spline, which is stored in a memory . 2. The maximum value and the minimum value of each control point section are a maximum value of each control point section and a maximum value of each control point section.
2. The method according to claim 1, wherein the linear shape is determined based on the minimum value.