CN105719347A - Recurrent curve surface construction method based on LW curve surface and device thereof - Google Patents

Abstract

The invention discloses a recurrent curve surface construction method based on LW curve surface and a device thereof. The recurrent curve surface construction method is characterized in that points of a curve can be calculated by adopting the recurrent algorithm, and then the points of the curve can be acquired; the derivative calculation of the points of the L curve can be carried out, and the higher derivative of the L curve can be acquired; by setting n+m vertexes and correcting node information, W curves of m segments can be generated; and the recurrent curve can be acquired according to the weights of the vertexes and the node information. The recurrent nature of the form can be expressed by adopting the curves of different forms, and the node parameters suitable for the recurrent curves can be defined, and then the object of unifying the curve surfaces of different forms can be expressed in a unified way. The conversion with different curve surface forms can be carried out, and the curve surfaces of different definition forms can be expressed by the computer using the same data format.

Description

A kind of Recurrent curve Surface Construction method based on LW curve and surface and device thereof

Technical field

The present invention relates to Computer-aided Design Technology field, particularly relate to a kind of Recurrent curve Surface Construction method based on LW curve and surface and device thereof.

Background technology

Computer-aided Geometric Design (ComputerAidedGeometricDesign, CAGD) it is close ties production practices and the new branch of science constantly improving development along with the needs of production practices, its range of application, except aviation, shipbuilding, this three big manufacturing industry of automobile, further relates to the technical fields such as architectural design, biological engineering, medical diagnosis, space material, electronic engineering, robot, clothes, shoes and hats modelling.Along with the development of computer graphics, it is also widely used for the fields such as computer vision, topography and geomorphology, military combat simulation, cartoon making, multimedia technology.Therefore along with Computer Graphic Demonstration is for the enhancing day by day of verity, real-time and interactivity requirements, along with geometry designs object draws close the day by day obvious of this trend towards multiformity, particularity and topological structure complexity, requirement for configuration design is also just more and more higher, then Curve On The Surface Modeling Technology obtain significant progress.

In configuration design, curve and surface can not describe with quadratic equation, usually only provides general shape or only knows that it passes through some space finite points, it is necessary to special method construct, and for a long time, the parameter curve representation of a surface is always up describing the main tool of geometry.Adopt the curve and surface of recursive form, compare the formula of the initial definition of curve and surface, be more suitable for being calculated in a computer, and calculate more stable.Existing a series of CAGD software adopts different formative methods, and the data exchange between them becomes very frequent, and therefore on L-curve curved surface basis, it is necessary for setting up consistent curve and surface representation.

For in the research of Recurrent curve, envelope define the L-curve that draws and get the attention due to its excellent characteristic.L-curve has several different representation, based on the form of Definition of L-curve as follows:

Definition 1:

$Q_i^r\left(t\right)=\left\{\begin{array}{c}{p}_i,r=0;i=0,\mathrm{1..}n\\ \frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}{Q}_{i-1}^{r-1}\left(t\right)+\frac{t-{\mathrm{\α}}_{i+r-1}}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}{Q}_i^{r-1}\left(t\right),r=1,\mathrm{2...}n,i=r,r+1,\mathrm{...}n\end{array}\right.;$

ObtainedFor n L-curve.Wherein

$0\≤\frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}},\frac{t-{\mathrm{\α}}_{i+r-1}}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\≤1;$

If

Then this L-curve is called W curve

Definition 2:

Given n+1 sings plane or summit spatially

${\mathrm{\ω}}_i^r\left(t\right)=\left\{\begin{array}{c}{\mathrm{\ω}}_i,r=0;i=0,\mathrm{1...}n\\ \frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}{\mathrm{\ω}}_{i-1}^{r-1}\left(t\right)+\frac{t-{\mathrm{\α}}_{i+r-1}}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}{\mathrm{\ω}}_i^{r-1}\left(t\right),r=1,\mathrm{2...}n,i=r,r+1,\mathrm{...}n\end{array}\right.;$

$Q_i^r\left(t\right)=\left\{\begin{array}{c}{p}_i,r=0;i=0,\mathrm{1...}n\\ \frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\frac{{\mathrm{\ω}}_{i-1}^{r-1}\left(t\right)}{{\mathrm{\ω}}_i^r\left(t\right)}{Q}_{i-1}^{r-1}\left(t\right)+\frac{t-{\mathrm{\α}}_{i+r-1}}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\frac{{\mathrm{\ω}}_i^{r-1}\left(t\right)}{{\mathrm{\ω}}_i^r\left(t\right)}{Q}_i^{r-1}\left(t\right),r=1,\mathrm{2...}n,i=r,r+1,\mathrm{...}n\end{array}\right.;$

ObtainedFor n reasonable L-curve.Wherein

$0\≤\frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}},\frac{t-{\mathrm{\α}}_{i+r-1}}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\≤1$

If $0\≤\frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\≤1,0\≤\frac{{\mathrm{\β}}_i-t}{{\mathrm{\β}}_i-{\mathrm{\α}}_{i+r-1}}\≤1;$

Then this reasonable L-curve is called reasonable W curve.

In prior art cannot the multi-form curve and surface of unified representation, different curve and surface forms cannot be changed simultaneously.

Summary of the invention

It is an object of the invention to overcome the deficiencies in the prior art, the invention provides a kind of Recurrent curve Surface Construction method based on LW curve and surface and device thereof, it is provided that different shaping technology Unified Form in data, it is achieved the data conversion of different shaping method.

In order to solve the problems referred to above, the present invention proposes a kind of Recurrent curve Surface Construction method based on LW curve and surface, and described method includes:

Adopt recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

Point on L-curve is carried out derivative calculations, it is thus achieved that the higher derivative of L-curve；

Set n+m summit and corresponding nodal information, produce m section W curve；

Weights and nodal information according to summit obtain Recurrent curve.

Preferably, the point on curve is calculated by described employing recursive algorithm, it is thus achieved that the step of the point on curve includes:

Initialize the data structure of curve；

Curve is carried out K iteration；

Obtaining upper triangular matrix, the value of this matrix is the point on required curve.

Correspondingly, the present invention also provides for a kind of Recurrent curve Surface Construction device based on LW curve and surface, and described device includes:

Point acquisition module, for adopting recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

Derivative acquisition module, for carrying out derivative calculations to the point on L-curve, it is thus achieved that the higher derivative of L-curve；

W curve generation module, is used for setting n+m summit and corresponding nodal information, produces m section W curve；

Recurrent curve acquisition module, obtains Recurrent curve for the weights according to summit and nodal information.

Preferably, described some acquisition module includes:

Initialization unit, for initializing the data structure of curve；

Iteration unit, for carrying out K iteration to curve；

Matrix acquisition module, is used for obtaining upper triangular matrix, and the value of this matrix is the point on required curve.

In embodiments of the present invention, utilize the recurrence essence of multi-form curve representation, the node parameter suitable by defining Recurrent curve, reach the purpose of the multi-form curve and surface of unified representation, can change from different curve and surface forms, facilitate computer to use same data form to represent the curve and surface under different definition form.

Accompanying drawing explanation

In order to be illustrated more clearly that the embodiment of the present invention or technical scheme of the prior art, the accompanying drawing used required in embodiment or description of the prior art will be briefly described below, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skill in the art, under the premise not paying creative work, it is also possible to obtain other accompanying drawing according to these accompanying drawings.

Fig. 1 is the schematic flow sheet of the Recurrent curve Surface Construction method based on LW curve and surface of the embodiment of the present invention；

Fig. 2 is the LW curve recursive calculation schematic diagram of the embodiment of the present invention；

Fig. 3 is the process schematic adopting recursive algorithm that the point on curve is calculated in the embodiment of the present invention；

Fig. 4 is the LW curved surface single calculation schematic diagram of the embodiment of the present invention；

Fig. 5 is the structure composition schematic diagram of the Recurrent curve Surface Construction device based on LW curve and surface of the embodiment of the present invention.

Detailed description of the invention

Below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is clearly and completely described, it is clear that described embodiment is only a part of embodiment of the present invention, rather than whole embodiments.Based on the embodiment in the present invention, the every other embodiment that those of ordinary skill in the art obtain under not making creative work premise, broadly fall into the scope of protection of the invention.

Fig. 1 is the schematic flow sheet of the Recurrent curve Surface Construction method based on LW curve and surface of the embodiment of the present invention, as it is shown in figure 1, the method includes:

S1, adopts recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

S2, carries out derivative calculations to the point on L-curve, it is thus achieved that the higher derivative of L-curve；

S3, sets n+m summit and corresponding nodal information, produces m section W curve；

S4, obtains Recurrent curve according to the weights on summit and nodal information.

Wherein, S1 farther includes:

Initialize the data structure of curve；

Curve is carried out K iteration；

Obtaining upper triangular matrix, the value of this matrix is the point on required curve.

In being embodied as, as shown in Figure 2 and Figure 3, including (1) initialization data structure: definition status number of times p=0；M sensing α [1], n point to the first row of β [1], two-dimensional array vex [n] [p] and preserve control point, and every string preserves iteration once rear calculated result thereafter, takes and determines s0≤t≤s1, and iteration starts；

(2) kth iteration:

A) initial i=0, m=i+k, n=i calculate:

$vex=\frac{\mathrm{\β}\[n\]-t}{\mathrm{\β}\[n\]-\mathrm{\α}\[m\]}vex\[i\]\[k-1\]+\frac{t-\mathrm{\α}\[m\]}{\mathrm{\β}\[n\]-\mathrm{\α}\[m\]}vex\[i+1\]\[k-1\]$

Result is saved in vec [i] [k]；

B) i++, m++, n++；

C) until i=n；Repeat a, exit current iteration；

(3) until k=p, step 2 is repeated；Exit algorithm；

The result vex matrix finally obtained is a upper triangular matrix, and wherein, the value on vex [n] [p] is the point on required curve.

In S2, on L-curve, the derivative calculations formula of point is as follows:

$Q^{,}\left(t\right)={\mathrm{na}}_0^n\left(L_0^{n-1}\right(t)-L_1^{n-1}(t\left)\right);$

Wherein $a_0^n=\frac{1}{\mathrm{\β}\[n\]-\mathrm{\α}\[2n\]};$

AndIt is saved in vex [0] [p-1] respectively, in vex [1] [p-1].

Therefore can calculating the first derivative of point on curve according to formula, recursively applying this formula can in the hope of the higher derivative of curve.

In S3, the splicing seriality of m section W curve needs to meet some requirements:

Consider two sections of curves therein:

$\left\{\begin{array}{c}{w}_0^n\left(t\right),t\∈\[s_0,s_1\]\\ w_1^n\left(t\right),t\∈\[s_1,s_2\]\end{array}\right.$

C is reached in stitching portion^n-1Continuous print condition is:

$-\frac{b_0^n}{a_0^n}=\frac{1-b_1^n}{a_1^n},a_i^j<0$

Or

$-\frac{1-b_0^n}{a_0^n}=-\frac{b_1^n}{a_1^n},a_i^j>0$

And

It is that strictly monotone is with increasing with subtrahend row.

According to above-mentioned condition, under data form, calculate a, b:

$a_i^j=\frac{-1}{\mathrm{\&beta;}\&lsqb;i\&rsqb;-\mathrm{\&alpha;}\&lsqb;i+j\&rsqb;}$

$b_i^j=\frac{\mathrm{\&beta;}\&lsqb;i\&rsqb;}{\mathrm{\&beta;}\&lsqb;i\&rsqb;-\mathrm{\&alpha;}\&lsqb;i+j\&rsqb;}$

Can draw and whether meet corresponding successional splicing condition.

In S4, nodal information then determines curve type, although Recurrent curve can represent a lot of curve, wherein comprises some conventional curves.

Parameter t ∈ [s₀,s₁] time,

Arranging nodal value α is definite value-s₁, nodal value β is definite value s₀, then curve is the L-curve that bezier form is of equal value:

Nodal value α is set for belonging to [s₀,s₁] ascending series and α [1]=s₀,

α [n+p-1]=s1, nodal value β is α [1], α [2] ... α [n], then curve is the Recurrent curve that Lagrange Interpolation is of equal value, note that while that Lagrange's interpolation curve is not L-curve, but still there is similar recursive form, therefore in the embodiment of the present invention, data structure is still suitable for.

Nodal value α is set for being arbitrarily incremented by node, nodal value

β [i]=-α [i+p-1], i=1,2 ..., n, then curve is a certain with nurbs curve section of L-curve of equal value.

Recurrent curve has natural advantage for the expression of divided method, because Curve Subdivision method is exactly substantially recursive interpolation.Each interpolation section is chosen different segmentations and just can draw different divided methods, essentially consist in choosing of blending functions, and nodal value is directly related and blending functions.Illustrate for DeRham algorithm (i.e. chamfer algorithm):

Initial condition: n+1 summit, nodal information；

Carry out an iteration:

A) nodal information is utilized to calculate chamfer parameter

B) it is calculated with a fixing parameter value t1, obtains n new summit；

C) it is calculated with a fixing parameter value t2 again, also obtains n new summit；

D) this 2n is sung set of vertices and obtain the new summit of 2n+1 altogether, here it is the control point of chamfer next time；

An iteration terminates.

If the weight factor in the data structure that the embodiment of the present invention is defined is established as non-definite value, so illustrate that represented curve is reasonable Recurrent curve, can be seen that from the definition of reasonable L-curve, weight factor and control point each step recurrence are all apply identical calculating conversion, therefore, can use homogeneous coordinates that from three-dimensional, control point coordinate is mapped to 4 dimensions, in the homogeneous coordinates of 4 dimensions, reasonable L-curve becomes non-reasonable L-curve, thus the most of computational methods under non-rational form can be applied, after obtaining result, reprojection returns 3 dimension spaces.

Adopt the reasonable L-curve form of equal value with secondary bezier form, it is possible to Precise Representation conic section, the wherein selection of three control point weight factors respectively w₀, w₁, w₂。

Set w₀=w₂, set the w of different relation₁To represent different types of conic section:

(W₁/w₀)²=1, then it represents that be parabola；

(W₁/w₀)²> 1, then it represents that be hyperbola；

(W₁/w₀)²< 1, then it represents that be oval.

From the definition of curved surface, L curved surface is the curved surface of tensor product form, adopts the recursive calculation algorithm of point on curve can obtain the point on curved surface, and 2 times application algorithm 1 can obtain the point on curved surface.Still first calculate v direction as first calculating u direction, have the number of times of boundary curve to determine.

As shown in Figure 4, initialization data structure, definition status number of times p=0, q=0；I points to α [1], j and points to β [1], a sensing φ [1], and b points toInitial (n+1) (m+1) the individual control point of the preservation of two-dimensional array vex [n] [m], takes and determines s₀≤ u≤s₁, z₀≤ v≤z₁；

If p is < q；

Use q, a, b, v, vex [n] [k] as q algorithm of initial condition application 1；

Using p, i, j, u, the result of second step, as q algorithm of initial condition application 1；

Recording a demerit of finally obtaining is exactly the point on curved surface.

Correspondingly, the embodiment of the present invention also provides for a kind of Recurrent curve Surface Construction device based on LW curve and surface, as it is shown in figure 5, this device includes:

Point acquisition module 1, for adopting recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

Derivative acquisition module 2, for carrying out derivative calculations to the point on L-curve, it is thus achieved that the higher derivative of L-curve；

W curve generation module 3, is used for setting n+m summit and corresponding nodal information, produces m section W curve；

Recurrent curve acquisition module 4, obtains Recurrent curve for the weights according to summit and nodal information.

Further, some acquisition module 1 includes:

Initialization unit, for initializing the data structure of curve；

Iteration unit, for carrying out K iteration to curve；

Matrix acquisition module, is used for obtaining upper triangular matrix, and the value of this matrix is the point on required curve.

In assembly of the invention embodiment, the function of each functional module referring to the flow processing in the inventive method embodiment, can repeat no more here.

In embodiments of the present invention, utilize the recurrence essence of multi-form curve representation, the node parameter suitable by defining Recurrent curve, reach the purpose of the multi-form curve and surface of unified representation, can change from different curve and surface forms, facilitate computer to use same data form to represent the curve and surface under different definition form.

One of ordinary skill in the art will appreciate that all or part of step in the various methods of above-described embodiment can be by the hardware that program carrys out instruction relevant and completes, this program can be stored in a computer-readable recording medium, storage medium may include that read only memory (ROM, ReadOnlyMemory), random access memory (RAM, RandomAccessMemory), disk or CD etc..

Additionally, the Recurrent curve Surface Construction method based on LW curve and surface and the device thereof that above the embodiment of the present invention are provided are described in detail, principles of the invention and embodiment are set forth by specific case used herein, and the explanation of above example is only intended to help to understand method and the core concept thereof of the present invention；Simultaneously for one of ordinary skill in the art, according to the thought of the present invention, all will change in specific embodiments and applications, in sum, this specification content should not be construed as limitation of the present invention.

Claims (4)

1. the Recurrent curve Surface Construction method based on LW curve and surface, it is characterised in that described method includes:

Adopt recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

Point on L-curve is carried out derivative calculations, it is thus achieved that the higher derivative of L-curve；

Set n+m summit and corresponding nodal information, produce m section W curve；

Weights and nodal information according to summit obtain Recurrent curve.

2. the Recurrent curve Surface Construction method based on LW curve and surface as claimed in claim 1, it is characterised in that the point on curve is calculated by described employing recursive algorithm, it is thus achieved that the step of the point on curve includes:

Initialize the data structure of curve；

Curve is carried out K iteration；

Obtaining upper triangular matrix, the value of this matrix is the point on required curve.

3. the Recurrent curve Surface Construction device based on LW curve and surface, it is characterised in that described device includes:

Point acquisition module, for adopting recursive algorithm that the point on curve is calculated, it is thus achieved that the point on curve；

Derivative acquisition module, for carrying out derivative calculations to the point on L-curve, it is thus achieved that the higher derivative of L-curve；

W curve generation module, is used for setting n+m summit and corresponding nodal information, produces m section W curve；

Recurrent curve acquisition module, obtains Recurrent curve for the weights according to summit and nodal information.

4. the Recurrent curve Surface Construction device based on LW curve and surface as claimed in claim 3, it is characterised in that described some acquisition module includes:

Initialization unit, for initializing the data structure of curve；

Iteration unit, for carrying out K iteration to curve；

Matrix acquisition module, is used for obtaining upper triangular matrix, and the value of this matrix is the point on required curve.