CN110335328A - A kind of curve plotting method based on B-spline, system and storage medium - Google Patents
A kind of curve plotting method based on B-spline, system and storage medium Download PDFInfo
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Abstract
The invention discloses a kind of curve plotting method based on B-spline, system and storage mediums, and wherein method includes: building B-spline expression formula, input interpolation point；Interpolation point is subjected to centripetal parametrization；Determine N basic function sequence node in B-spline expression formula；Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；It calculates the solution equation based on matrix and obtains curve controlled point sequence；Utilize the point-rendering curve in control point sequence.The present invention is based on the drafting that B-spline function carries out curve, the description to general edge contour can be preferably adapted to, and expression formula is polynomial of lower degree, calculation amount is small；The drafting of complex curve can be realized by setting interpolation point in ability to express user based on batten, easy to operate；The slickness requirement that identical vector reaches closed curve is added based on first and last endpoint.
Description
Technical field
The present embodiments relate to Computer-aided Geometric Design technical field, especially a kind of curve based on B-spline is drawn
Method, system and storage medium processed.
Background technique
CAGD is widely used in industrial production and manufacturing, Geometric Modeling.Mathematical method among these has formed complete set
System (NURBS) generallyd use by industry.Interpolation technique has long history as a kind of technology, there is various interpolation shapes
Formula.Low order piecewise polynomial is generally required in work to express the curve that we want, and curve will meet specific slickness
It is required with boundary.
It is not the curve of straight line for target area edge in current research, the polygon broken line generally used needs
Many interpolation points are drawn, and cannot be well close to cambered contour edge.And broken line slickness is inadequate.Further for diameter
To the Drawing of Curve for having same shape after scaling, existing broken line segmented mode implements difficulty.
Summary of the invention
In order to solve at least one above-mentioned technical problem, the invention proposes a kind of curve plotting method based on B-spline,
System and storage medium.
To achieve the goals above, first aspect present invention provides a kind of curve plotting method based on B-spline, packet
It includes:
B-spline expression formula is constructed, interpolation point is inputted；
Interpolation point is subjected to centripetal parametrization；
Determine N basic function sequence node in B-spline expression formula；
Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；
It calculates the solution equation based on matrix and obtains curve controlled point sequence；
Utilize the point-rendering curve in control point sequence.
In the present solution, the B-spline expression formula is as follows:
Wherein, p is batten number, PiFor control point, Ni,pFor p B-spline basic function.
In the present solution, the interpolation point is denoted as Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, institute
State condition formula are as follows:
Wherein, k=0,1....n.
In the present solution, described carry out centripetal parametrization for interpolation point, i.e., so that interpolation point QkWith Qk-1Meet following formula:
Abbreviation obtains
In the present solution, N basic function sequence node in the determining B-spline expression formula, specifically includes: p B-spline basic function
It is expressed as follows by given sequence node:
{ukK=0,1 ..., m
M=n+p+1
Wherein, n corresponds to interpolation point Q subscript number, when setting n >=3, p=3, and interpolation point corresponding node meets formula:
u0=...=u3=0, um-p=...=um=1
Wherein, n=1, p=1 are set；N=2, p=2.
In the present solution, the solution equation is expressed as follows:
The tangent vector increased at first and last endpoint in the solution equation is denoted as D respectively0, Dn, i.e., the described B-spline expression formula
Become:
Then, node ukCorresponding sequence are as follows:
u0=...=um=0, um-p=...=um=1
Wherein, m=n+p+3；
As p=3, ukIt indicates are as follows:
u0=...=u3=0, un-3=...=un+6=1
Then, the solution equation turns to tridiagonal matrix and is expressed as follows:
Wherein,
In the present solution, after obtaining control point sequence, it is identical by setting first and last endpoint, make curve Natural closure, by setting
Determine the equal closure and scaling for carrying out curve of tangent vector at first and last end.
Second aspect of the present invention additionally provides a kind of Drawing of Curve system based on B-spline, the system include: memory,
Processor includes the curve plotting method program based on B-spline, the Drawing of Curve side based on B-spline in the memory
Method program realizes following steps when being executed by the processor:
B-spline expression formula is constructed, interpolation point is inputted；
Interpolation point is subjected to centripetal parametrization；
Determine N basic function sequence node in B-spline expression formula；
Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；
It calculates the solution equation based on matrix and obtains curve controlled point sequence；
Utilize the point-rendering curve in control point sequence.
In the present solution, the interpolation point is denoted as Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, institute
State condition formula are as follows:
Wherein, k=0,1....n.
A kind of computer readable storage medium is also disclosed in third aspect present invention, wraps in the computer readable storage medium
The curve plotting method program based on B-spline is included, when the curve plotting method program based on B-spline is executed by processor,
The step of realizing the curve plotting method as described in any one of the above embodiments based on B-spline.
The present invention is based on the draftings that B-spline function carries out curve, can preferably adapt to the description to general edge contour, and
Expression formula is polynomial of lower degree, and calculation amount is small；Complexity can be realized by setting interpolation point in ability to express user based on batten
The drafting of curve, it is easy to operate；The slickness requirement that identical vector reaches closed curve is added based on first and last endpoint.
Detailed description of the invention
Fig. 1 shows a kind of flow chart of the curve plotting method based on B-spline of the present invention；
Fig. 2 shows a kind of block diagrams of the Drawing of Curve system based on B-spline of the invention.
Specific embodiment
To better understand the objects, features and advantages of the present invention, with reference to the accompanying drawing and specific real
Applying mode, the present invention is further described in detail.It should be noted that in the absence of conflict, the implementation of the application
Feature in example and embodiment can be combined with each other.
In the following description, numerous specific details are set forth in order to facilitate a full understanding of the present invention, still, the present invention may be used also
To be implemented using other than the one described here other modes, therefore, protection scope of the present invention is not by described below
Specific embodiment limitation.
The embodiment of the present application can be applied to the electronic equipments such as terminal device, computer system and server, can be with crowd
Mostly other general or dedicated computing system environment or configuration operate together.Suitable for terminal device, computer system with
And the example of well-known terminal device, computing system, environment and/or configuration that the electronic equipments such as server are used together,
Including but not limited to: personal computer system, server computer system, thin client, thick client computer, hand-held or above-knee set
It is standby, microprocessor-based system, set-top box, programmable consumer electronics, NetPC Network PC, little type Ji calculate machine Xi Tong ﹑
Large computer system and the distributed cloud computing technology environment including above-mentioned any system, etc..
The electronic equipments such as terminal device, computer system and server can be in the computer executed by computer system
It is described under the general context of system executable instruction (such as program module).In general, program module may include routine, program,
Target program, component, logic and data structure etc., they execute specific task or realize specific abstract data class
Type.Computer system/server can be implemented in distributed cloud computing environment, and in distributed cloud computing environment, task is
Shoumo is executed by the remote processing devices being linked through a communication network.In distributed cloud computing environment, program module can
It include on the Local or Remote computing system storage medium of storage equipment to be located at.
Fig. 1 shows a kind of flow chart of the curve plotting method based on B-spline of the present invention.
As described in Figure 1, the invention discloses a kind of curve plotting methods based on B-spline, comprising:
S102 constructs B-spline expression formula, inputs interpolation point；
Interpolation point is carried out centripetal parametrization by S104；
S106 determines N basic function sequence node in B-spline expression formula；
S108 brings interpolation point into B-spline expression formula using the N basic function sequence node and obtains the solution of difference condition
Equation；
S110 calculates the solution equation based on matrix and obtains curve controlled point sequence；
S112 utilizes the point-rendering curve in control point sequence.
In the present solution, the B-spline expression formula is as follows:
Wherein, p is batten number, PiFor control point, Ni,pFor p B-spline basic function.
It should be noted that the basic function is exactly the fixed form of a function, that is, function only can be in this letter
The function changed on the basis of number without losing, in mathematics, basic function is the element of one group of special base of function space.It is right
Continuous function in function space can be expressed as a series of linear combination of basic functions, be like every in vector space
The linear combination that a vector can be expressed as base vector is the same.
B-spline basic function is the k rank piecewise polynomial that the parameter of a non-decreasing is determined by the sequence of t.
Let t be the set of n+1 non-decreasing number, t0≤t1≤t2≤...≤tn.Ti is known as node (knots), collection
T is closed to be known as knot vector (knot vector), and half open interval [t, ti+1) it is i-th of node interval (knot span).
If a node occurs k times, this node is the multiple section that a multiplicity (multiplicity) is k
Point, is written as ti (k).Otherwise, if ti only occurs once, it is a simple node.If node is equidistant, knot vector or
Sequence node is referred to as uniform；Otherwise it is heterogeneous.
In the present solution, the interpolation point is denoted as Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, institute
State condition formula are as follows:
Wherein, k=0,1....n.
It should be noted that the method for interpolation point parametrization includes uniformity parameter, Chord Length Parameterization, centripetal parameter
Change, using centripetal parametrization in the present embodiment, the centripetal force that the centripetal parametrization is set on one section of curve arc with
Curve, which is cut, swears that the corner from segmental arc beginning to the end is directly proportional, in addition some letters
Change it is assumed that obtaining centripetal parameterization.This method is particularly suitable for non-homogeneous data point distribution.
In the present solution, described carry out centripetal parametrization for interpolation point, i.e., so that interpolation point QkWith Qk-1Meet following formula:
Abbreviation obtains
In the present solution, N basic function sequence node in the determining B-spline expression formula, specifically includes: p B-spline basic function
It is expressed as follows by given sequence node:
{ukK=0,1 ..., m
M=n+p+1
Wherein, n corresponds to interpolation point Q subscript number, when setting n >=3, p=3, and interpolation point corresponding node meets formula:
u0=...=u3=0, um-p=...=um=1
Wherein, n=1, p=1 are set；N=2, p=2.
It should be noted that using the distribution feelings for determining that the method knot vector of sequence node can react well are invented
Condition, and obtained solution matrix is the band matrix of positive definite, and half-band width is less than p.
In the present solution, the solution equation is expressed as follows:
The tangent vector increased at first and last endpoint in the solution equation is denoted as D respectively0, Dn, i.e., the described B-spline expression formula
Become:
Then, node ukCorresponding sequence are as follows:
u0=...=um=0, um-p=...=um=1
Wherein, m=n+p+3；
It should be noted that the constraint condition met at this time is,
As p=3, ukIt indicates are as follows:
u0=...=u3=0, un+3=...=un+6=1
Then, the solution equation turns to tridiagonal matrix and is expressed as follows:
Wherein,
In the present solution, after obtaining control point sequence, it is identical by setting first and last endpoint, make curve Natural closure, by setting
Determine the equal closure and scaling for carrying out curve of tangent vector at first and last end.
Detailed process are as follows: vector is led for the k rank of batten C, expression formula has:
It is corresponding to askLeading vector positive direction is the increased direction u, and find out corresponding u leads vector C (1) (u), then does
90 degree are rotated counterclockwise, normal vector positive direction is acquiredHere normal vector is unit, according to given contraction
Corresponding points after the available song on-line parameters u contraction of size:
There is the foundation calculated for the point after the offset of certain point on complex curve through the above way.
It obtains acquiring area by the B-spline function expression formula to get to B-spline function expression formula after control point sequence
Between in [0,1] any parameter u point, according to the point-rendering curve of the parameter u.
When screen carries out Drawing of Curve, usually segment splices, therefore can choose dense argument sequence, obtains institute
Corresponding screen coordinate reconnects each point.
It should be noted that the Spline Node sequence difference of selection leads to different spline Basis using B-spline interpolation method
Function or additional constraint condition are different, cause solution matrix different, and the Splines Interpolation Curve expression acquired is also different,
The curve of drafting also has difference slightly.But curve can be met and pass through this basic demand of the interpolation point of user's input.
Fig. 2 shows a kind of block diagrams of the Drawing of Curve system based on B-spline of the invention.
As shown in Fig. 2, second aspect of the present invention additionally provides a kind of Drawing of Curve system based on B-spline, the system packet
Include: memory 21, processor 22 include the curve plotting method program based on B-spline in the memory, described to be based on B sample
The curve plotting method program of item realizes following steps when being executed by the processor:
B-spline expression formula is constructed, interpolation point is inputted；
Interpolation point is subjected to centripetal parametrization；
Determine N basic function sequence node in B-spline expression formula；
Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；
It calculates the solution equation based on matrix and obtains curve controlled point sequence；
Utilize the point-rendering curve in control point sequence.
In the present solution, the interpolation point is denoted as Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, institute
State condition formula are as follows:
Wherein, k=0,1....n.
In the present solution, the interpolation point is denoted as Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, institute
State condition formula are as follows:
Wherein, k=0,1....n.
It should be noted that the method for interpolation point parametrization includes uniformity parameter, Chord Length Parameterization, centripetal parameter
Change, using centripetal parametrization in the present embodiment, the centripetal force that the centripetal parametrization is set on one section of curve arc with
Curve, which is cut, swears that the corner from segmental arc beginning to the end is directly proportional, in addition some letters
Change it is assumed that obtaining centripetal parameterization.This method is particularly suitable for non-homogeneous data point distribution.
In the present solution, described carry out centripetal parametrization for interpolation point, i.e., so that interpolation point QkWith Qk-1Meet following formula:
Abbreviation obtains
In the present solution, N basic function sequence node in the determining B-spline expression formula, specifically includes: p B-spline basic function
It is expressed as follows by given sequence node:
{ukK=0,1 ..., m
M=n+p+1
Wherein, n corresponds to interpolation point Q subscript number, when setting n >=3, p=3, and interpolation point corresponding node meets formula:
U0=...=u3=0, um-p=...=um=1
Wherein, n=1, p=1 are set；N=2, p=2.
It should be noted that using the distribution feelings for determining that the method knot vector of sequence node can react well are invented
Condition, and obtained solution matrix is the band matrix of positive definite, and half-band width is less than p.
In the present solution, the solution equation is expressed as follows:
The tangent vector increased at first and last endpoint in the solution equation is denoted as D respectively0, Dn, i.e., the described B-spline expression formula
Become:
Then, node ukCorresponding sequence are as follows:
u0=...=um=0, um-p=...=um=1
Wherein, m=n+p+3；
It should be noted that the constraint condition met at this time is,
As p=3, ukIt indicates are as follows:
u0=...=u3=0, un+3=...=un+6=1
Then, the solution equation turns to tridiagonal matrix and is expressed as follows:
Wherein,
In the present solution, after obtaining control point sequence, it is identical by setting first and last endpoint, make curve Natural closure, by setting
Determine the equal closure and scaling for carrying out curve of tangent vector at first and last end.
Detailed process are as follows: vector is led for the k rank of batten C, expression formula has:
It is corresponding to askLeading vector positive direction is the increased direction u, and find out corresponding u leads vector C(1)(u), it then does inverse
Clockwise is rotated by 90 °, and acquires normal vector positive directionHere normal vector is unit, according to given contraction size
Corresponding points after available song on-line parameters u contraction:
There is the foundation calculated for the point after the offset of certain point on complex curve through the above way.
It obtains acquiring area by the B-spline function expression formula to get to B-spline function expression formula after control point sequence
Between in [0,1] any parameter u point, according to the point-rendering curve of the parameter u.
When screen carries out Drawing of Curve, usually segment splices, therefore can choose dense argument sequence, obtains institute
Corresponding screen coordinate reconnects each point.
It should be noted that the Spline Node sequence difference of selection leads to different spline Basis using B-spline interpolation method
Function or additional constraint condition are different, cause solution matrix different, and the Splines Interpolation Curve expression acquired is also different,
The curve of drafting also has difference slightly.But curve can be met and pass through this basic demand of the interpolation point of user's input.
A kind of computer readable storage medium is also disclosed in third aspect present invention, wraps in the computer readable storage medium
The curve plotting method program based on B-spline is included, when the curve plotting method program based on B-spline is executed by processor,
The step of realizing the curve plotting method as described in any one of the above embodiments based on B-spline.
The present invention is based on the draftings that B-spline function carries out curve, can preferably adapt to the description to general edge contour, and
Expression formula is polynomial of lower degree, and calculation amount is small；Complexity can be realized by setting interpolation point in ability to express user based on batten
The drafting of curve, it is easy to operate；The slickness requirement that identical vector reaches closed curve is added based on first and last endpoint.
In several embodiments provided herein, it should be understood that disclosed device and method can pass through it
Its mode is realized.Apparatus embodiments described above are merely indicative, for example, the division of the unit, only
A kind of logical function partition, there may be another division manner in actual implementation, such as: multiple units or components can combine, or
It is desirably integrated into another system, or some features can be ignored or not executed.In addition, shown or discussed each composition portion
Mutual coupling or direct-coupling or communication connection is divided to can be through some interfaces, the INDIRECT COUPLING of equipment or unit
Or communication connection, it can be electrical, mechanical or other forms.
Above-mentioned unit as illustrated by the separation member, which can be or may not be, to be physically separated, aobvious as unit
The component shown can be or may not be physical unit；Both it can be located in one place, and may be distributed over multiple network lists
In member；Some or all of units can be selected to achieve the purpose of the solution of this embodiment according to the actual needs.
In addition, each functional unit in various embodiments of the present invention can be fully integrated in one processing unit, it can also
To be each unit individually as a unit, can also be integrated in one unit with two or more units；It is above-mentioned
Integrated unit both can take the form of hardware realization, can also realize in the form of hardware adds SFU software functional unit.
Those of ordinary skill in the art will appreciate that: realize that all or part of the steps of above method embodiment can pass through
The relevant hardware of program instruction is completed, and program above-mentioned can store in computer-readable storage medium, which exists
When execution, step including the steps of the foregoing method embodiments is executed；And storage medium above-mentioned includes: movable storage device, read-only deposits
Reservoir (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), magnetic disk or
The various media that can store program code such as CD.
If alternatively, the above-mentioned integrated unit of the present invention is realized in the form of software function module and as independent product
When selling or using, it also can store in a computer readable storage medium.Based on this understanding, the present invention is implemented
Substantially the part that contributes to existing technology can be embodied in the form of software products the technical solution of example in other words,
The computer software product is stored in a storage medium, including some instructions are used so that computer equipment (can be with
It is personal computer, server or network equipment etc.) execute all or part of each embodiment the method for the present invention.
And storage medium above-mentioned includes: that movable storage device, ROM, RAM, magnetic or disk etc. are various can store program code
Medium.
The above description is merely a specific embodiment, but scope of protection of the present invention is not limited thereto, any
Those familiar with the art in the technical scope disclosed by the present invention, can easily think of the change or the replacement, and should all contain
Lid is within protection scope of the present invention.Therefore, protection scope of the present invention should be based on the protection scope of the described claims.
Claims (10)
1. a kind of curve plotting method based on B-spline, which is characterized in that the described method includes:
B-spline expression formula is constructed, interpolation point is inputted；
Interpolation point is subjected to centripetal parametrization；
Determine N basic function sequence node in B-spline expression formula；
Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；
It calculates the solution equation based on matrix and obtains curve controlled point sequence；
Utilize the point-rendering curve in control point sequence.
2. a kind of curve plotting method based on B-spline according to claim 1, which is characterized in that the B-spline expression
Formula is as follows:
Wherein, p is batten number, PiFor control point, NI, pFor p B-spline basic function.
3. a kind of curve plotting method based on B-spline according to claim 2, which is characterized in that the interpolation point note
For Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, the condition formula are as follows:
Wherein, k=0,1....n.
4. a kind of curve plotting method based on B-spline according to claim 3, which is characterized in that described by interpolation point
Centripetal parametrization is carried out, i.e., so that interpolation point QkWith Qk-1Meet following formula:
Abbreviation obtains
5. a kind of curve plotting method based on B-spline according to claim 4, which is characterized in that the determining B-spline
N basic function sequence node in expression formula, specifically includes: p B-spline basic function is expressed as follows by given sequence node:
{ukK=0,1..., m
M=n+p+1
Wherein, n corresponds to interpolation point Q subscript number, when setting n >=3, p=3, and interpolation point corresponding node meets formula:
u0=... u3=0, um-p=...=um=1
Wherein, n=1, p=1 are set；N=2, p=2.
6. a kind of curve plotting method based on B-spline according to claim 5, which is characterized in that the solution equation
It is expressed as follows:
The tangent vector increased at first and last endpoint in the solution equation is denoted as D respectively0, Dn, i.e., the described B-spline expression formula becomes:
Then, node ukCorresponding sequence are as follows:
u0=...=um=0, um-p=...=um=1
Wherein, m=n+p+3；
As p=3, ukIt indicates are as follows:
u0=...=u3=0, un+3=...=un+6=1
Then, the solution equation turns to tridiagonal matrix and is expressed as follows:
Wherein,
7. a kind of curve plotting method based on B-spline according to claim 6, which is characterized in that obtain control point sequence
It is identical by setting first and last endpoint after column, make curve Natural closure, the equal progress curve of the tangent vector by setting first and last end
Closure and scaling.
8. a kind of Drawing of Curve system based on B-spline, which is characterized in that the system includes: memory, processor, described to deposit
It include the curve plotting method program based on B-spline in reservoir, the curve plotting method program based on B-spline is by the place
Reason device realizes following steps when executing:
B-spline expression formula is constructed, interpolation point is inputted；
Interpolation point is subjected to centripetal parametrization；
Determine N basic function sequence node in B-spline expression formula；
Interpolation point is brought into B-spline expression formula using the N basic function sequence node obtains the solution equation of difference condition；
It calculates the solution equation based on matrix and obtains curve controlled point sequence；
Utilize the point-rendering curve in control point sequence.
9. a kind of Drawing of Curve system based on B-spline according to claim 8, which is characterized in that the interpolation point note
For Qk, the interpolation point parametrization is to solve the C (u) for meeting condition formula, the condition formula are as follows:
Wherein, k=0,1....n.
10. a kind of computer readable storage medium, which is characterized in that include being based on B sample in the computer readable storage medium
The curve plotting method program of item when the curve plotting method program based on B-spline is executed by processor, realizes such as right
It is required that the step of curve plotting method described in any one of 1 to 7 based on B-spline.
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| CN111260724A (en) * | 2020-01-07 | 2020-06-09 | 王伟佳 | Example segmentation method based on periodic B spline |
| CN112330770A (en) * | 2020-10-21 | 2021-02-05 | 同济大学 | Automatic generation system for engineering curve group |
| CN112666949A (en) * | 2020-12-24 | 2021-04-16 | 武汉理工大学 | Ship path planning method, system and storage medium |
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