JPH02109182A - Free curve generating method - Google Patents

Abstract

PURPOSE:To obtain a free curve brought into contact with the free curve with a simple operation by obtaining intersections of an extended tangent vector and the free curve, and regenerating a control line vector so as to pass a point on the free curve between the intersections. CONSTITUTION:First and second intersection Q1 and Q2 with a desired free curve Ks can be obtained extending a first tangent vector a1 advancing from a first node P0 to a first control point P1 and a second tangent vector a2 advancing from a second node P3 to a second control point P2. And by generating a control vector Pin-P0 by extending the vector a1 further, the title method is taken so that a new free curve expressed by using a new control point Pin decided by the vector Pin-P0, the nodes P0 and P3, and the control point P2, or a new free curve KSGN expressed by using the new control point Pin decided by the victor Pin-P0, the nodes P0 and P3, and the intersection Q2 can pass a transit point PT on the free curve Ks between the intersections Q1 and Q2.

Description

[Detailed description of the invention]

The present invention will be explained in the following order. A: Industrial field of application B: Outline of the invention C: Conventional technology: Problems to be solved by the invention Eff [Means for solving the problem (Figs. 1 and 5) Figure) G Example (G1) Principle of free curve (Figure 2) (G2) Example of free curve creation (Figures 1 to 7) (G
3) Other embodiments H Effects of the invention A Field of industrial application The present invention relates to a free curve creation method, for example CAD (c
This method is suitable for use when deforming the shape of a free curve generated in a computer aided design. B. Summary of the Invention The present invention provides a method for creating a free curve by obtaining an intersection point between an extended tangent vector and a curve segment, and regenerating a control line vector so as to pass through a point on the free curve between the intersection points. , a free curve tangent to the free curve can be obtained with a simple operation. C Conventional technology For example, when designing the shape of an object with a free-form surface using a CAD method (geowIetric mod
e]jng), designers generally use 3 that the curved surface should pass through.
By specifying multiple points in a dimensional space (these are called nodes) and having a computer calculate a boundary curve network connecting the specified multiple nodes using a predetermined vector function, it can be expressed in a so-called wire frame. Create a curved surface. In this way, it is possible to form a large number of framework spaces surrounded by boundary curves (such processing is hereinafter referred to as framework processing). The boundary curve network formed by such framework processing itself represents the rough shape that the designer intends to design, and a curved surface that can be expressed by a predetermined vector function is calculated using interpolation using the boundary curves surrounding each framework space. If it is possible to do so, it is possible to generate a free-form surface (which cannot be defined by a quadratic function) that is designed by the designer as a whole. Here, the curved surfaces stretched in each framework space form basic elements that constitute the entire curved surface, and these are called batches. Conventionally, in this type of CAD system, as a vector function expressing a boundary curve network, easy-to-calculate vector functions such as the Bezier equation, B-suf'' line (B-5pl
A third-order tensor product formed by the formula 1ne) is used, and is considered to be optimal for mathematically expressing a free-form surface that does not have any special features in terms of shape, for example. In other words, a free-form surface that has no special features in shape is
When points given in space are projected onto the xy plane, the projected points are often arranged regularly in a matrix, and when the number of projected points is expressed as mxn, the framework space It is known that it can be easily stretched using a quadrilateral patch expressed by a cubic Bezier equation. However, when this conventional mathematical expression is applied to a curved surface with characteristic shapes (for example, a curved surface with a greatly distorted shape), it is difficult to connect the patches, and it requires advanced mathematical calculation processing. Since it is necessary to execute the method, there is a problem that the calculation processing by the computer becomes complicated and enormous, and the calculation time becomes long. As a way to solve this problem, we find internal control points that satisfy the condition of tangent plane continuity for the shared boundaries of adjacent framework spaces, and use a vector function that represents the free-form surface determined by the internal control points to A method of applying patches made of curved surfaces has been proposed (Japanese Patent Application No. 60-277)
No. 448, Patent Application No. 1984-290849, Patent Application No. 1983-
No. 298638, Japanese Patent Application No. 15396, 1983, Japanese Patent Application No. 1983
1-33412, Japanese Patent Application No. 61-59790, Japanese Patent Application No. 61-64560, Japanese Patent Application No. 61-96368, Japanese Patent Application No. 61-69385). Problems that the invention aims to solve By the way, in reality, designers' design work usually involves repeating local corrections to move step by step closer to the shape that the designer envisions. The free-form surface generated as described above is also repeatedly modified to approximate the shape envisioned by the designer. In other words, as a method for modifying such a free-form surface, the designer cuts the free-form surface at a desired cross section on the display screen, visually confirms the cross-sectional shape of the resulting free-form surface, and locally corrects the details. After that, a method is used to generate a free-form surface again based on the cross-sectional shape (
Patent Application No. 1982-276925, Patent Application No. 1982-27869
No. 4). However, when a designer uses such a method to modify a free-form surface, he or she may wish to modify the shape of the original cross-sectional curve so that it is in contact with a predetermined free curve. In such a case, conventionally, nodes and internal control points, etc. had to be re-set in an error-oriented manner for each curve segment of the free curve formed by the cross-sectional curve, which is a problem that is difficult to design in practice. was there. The present invention has been made in consideration of the above points, and aims to propose a free curve creation method that can obtain a free curve tangent to a predetermined straight line with simple operations. E Means for Solving the Problem In order to solve the problem, in the first invention, the first and second nodes P0 and P, and the first and second node P
G on a free curve expressed by a predetermined vector function R(t) using the first and second control points P and P2 set between 0 and 23. , the first node P. Extend the first tangent vector a from the point P to the first control point P, and the second tangent vector at from the second node P to the second control point P2 to form a desired free curve, The first and second intersection points Q+ and Q! By further extending the first tangent vector a to generate a control line vector PIN po, a new control point P0 determined by the control line vector PIN PO is obtained.
, first and second nodes P0 and P, and second control point P2
or a new free curve expressed using a new control point P0 determined by the control line vector P+sPa, the first and second nodes P0 and P3, and the above second intersection Q2. A free curve. is the free curve between the first and second intersection points Q1 and 07,
Make sure to pass through the upper passing point Pa. Furthermore, in the second invention, the first and second nodes P0
And using the first and second control points P and P2 set between P3 and the first and second nodes P0 and 23, a free curve expressed by a predetermined vector function R(t), G.
, a first tangent vector a1 from the first node P0 to the first control point P1, and a second tangent vector a1 from the second node P,
Extend the second tangent vector a toward the control point P2 to obtain the first and second intersection points Q and Q2 with the desired free curve, and further extend the first tangent vector a. By generating a control line vector PIN PO, a new control point is created on the free curve between the first and second intersection Q and Qt, passing through the upper passing point P and determined by the control line vector PIN po. Plus a new free curve expressed using the first and second nodes P0 and P3 and the second control point P2, or the free curve K between the first and second intersection points Q1 and 02. Passes through the upper passing point PT and the control line vector PINP0
A new control point PIN, first and second nodes P determined by . A new free curve KffGN expressed using P3 and the second intersection Q2 is generated, and a new free curve is created. is divided at il1 point Pi, and the divided free curve KSGN1%
KSGN2 is transformed so that the tangent line coincides with the free curve at the passing point P1. A free curve K3GN that is tangent to the free curve at P7 can be obtained. Furthermore, the free curve K passing through the passing point PT! After dividing 9Gl+ at the corresponding passing point PT, the divided free curve Ksar
tIzKs. 2 so that the tangent line coincides with the free curve at the passing point PT, it is possible to obtain a free curve that is in contact with the free curve at the passing point P as a whole with a simple operation. After obtaining the intersection points Q1 and G2 of the tangent vectors a and G2 extended by F action and the free curve, and obtaining the upper passing point PT on the free curve between the intersection points Ql and 02, further tangent vector a1 If one of the two control line vectors of the free curve is generated by extending the control line vector PINPO, then the control line vector P +NPo-(?Determined free curve NI
AKscN'li: It is possible to set the vehicle to pass through the corresponding passing point P1 with a simple operation. To the extent that this is practical, an embodiment of the present invention will be described in detail below with reference to the drawings. (G1) Principle of free curve In this example, the free curve is as shown in Figure 2.
Nodes Po and P3 (P(
010 and P(310, P(011 and P(111%P
,. , 2 and P(312).
at') are connected. This curve segment Ksc is a cubic Bezier (bezi
Using the following formula, R(t) = (1-t + tE)3P0...
... Parametric spatial curve R (1)
t). Here, t is the curve segment KsG from one node P0
It is a parameter that changes from a value of 0 to a value of 1 as expressed by the following equation (2) in the direction along the direction until reaching the other node P. In this way, the curve segment KsG expressed by the cubic Bezier equation is divided into nodes p and P1 by the shift operator E.
By specifying two control points p and P2 between them, each point on the curve segment can be expressed as a position vector R(t) from the origin 0 in xyz space using the following expansion formula: be done. Here, the shift operator E has the following relationship with respect to the control point P on the curve segment Ksc. Therefore, by expanding equation (1) and substituting the relationship in equation (4), we get the following equation Rct)=(1 t+tE)"P.=((1-t)3+3(1-t)"tE+3
(1-t)t"B"+t"E3)p. -(1-t)po+ 3 (1 t)2 M E P. +3(1-t)t"E"P, +t3E3P. -(l t )”Pal 3(1t)”t P+→
3(1-tjt")", upper t 3P J...
... (6) It is possible to perform the calculation as shown below, and as a result, (3) 2 is obtained. Thus, each curve segment Ksao, K of the free curve G
sc+, KsG□, respectively 2 based on formula (3)
one node and a control point P,. ,0~2,P(1,.~2
, P(21o~yo and P(31゜2.), and each curve segment, member l'KsG., K3+;;
, by expressing a free curve using KSG2 baskets, 1
, 9GO, KSGI −, respectively for each curve segment.
Nodes and control points P (0111-,
, , Z, P(1)O-2, P(Z'l0-2 and P(1
10 to 2 can be used to express a free curve. (G2) Example of free curve creation In this example, the central processing unit (CP (J) of the free curve creation device executes the curve deformation processing program shown in Fig. 1 to create a predetermined curve segment). 1. Transform the curve segment) Ksc+ so that it is tangent to the free curve "4".1. This allows the free curves to touch each other. C)) U is ノ, step SPI to Stella T
” Proceeds to S P 2 and waits for the input of the free curve to be deformed. Incidentally, as shown in Figure 2, the free curve has a common node P, (P L3 to) with the adjacent curve segment.
, P +o+z (P uz) for each curve segment. , KsGl-, KSG2, for example, three curved segments KsGo connected such that their tangents coincide.
% Ksc+ and Ksrrz, so that the curve shape of the entire free curve changes smoothly. Here, in CP LJ, when the designer moves a cursor over the first curve segment KsG+ using a mouse, for example, and then clicks the mouse, the CP LJ receives an input of a curve segment to be transformed. Subsequently, CP LJ, in snap SI) 3,
After receiving the input of the curve segment Ks that serves as a reference for the deformation of the curve segment F(sat), the process moves to step SP4.Here, as shown in FIG. 3, in the CPU, the nodes P, , P,,P,.13,Pt3)S and control point PH,,Pg,Ptn5
, PT! After coordinate transformation is performed so that the plane passing through +s becomes the xy plane, the curve segment KsGI and its nodes P o , P z , P t. 13, PH, s and control points P + , P z , P t+□, P, t□ are displayed together. In fact, a curve segment expressed by a quadratic or cubic Bezier equation)
When resetting and deforming the control points p, ,p for sa+, a vector from the node P0 to the control point P (hereinafter referred to as the control line vector of the node P0) and the node P3
and the control line vectors are the nodes P, respectively. and P, representing the tangent of the curve segment KSG+ in the adjacent curve segment) KEG. and Kst, the curve segment while maintaining the state where the tangent line coincides with z) KS
In order to transform the GI, the position of the control point to be reset is limited to an extension of the control line vector. Furthermore, at this time, the intersection of the straight line extending from node P0 to control point P1 and the straight line extending from node P to control point P2 is set as Ql, and the line segment connecting the intersection Qx and nodes Pa and Pt is drawn. , and, the node P. from control point P,
If the control points P, , P2 are reset on the straight line extending from the node P to the control point P, and the control points P, , P2 are reset on the line segment and on the straight line other than LA, There is a problem that the curve segment expressed by the reset control points p, ,p, draws a loop, and in order to perform the deformation process while maintaining the rough shape of the curve segment KsGI, it is practically necessary to reset the curve segment. The positions of the control points are limited to the line segment and L5□. Therefore, in the curve segment KSG+, the node P
It is possible to deform the curve segment so that it protrudes from the triangular area connecting the intersection point Q8 with 0 and P3.In order to deform the curve segment so that it is in contact with and Lx to form intersections Q and Q2 (i.e., curve segments that intersect the area of the triangle). Therefore, in this example, the curve segment has G
By resetting the control points P, , P2 of , on the line segments L1 and L2, the state in which the tangents coincide with the adjacent curve segments K5GG and KSG2 is maintained, and the rough shape of the curve segment Ksa+ is maintained. The curve segment (Ksc+) is deformed step by step. That is, in step SP4, the CPU obtains coordinate data of the intersection point Q with the curve segment Ks by extending the unit vector a of the control line vector (that is, consisting of a tangential vector) of the control line vector of the node P0. At the same time, the CPU obtains a parameter representing the position vector of the intersection point Q from the vector function R(t)* representing the curve segment Ks, and proceeds to the subsequent step SP5. Here, the CPU calculates, regarding the control line vector of node P3,
After extending the unit vector a2 of the control line vector to obtain the coordinate data of the intersection point Q2 with the curve segment) Ks, the parameter t expressing the position vector of the intersection point Q2 is obtained as in step SP4. Next, as shown in FIG. 4, in step SP6, the CPU obtains the parameter t expressed by the following relational expression and the average value of [2, from the parameters t1 and t2 of the intersection points Q and Q2, curve segment K, with the parameter on top

Set a passing point P□ expressed as [, That is, from equation (3), the nodes Pf01S, P(311
In the curve segment l- expressed by control points PfllS, P(215), since it is expressed by the vector function R(t)S of the following formula, (8
) By substituting the parameter t into the equation, the passing point P
t is obtained. Furthermore, in the CPU, the curve segment K. For the number of vectors R(t)s expressing
Substitute. Thus, in the differential expression of the vector function R(t)s by the parameter t, it is possible to express the tangent vector of the curve segment )Ks expressed by the vector function R(t)s, and by substituting the parameter t3. As a result, the tangent vector function of the curve segment Ks at the passing point Pt can be obtained. Thus, in this embodiment, the control points are reset on the line segments Ll and L2, and the curve segment KsGI is deformed in stages to obtain a curved segment that passes through the iIl point PT, and then this is divided, By reconnecting using the tangent vector aT, to the curved segment at the corresponding passing point pt,
Transform it so that it touches . In this way, within the triangular area determined by the line segments L+ and L2, the curve segment can be transformed by 3,1, thereby deforming the adjacent curve segment KSG. And, G□
and the curve segment K
It is possible to perform deformation processing while maintaining the rough shape of se+. Therefore, in step SP8, the CPU sets control points at ζ=21 intersection Q and Q2 instead of control points P and P2, as shown in FIG. A curve segment KSON+ close to point P1 is generated. By the way, node P. and extend 1 to the control line vector of P3
Using the intersection points Q and Q2 of the curve segment) KSG
When generating NI, the original curve segment Ksc+ and the node P. Since the tangents at and P3 can be made to match, it is possible to maintain a state in which the tangents match the adjacent curve segments, similar to the original curve segment Ksc,l. Thus, in the curve segment K SGN+, (
3), the following formula R(t) = (1-t) 3Po + 3 (i
-t) 2 t Q+3 (1t)t2Qz +
It is expressed as u, 'P, l... (10) and has control line vectors of vectors Q+Po and QtP3. Next, CP[J extends the control line vector Q+ Po among the two control line vectors QI PO and Q2 P:I of the curve segment Kscs+, and places the control point P18 at the point determined by the extended control line vector Pus Pa.
Set. At this time, the amount of extension of the control line vector Q+ Pa is determined so that the curve segment K SGN expressed by the control point PIH passes through the passing point P, and thereby the curvature segment]・KsGN that passes through the passing point P is determined. generate. In this case, the newly set control point PIN is the control line vector Q, -P. Expressing the amount of extension using the unit hectare a and the scalar amount u, the following formula % Formula % (11) Therefore, the curve segment K5 represented by the control point PIN, nodes P0 and P8, and control point Q2. G N is a parameter u that changes between the value O and 1 instead of the parameter L.
From equation (10) using c and equation (11),
The following formula R(u,, uc)-(] 1l-)31'. →-3(1um)2ua(p6+a・uc)
+3(1,ua)ua''Qz +u,'P3 ...... (12) Therefore, in order for the relevant curve segment Ks to pass through the passing point PT, the following formula % formula % (13 ) should hold, and if we obtain a scalar quantity uc that satisfies this relationship, we can obtain a new control point PIN from equation (11).
coordinate data can be obtained. By the way, in practice, when obtaining coordinate data in a three-dimensional space from equation (13), it is necessary to solve a nonlinear cubic equation, and when performing this type of calculation processing, it is necessary to perform repeated calculations. There is a problem that the processing time becomes long. Therefore, in this embodiment, only coordinate data about the y-axis and the y-axis are obtained by performing coordinate transformation on the curve segment in step SP4 so that the plane passing through GI and 2 becomes the xy plane. Furthermore, by linearizing equation (13) and performing arithmetic processing using Newton-Raphson's method, the arithmetic processing can be performed in a short time. In other words, by performing Taylor expansion on equation (13) and eliminating its quadratic and subsequent terms, the following equation (%) is obtained, and by transforming this, the following equation = P r R (u,. J Therefore, by decomposing equation 15) into X-axis and y-axis components and adding subscripts to each component, it is possible to form a matrix as shown below. . Here, for each component of the matrix, from (12), the following formula % formula %) (1 Thus, by decomposing formulas (17), (18) and (19) into X-axis and y-axis components, (16) After assigning to 2, 11,
As well as assigning initial values to and uc, sequentially ΔU and Δu
Substitute c. As a result, (16) Δu3 and Δuc that converge are obtained, and the scalar quantity uc and balance meta U can be obtained from the values ΔU and Δu and the initial values of u8 and uc. Thus, based on the scalar Muc, a new control point PIl+ can be set, and by obtaining the control point P and the curve segment KSGN represented by the control point Q2 and the nodes Po and Ps, the passing point P can be set. to the curved segment that passes through. can be obtained. By the way, for both the two control points Q and Q2,
A method of resetting the control points and deforming the curve segment so that it passes through the passing point P is also considered, but in practice, in this method, it is difficult to unambiguously determine the control points to be reset. However, by resetting only one control point Q1 out of the two control points Q and Q2 as in this embodiment, a free curve passing through the passing point Pi can be easily and reliably obtained. As shown in FIG. 6, CP tJ is the new control point PIN
Once obtained, the process proceeds to step SP9, where the curve segment Ks is obtained. Divide by the passing point P. That is, the curve segment Ksc++ is determined by the node P using the parameter obtained from equation (16). and passing point Pa as node P(.l IN % P (x,
The curved segment KNNI formed by IN and the iI passing point. and the node P3 are divided into the curve segments KNN! formed by the nodes P(., 2N, a3, 2N).Then, the CPU moves to step 5PIO and divides the curve segments KuN+ and KNN! as shown in FIG. K) Common node P of IN□. The control point P TH11 is set so that the control 1g line vector at lIN%P(.12N coincides with the tangent vector aT of the curve segment at the passing point P.
8% P (by resetting 11288, the curve segment passing through the passing point Pa) K□ and +<H□ are transformed so that they touch the curve segment Ks at the passing point P1. In this way, the curve segment KSGW that has been deformed so as to pass through the passing point P□ between the intersection points Q1 and 02 is transformed to the ill passing point P
After dividing by T and 7, by reconnecting the curve segment so that the tangent line coincides with 5 at the corresponding passing point PT, we can obtain 2 to the curve segment, to the curve segment tangent to , and to 2, thus As a whole, we can obtain a free curve that is tangent to the curve segment. CP Ll is then filled with nodes P +oz++ % P +5zN(P (
Q12N ), 711211 and control point P (11IN
,, P (2) lNN%P ++yzsN-,P
After converting the +z+zs coordinate data into the original three-dimensional space - coordinate data, it is stored in the beta base, and the process moves to step 5PII to end the processing procedure. In the above method, the node P of the curve segment) Ksc+
. , P3 is extended to obtain the intersection points Q and G2 with the curve segment) K s (Fig. 3), and a curve formed by using the intersection points Q1 and G2 as control points. Segment KSG□ is generated (FIG. 5). Subsequently, the curve segment KsGNl has the intersection point Q and 0
After being deformed so as to pass through the passing point PT between 2 (5th
(Fig. 6), and is divided at the passing point PT (Fig. 6). The divided curve segments KNNI and KNN2 are
At the passing point P, it is reconnected so that it coincides tangent with the tangent hector at of the curve segment l' K s, thus adding 50. The curve segment K
A deformation process can be applied to the curve segment tangent to s and to Kt (FIG. 7). According to the above method, the intersection points Q and G2 of the extended control line vector and the curve segment Ks are obtained, and one of the control line vectors is further extended to obtain the passing point P1 between the intersection points Q1 and 02. After generating a curve segment KscN that passes through, by reconnecting the curve segments K and G, one of the curve segments connected so that the tangent lines coincide with each other at a common point of contact, the tangent lines coincide with each other. While maintaining the condition, the predetermined curve segment K s
In this way, a free curve that is in contact with a desired free curve can be obtained with a simple operation. (G3) Other embodiments (1) In the above embodiment, the intersections Q1 and G2
A case has been described in which the curve segment KscN+ is generated using the control point KscN+, and then the curve segment KSGNI is transformed so as to pass through the passing point PT. However, the present invention is not limited to this, and the original curve segment is , may be transformed so that it directly passes through the passing point PT. In this case, the curve segment passing through the passing point PT is expressed by a control point determined by the extended control line vector, nodes P0 and P3 of the original curve segment, and the remaining control point P2. (2) Furthermore, in the above embodiment, the passing point P1
A case has been described in which the curved segment KsGN that has been deformed so as to pass through is divided into two and reconnected, but the present invention is not limited to this. Point P. (curve segment) KS, the reconnection process may be omitted. (3) Furthermore, in the above-described embodiment, the parameters 1 . We have described the case where the curve segment Ksc+ is transformed so that it passes through the passing point PT expressed by the vector function R(L) using the parameter t and the average value t of t2. The present invention is not limited to this, but the point is that the passing point P is on the curve segment Ks between the intersection Q and 02.
, and set the relevant passing point PT to the parameter -
and t2 may be used for setting. (4) Further, in the above-described embodiments, a case has been described in which a curve segment expressed by a cubic Bezier equation is transformed, but the order of the equation is not limited to this, and may be quartic or higher. (5) Furthermore, in the above embodiment, one of the curved segments connected such that the tangents coincide at a common node.
In the above description, one curve segment is transformed so that one curve segment is tangent to another curve segment = 34, but the present invention is not limited to this. Furthermore, it can be widely applied when deforming independent curve segments. H Effects of the Invention As described above, according to the present invention, the intersection point between the extended control line vector and the curve segment is obtained, and the control line vector is further controlled. [By extending one of the one-line vectors and generating a curve segment that passes through a passing point on the curve segment between the intersection points, the curve segment can be transformed so that it touches a predetermined curve segment with a simple operation. In this way, the free curve can be transformed so as to be in contact with the desired free curve. is a route map for explaining deformation of curved segments. K+ X Kz % Ksc++ −, Ksc
z % Kscx s KsGN −. K3ONls KNNI% KN)lz・・・・・・
Curve segment l-5PG-: l-, P (ell-3
% P (n+-s-, P (o++++ -28%P
(3118~2N...node, PT...
... Passing point, P(Ill~3, P(21+-3, P
(1118-28%PfZth, ~2□, P(211NS
, P(1)21□ , Q4, Q2... Control point.

Claims (2)

[Claims] (1) In a free curve expressed by a predetermined vector function using first and second nodes and first and second control points set between the first and second nodes, By extending a first tangent vector from the first node to the first control point and a second tangent vector from the second node to the second control point, By obtaining the first and second intersection points and further extending the first tangent vector to generate a control line vector, a new control point determined by the control line vector, the first and second nodes, and the above A new free curve expressed using the second control point, a new control point determined by the control line vector, or a new free curve expressed using the first and second nodes and the second intersection point. A method for creating a free curve, characterized in that the free curve passes through a passing point on the free curve between the first and second intersection points. (2) In the free curve expressed by a predetermined vector function using the first and second nodes and the first and second control points set between the first and second nodes, By extending a first tangent vector from the first node to the first control point and a second tangent vector from the second node to the second control point, 1 and a second intersection point, and further extend the first tangent vector to generate a control line vector, passing through a passing point on the free curve between the first and second intersection points, and A new control point determined by the control line vector, a new free curve expressed using the first and second nodes and the second control point, or the free curve between the first and second intersection points. Generate a new free curve that passes through the passing point on the curve and is expressed using a new control point determined by the control line vector, the first and second nodes, and the second intersection point, and A method for creating a free curve, comprising: dividing a free curve at the passing point, and deforming the divided free curve so that a tangent to the free curve coincides with the free curve at the passing point.