CN109840930A - Curve constructing method based on the control of geometrical characteristic point - Google Patents
Curve constructing method based on the control of geometrical characteristic point Download PDFInfo
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Abstract
Based on the curve constructing method of geometrical characteristic point control, it is related to the construction of plane curve.Interpolation curve is constructed using cubic Bézier curve, this method provides volume of data point in the plane first, and specified data point is cusp, annulus from intersection point or inflection point, excessively each point can interpolation go out a cubic Bézier curve, while allowing geometrical characteristic point present in interpolation point.Then these curved sections are stitched together, to obtain a piecewise interpolation curve, and are at least up to G at splice point1Continuously, i.e., two sections of curve curvature directions at splice point are conllinear, even can achieve G in some cases2Continuously, i.e. two sections of curves amount of curvature and direction at splice point is all identical.
Description
Technical field
The present invention relates to the constructions of plane curve, construct accordingly more particularly, to using a series of interpolation points provided
Curved section, and by they be spliced into special pattern based on geometrical characteristic point control curve constructing method.
Background technique
Curve construction has very long history in field of Computer Graphics, it is widely used in drawing, sketch
The fields such as design and animation.Currently, the method for curve construction mainly has interpolation method, fitting process and to Bézier curve control
The direct operation on vertex processed.
Wherein, interpolation method is the very important method of one kind of curve construction, using interpolation method can construct across with
Specify the curve of data point in family.But during curve construction, generate sometimes cusp, annulus from intersection point and inflection point this
A little particular points, we term it geometrical characteristic points.The curve that previous interpolation method constructs not can control these geometrical characteristics
The position of point, the curve that some method interpolation come out even can not determine whether that these geometrical characteristic points can be generated.
Summary of the invention
The purpose of the present invention is to provide constructing corresponding curved section using a series of interpolation points for providing, and by they
It is spliced into the curve constructing method based on the control of geometrical characteristic point of special pattern.
The present invention constructs interpolation curve using cubic Bézier curve, and this method provides a series of numbers in the plane first
Strong point, and specified data point is cusp, annulus from intersection point or inflection point, excessively each point can interpolation go out a B é zier three times
Curve, while allowing geometrical characteristic point present in interpolation point.Then these curved sections are stitched together, to obtain a segmentation
Interpolation curve, and G is at least up at splice point1Continuously, i.e., two sections of curve curvature directions at splice point are conllinear, certain feelings
It even can achieve G under condition2Continuously, i.e. two sections of curves amount of curvature and direction at splice point is all identical.
Before introducing detailed process, relevant theoretical knowledge is first provided:
1, the general parameters equation of the cubic Bézier curve in plane can be written as follow form:
Wherein,T is the parameter coordinate of curve.
2, on curve every bit curvature are as follows:
The molecule of curvature expression formula is a quadratic equation with one unknown, be can be obtained after arranging to it:
Det (Q'(t), Q " (t))=2 (At2+ Bt+C)=2F (t)
Wherein, A=3det (P2,P3), B=3det (P1,P3), C=det (P1,P2)
Then △=B2-4AC
It is of the present invention based on geometrical characteristic point control curve constructing method the following steps are included:
1) curve to be drawn of selection is closure or non-closed, according to the difference of selection, is closed or non-closed,
Then different systems of linear equations is constructed；
2) volume of data point is provided in the plane as interpolation point, and the volume of data point is denoted as { vi| i=
1 ..., N }, and the type of every kind of data point is specified, including cusp, annulus, from intersection point, inflection point or endpoint etc., the cusp is bent
Curvature of the line at the data point is not present；The annulus is that curve passes through the data point twice from intersection point；The inflection point or end
It is zero that point, which is curvature of the curve at the data point, and the symbol of curvature changes at the data point；For non-closed
There are two endpoints in curve；
3) according to each data point viAnd consecutive number strong point vi-1, vi+1Position calculating parameter coordinateCalculation formula is such as
Under:
4) each data point interpolation goes out a cubic Bézier curve, and allows geometrical characteristic point: cusp, annulus selfing
Point or inflection point are as follows for different characteristic point difference present in interpolation point:
Cusp:
As △=B2Occurs unique cusp when -4AC=0, on curve, it is assumed that the parameter coordinate of cusp is t*, then t*
It is the root of equation F (t), it may be assumed that
By △=B2- 4AC=0 andIt is available:
Interpolation point is v, then has:
Annulus is from intersection point:
Cubic Bézier curve generates annulus in inside, indicates the corresponding two parameter coordinates in certain pointAndThe point is annulus from intersection point, while being also interpolation point v, then has:
With
Wherein, it enables
Inflection point:
As △ > 0, there are two roots by equation F (t), respectively correspond the parameter coordinate with two inflection points, enable one of root t*
∈ (0,1), it may be assumed that
F(t*)=0
New parameter h is introduced, so that:
Wherein,At this point, another root of equation F (t) is always located in section [0,1]
Except；
By F (t*The He of)=0Obtain:
Interpolation point is v, then has:
In addition, that is, non-cusp, annulus obtain a B é three times from the data point of intersection point and inflection point by the general point of interpolation
Zier curve, this curve do not have cusp, annulus from intersection point and inflection point:
As △ > 0, there are two roots by equation F (t), and curve will appear two inflection points, if two roots of equation is allowed to be all located at area
Between except [0,1], obtain one without cusp, annulus from the cubic Bézier curve of intersection point and inflection point, specific practice is as follows:
Enabling t=-1 is one of root of equation F (t), it may be assumed that
F (- 1)=A-B+C=0
With season symmetry axisBetween straight line t=0 and straight line t=1, and enable the parameter coordinate of interpolation point:
By F (- 1)=A-B+C=0 andIt obtains:
3(1+2t*)P3-2t*P2-P1=0
In addition, interpolation point is v, then have:
5) curved section for obtaining step 4) is stitched together, and meets the following condition of continuity:
G0It is continuous:
Pi,0+Pi,1+Pi,2+Pi,3=Pi+1,0
G1It is continuous:
3Pi,0+2Pi,1+Pi,2-3Pi+1,0=-Pi+1,1λi
G2It is continuous:
κi(1)=κi+1(0)
Wherein, i indicates i-th section of curve；
Four kinds of curved sections: cusp, annulus from intersection point, inflection point and without cusp, annulus from three kinds of intersection point, inflection point points three times
Any two kinds of curved sections in Bézier curve section are stitched together, available 16 kinds of joining methods, below with cusp+cusp
For derived in detail, other situations can be processed similarly:
(5.1) according to the correlation formula and G of cusp in step 4)0Continuous condition, available Pi,1、Pi,2And Pi,3
Expression formula:
(5.2) similarly, P in available i+1 curvei+1,1Expression formula:
(5.3) by P obtained in step (5.1) and (5.2)i,1、Pi,2And Pi+1,1Expression formula bring G into1Continuous condition
In, P can be obtainedi,0、Pi+1,0And Pi+2,0Between relational expression；
6) system of linear equations is constructed；
In step 6), the specific method of the construction system of linear equations can are as follows: draws closed curve and non-closed curve
The system of linear equations of construction:
(6.1) closed curve:
N number of data point is provided, N curve can be obtained with interpolation to get P in N number of step (5.3) is arrivedi,0、Pi+1,0And Pi+2,0
Between relational expression, be uniformly written as follow form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N
Wherein, Ei,1,Ei,2,Ei,3And CiBe byAnd viType determine.
(6.2) non-closed curve:
N number of data point is equally provided, enables first and two endpoints of the last one data point as curve, it in this way can be with
Interpolation obtains N-2 curve to get the P into N-3 step 5)i,0,Pi+1,0And Pi+2,0Between relational expression, be uniformly written as follow
Form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N-3
Wherein, P1,0=v1,PN-1,0=vN。
7) solution procedure 6) in system of linear equations, obtain all splice point Pi,0Coordinate, closed curve i=1 ...,
N, non-closed curve i=1 ..., N-3, then by the available P of expression formula in step 6)i,1, Pi,2Pi,3；
8) condition is utilized | κi(1) |=| κi+1(0) | improve the continuity of curve, the P that step 7) is obtainedi,0, Pi,1Pi,2
And Pi,3It is brought intoObtain new λi, step 7) and step 8) are repeated, directly
Arrive | | κi(1)|-|κi+1(0) | | < ε, ε are previously given threshold value, take ε=10-10；
9) after the iteration stopping in step 8), the P that is obtained using step 7)i,0,Pi,1,Pi,2And Pi,3Draw curve.
The present invention using cusp, annulus can be generated inside cubic Bézier curve from the characteristic of intersection point and inflection point, interpolation in
A series of data point of specified types obtains corresponding curved section, they are stitched together, and obtains specific figure, and whole
At least meet G on body1Continuously.
Curve designed by the present invention can apply to Art Design, facilitate those skilled in the art to the structure of specific curves
It makes, has the advantage that
First: geometrical characteristic point (cusp, annulus from intersection point, inflection point) is only present at interpolation point or at tie point.
Second: the curve of generation at least meets G1, when the sign of curvature at tie point does not change, can guarantee curve
Reach G2Continuously.
Third: curve has good locality, and when some interpolation point occurs mobile, curve only occurs near the point
Deformation.
4th: by the regulation to parameter, different size of annulus and inflection point curve of different shapes can be obtained.
Detailed description of the invention
Fig. 1 is that (hollow triangle indicates that cusp, hollow round expression annulus are empty from intersection point to the data point of the invention inputted
Heart five-pointed star indicates that inflection point, hollow square indicate the general point of non-cusp, annulus from intersection point and inflection point, filled square expression
Endpoint).
Fig. 2 is curve of the present invention according to input point-rendering.
Fig. 3 is that the present invention removes the curve drawn after data point.
Fig. 4 is the schematic diagram that curve designed by the present invention has locality.
Fig. 5 is as all αiAnd βiAll value 0.3 when, the annulus schematic diagram of drafting.
Fig. 6 is as all αiAnd βiAll value 0.4 when, the annulus schematic diagram of drafting.
Fig. 7 is to work as hiWhen value 0.5, the curve synoptic diagram of drafting.
Fig. 8 is to work as hiWhen value 1, the curve synoptic diagram of drafting.
Fig. 9 is to work as hiWhen value -0.5, the curve synoptic diagram of drafting.
Specific embodiment
In order to make the objectives, technical solutions, and advantages of the present invention clearer, with reference to the accompanying drawings and embodiments, right
The present invention is further described.
The present invention the following steps are included:
1) curve to be drawn of selection is closure or non-closed, according to the difference of selection, is closed or non-closed,
Then different systems of linear equations is constructed；
2) volume of data point is provided in the plane as interpolation point, and the volume of data point is denoted as { vi| i=1 ...,
N }, and the type of every kind of data point is specified, including cusp, annulus, from intersection point, inflection point or endpoint etc., the cusp is curve at this
Curvature at data point is not present；The annulus is that curve passes through the data point twice from intersection point；The inflection point or endpoint are bent
Curvature of the line at the data point is zero, and the symbol of curvature changes at the data point；Non-closed curve is gone out
Existing two endpoints；
3) according to each data point viAnd consecutive number strong point vi- 1, vi+1Position calculating parameter coordinateCalculation formula is such as
Under:
4) each data point interpolation goes out a cubic Bézier curve, and allows geometrical characteristic point: cusp, annulus selfing
Point or inflection point are as follows for different characteristic point difference present in interpolation point:
Cusp:
As △=B2Occurs unique cusp when -4AC=0, on curve, it is assumed that the parameter coordinate of cusp is t*, then t*
It is the root of equation F (t), it may be assumed that
By △=B2- 4AC=0 andObtain:
Interpolation point is v, then has:
Annulus is from intersection point:
Cubic Bézier curve generates annulus in inside, indicates the corresponding two parameter coordinates in certain pointAndThe point is annulus from intersection point, while being also interpolation point v, then has:
With
Wherein, it enables
Inflection point:
As △ > 0, there are two roots by equation F (t), respectively correspond the parameter coordinate with two inflection points, enable one of root t*
∈ (0,1), it may be assumed that
F(t*)=0
New parameter h is introduced, so that:
Wherein,At this point, another root of equation F (t) is always located in section [0,1]
Except；
By F (t*The He of)=0It is available:
Interpolation point is v, then has:
In addition, that is, non-cusp, annulus obtain a B é three times from the data point of intersection point and inflection point by the general point of interpolation
Zier curve, this curve do not have cusp, annulus from intersection point and inflection point:
As △ > 0, there are two roots by equation F (t), and curve will appear two inflection points, if two roots of equation is allowed to be all located at area
Between except [0,1], obtain one without cusp, annulus from the cubic Bézier curve of intersection point and inflection point, specific practice is as follows:
Enabling t=-1 is one of root of equation F (t), it may be assumed that
F (- 1)=A-B+C=0
With season symmetry axisBetween straight line t=0 and straight line t=1, and enable the parameter coordinate of interpolation point
By F (- 1)=A-B+C=0 andIt obtains:
3(1+2t*)P3-2t*P2-P1=0
In addition, interpolation point is v, then have:
5) curved section for obtaining step 4) is stitched together, and meets the following condition of continuity:
G0It is continuous:
Pi,0+Pi,1+Pi,2+Pi,3=Pi+1,0
G1It is continuous:
3Pi,0+2Pi,1+Pi,2-3Pi+1,0=-Pi+1,1λi
G2It is continuous:
κi(1)=κi+1(0)
Wherein, i indicates i-th section of curve；
It has been previously noted four kinds of curved sections and (has separately included cusp, annulus from intersection point, inflection point and without above-mentioned three kinds of points
Cubic Bézier curve section), any two kinds of curved sections are stitched together, available 16 kinds of joining methods, below with cusp+point
It is derived in detail for point, other situations can be processed similarly:
(5.1) according to the correlation formula and G of cusp in step 4)0Continuous condition, available Pi,1、Pi,2And Pi,3
Expression formula:
(5.2) similarly, P in available i+1 curvei+1,1Expression formula:
(5.3) by P obtained in step (5.1) and (5.2)i,1、Pi,2And Pi+1,1Expression formula bring G into1Continuous condition
In, P can be obtainedi,0、Pi+1,0And Pi+2,0Between relational expression；
6) system of linear equations is constructed, specific method is the system of linear equations for drawing closed curve and non-closed curve construction:
(6.1) closed curve:
N number of data point is provided, N curve can be obtained with interpolation to get P in N number of step (5.3) is arrivedi,0、Pi+1,0And Pi+2,0
Between relational expression, be uniformly written as follow form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N
Wherein, Ei,1,Ei,2,Ei,3And CiBe byAnd viType determine.
(6.2) non-closed curve:
N number of data point is equally provided, enables first and two endpoints of the last one data point as curve, it in this way can be with
Interpolation obtains N-2 curve to get the P into N-3 step 5)i,0,Pi+1,0And Pi+2,0Between relational expression, be uniformly written as follow
Form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N-3
Wherein, P1,0=v1,PN-1,0=vN。
7) solution procedure 6) in system of linear equations, obtain all splice point Pi,0Coordinate, closed curve i=1 ...,
N, non-closed curve i=1 ..., N-3, then by the available P of expression formula in step 6)i,1, Pi,2Pi,3；
8) condition is utilized | κi(1) |=| κi+1(0) | improve the continuity of curve, the P that step 7) is obtainedi,0, Pi,1Pi,2
And Pi,3It is brought intoObtain new λi, step 7) and step 8) are repeated, directly
Arrive | | κi(1)|-|κi+1(0) | | < ε, ε are previously given threshold value, take ε=10-10；
9) after the iteration stopping in step 8), the P that is obtained using step 7)i,0,Pi,1,Pi,2And Pi,3Draw curve.
Specific embodiment is given below.
Before the present embodiment is described in detail, it should be pointed out that the present embodiment was demonstrated is non-closed song
The building method of line, for closed curve, user only needs that construction can be completed using identical process.
Step 1: 18 data points of input, and the type (Fig. 1) of each point is specified, solid black square indicates endpoint,
Triangle indicates cusp, and circle indicates annulus from intersection point, and five-pointed star indicates that inflection point, hollow square indicate general point.
Step 2: initiation parameter, λi=1, calculate the corresponding parameter coordinate of each interpolation pointValue, for annulus be selfed
Point and inflection point, user input parameter alphai,βiAnd hiValue.
Step 3: solving system of linear equations, obtain splice point Pi,0Coordinate, i=1 ..., N-3
Step 4: calculating other control points Pi,1,Pi,2,Pi,3Coordinate.
Step 5: updating λi。
Step 6: judging maxi||κi(1)|-|κi+1(0)||<10-10Whether true, establishment then performs the next step, and otherwise returns
Return step 3.
Step 7: according to the P calculatedi,0, Pi,1, Pi,2Pi,3It draws curve (Fig. 2), removes the song obtained after data point
Line (Fig. 3).
Step 8: inspection obtains the properties of curve
Locality:
By the control point in the mobile upper right corner on the basis of virgin curve, then repaint curve (Fig. 4)
The regulation of annulus size:
As all αiAnd βiAll value 0.3 when, draw curve (Fig. 5)
As all αiAnd βiAll value 0.4 when, draw curve (Fig. 6)
The shape of curve where inflection point is regulated and controled:
Work as hiWhen value 0.5, draw curve (Fig. 7)
Work as hiValue 1 is drawn curve (Fig. 8)
Work as hiWhen value -0.5, draw curve (Fig. 9).
Claims (2)
1. the curve constructing method based on the control of geometrical characteristic point, it is characterised in that the following steps are included:
1) curve to be drawn of selection is closure or non-closed, according to the difference of selection, is closed or non-closed, then
Construct different systems of linear equations；
2) volume of data point is provided in the plane as interpolation point, and the volume of data point is denoted as { vi| i=1 ..., N },
And the type of every kind of data point is specified, including cusp, annulus are from intersection point, inflection point or endpoint；The cusp is curve in the data
Curvature at point is not present；The annulus is that curve passes through the data point twice from intersection point；The inflection point or endpoint are that curve exists
Curvature at the data point is zero, and the symbol of curvature changes at the data point；Occur two for non-closed curve
A endpoint；
3) according to each data point viAnd consecutive number strong point vi-1, vi+1Position calculating parameter coordinateCalculation formula is as follows:
4) each data point interpolation goes out a cubic Bézier curve, and allows geometrical characteristic point: cusp, annulus from intersection point or
Inflection point is as follows for different characteristic point difference present in interpolation point:
Cusp:
As △=B2Occurs unique cusp when -4AC=0, on curve, it is assumed that the parameter coordinate of cusp is t*, then t*It is also equation
The root of F (t), it may be assumed that
By △=B2- 4AC=0 andObtain:
Interpolation point is v, then has:
Annulus is from intersection point:
Cubic Bézier curve generates annulus in inside, indicates the corresponding two parameter coordinates in certain pointAnd
The point is annulus from intersection point, while being also interpolation point v, then has:
With
Wherein, it enables
Inflection point:
As △ > 0, there are two roots by equation F (t), respectively correspond the parameter coordinate with two inflection points, enable one of root t*∈(0,
1), it may be assumed that
F(t*)=0
New parameter h is introduced, so that:
Wherein,At this point, another root of equation F (t) is always located in except section [0,1]；
By F (t*The He of)=0Obtain:
Interpolation point is v, then has:
In addition, that is, non-cusp, annulus obtain a B é zier three times from the data point of intersection point and inflection point by the general point of interpolation
Curve, this curve do not have cusp, annulus from intersection point and inflection point:
As △ > 0, there are two roots by equation F (t), and curve will appear two inflection points, if two roots of equation is allowed to be all located at section
Except [0,1], one is obtained without cusp, annulus from the cubic Bézier curve of intersection point and inflection point, specific practice is as follows:
Enabling t=-1 is one of root of equation F (t), it may be assumed that
F (- 1)=A-B+C=0
With season symmetry axisBetween straight line t=0 and straight line t=1, and enable the parameter coordinate of interpolation point
By F (- 1)=A-B+C=0 andIt obtains:
3(1+2t*)P3-2t*P2-P1=0
In addition, interpolation point is v, then have:
5) curved section for obtaining step 4) is stitched together, and meets the following condition of continuity:
G0It is continuous:
Pi,0+Pi,1+Pi,2+Pi,3=Pi+1,0
G1It is continuous:
3Pi,0+2Pi,1+Pi,2-3Pi+1,0=-Pi+1,1λi
G2It is continuous:
κi(1)=κi+1(0)
Wherein, i indicates i-th section of curve；
Four kinds of curved sections: cusp, annulus are from intersection point, inflection point and without cusp, annulus from intersection point, the B é three times of three kinds of points of inflection point
Any two kinds of curved sections in zier curved section are stitched together, and obtain 16 kinds of joining methods:
(5.1) according to the correlation formula and G of cusp in step 4)0Continuous condition, obtains Pi,1、Pi,2And Pi,3Expression formula:
(5.2) similarly, P in i+1 curve is obtainedi+1,1Expression formula:
(5.3) by P obtained in step (5.1) and (5.2)i,1、Pi,2And Pi+1,1Expression formula bring G into1In continuous condition, obtain
To Pi,0、Pi+1,0And Pi+2,0Between relational expression；
6) system of linear equations is constructed；
7) solution procedure 6) in system of linear equations, obtain all splice point Pi,0Coordinate, closed curve i=1 ..., N be non-
Closed curve i=1 ..., N-3, then P is obtained by the expression formula in step 6)i,1, Pi,2Pi,3；
8) condition is utilized | κi(1) |=| κi+1(0) | improve the continuity of curve, the P that step 7) is obtainedi,0, Pi,1Pi,2And Pi,3
It is brought intoObtain new λi, step 7) and step 8) are repeated, until | |
κi(1)|-|κi+1(0) | | < ε, ε are previously given threshold value, take ε=10-10；
9) after the iteration stopping in step 8), the P that is obtained using step 7)i,0,Pi,1,Pi,2And Pi,3Draw curve.
2. the curve constructing method as described in claim 1 based on the control of geometrical characteristic point, it is characterised in that in step 6), institute
State construction system of linear equations method particularly includes: draw the system of linear equations of closed curve and non-closed curve construction:
(6.1) closed curve:
N number of data point is provided, interpolation obtains N curve to get P in N number of step (5.3) is arrivedi,0、Pi+1,0And Pi+2,0Between pass
It is formula, is uniformly written as follow form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N
Wherein, Ei,1,Ei,2,Ei,3And CiBe byλi,αi,βi,hi,viAnd viType determine；
(6.2) non-closed curve:
N number of data point is equally provided, first and two endpoints of the last one data point as curve are enabled, interpolation obtains N-2
Curve is to get the P into N-3 step 5)i,0,Pi+1,0And Pi+2,0Between relational expression, be uniformly written as follow form:
Ei,1Pi,0+Ei,2Pi+1,0+Ei,3Pi+2,0=Ci, i=1 ..., N-3
Wherein, P1,0=v1,PN-1,0=vN。
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| CN117351109A (en) * | 2023-09-05 | 2024-01-05 | 中交第二公路勘察设计研究院有限公司 | Method for reconstructing section curve of shield tunnel |
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| CN117351109A (en) * | 2023-09-05 | 2024-01-05 | 中交第二公路勘察设计研究院有限公司 | Method for reconstructing section curve of shield tunnel |
| CN117351109B (en) * | 2023-09-05 | 2024-06-07 | 中交第二公路勘察设计研究院有限公司 | Method for reconstructing section curve of shield tunnel |
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