CN105243234A - Perimeter-maintaining curve subdivision method and system - Google Patents

Abstract

The invention discloses a perimeter-maintaining curve subdivision method and system. The method comprises the following steps: obtaining information of an un-subdivided curve; carrying out four-point triple subdivision processing of fusion approach and interpolation on the un-subdivided curve, and obtaining a curve subdivision mode; modifying parameters of the curve subdivision mode and enabling second derivatives of the curve in the curve subdivision mode to be continuous; adjusting the parameters within a parameter range with continuous second derivatives of the curve, and determining an perimeter error of the curve before and after a subdivision to be within a threshold range; and outputting a perimeter-maintaining subdivided curve of fusion approach and interpolation. According to the perimeter-maintaining curve subdivision method and system, the invariable perimeter before and after the subdivision can be achieved by a fusion approach and interpolation subdivision method; the smoothness of the curve can be improved; and the use experience of a user is improved.

Description

A kind of Curve Subdivision method and system of protecting girth

Technical field

The present invention relates to technical field of image processing, particularly relate to a kind of Curve Subdivision method and system of protecting girth.

Background technology

In CAGD (computer-aided geometry graphic designs) field, segmentation is a kind of method of effective Generating Smooth Curves and curved surface, because it can process randomly topologically structured.In fact divided method defines from the controlling polygon of an initial any given control mesh or a surperficial curve, according to the refining rule of particular design, makes restricted curve or surface can reach certain smoothness and continuity; Close classification only need store discrete point range (from discrete control vertex to discrete display point), can improve curve or the curved surface speed in computer calculate, generation and display significantly; Thus paid attention to widely, in computer animation, visualization in scientific computing, manufacturing reverse-engineering and Medical Image Processing, had application.

In each subdivision rules, new vertex position can come from the row vector of previous summit effect patch matrix, and therefore, its position can be counted as the summation of the certain weights on old summit in the past; In actual applications, subdivision scheme can be divided into two classes: a class is interpolation subdividing, and another kind of is approach segmentation, both has the advantage of oneself; About interpolation subdividing scheme, because it will through all initially given reference mark, have a lot of constraints, this subdivision is applicable to specific commercial Application; Approach subdivision scheme and can be advantageously used in intersection operation, and produce a restrictive curve there is higher flatness and continuity; Therefore, disclose the substantial connection between two schemes, can in conjunction with approaching the advantage with interpolation subdividing scheme by their mixing, and the method producing a mixing has application prospect widely, if it enough simply calculates; But it remains the effective and simple mode that lacks integration and approximate interpolation subdividing scheme.

Especially, much segmenting occasion, particularly dress designing field, except to except the smooth limit curve of generation and curved surface, also requiring that its girth is constant before and after segmentation.

Summary of the invention

The object of the invention is to overcome the deficiencies in the prior art, the invention provides a kind of Curve Subdivision method of protecting girth, constant for adopting fusion to approach the girth that can realize segmenting front and back with the divided method of interpolation, the smoothness of curve can be improved again, improve the experience sense of user.

In order to solve the above problems, the present invention proposes a kind of Curve Subdivision method of protecting girth, described method comprises:

Obtain the information of non-Subdivision Curves;

The described curve do not segmented is merged and approaches and 4 triple Subdividing Processing of interpolation, obtain Curve Subdivision pattern;

The parameter of described Curve Subdivision pattern is modified, makes the curve second order under described Curve Subdivision pattern lead continuously;

Lead in continuous print parameter area at described curve second order and adjust parameter, determine that the girth error segmenting front and back is in threshold range;

Export to merge and approach the Subdivision Curves with guarantor's girth of interpolation.

Preferably, the information of described non-Subdivision Curves comprises the threshold value of the girth error allowed before and after the girth acquisition segmentation of the shape of non-Subdivision Curves, the girth of non-Subdivision Curves, the control vertex of non-Subdivision Curves and described non-Subdivision Curves.

Preferably, described fusion the described curve do not segmented approaches and 4 triple Subdividing Processing steps of interpolation comprise:

According to the control vertex of described non-Subdivision Curves after k+1 segmentation, obtain new control vertex the form of described triple segmentation is defined as:

$p_i^{k+1}=\underset{j\∈z}{\Σ}{a}_{i-3j}{p}_j^k,i\∈z;$

Coefficient a={a is set _i∈ R, i ∈ z}; Described form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\∈z}{\Σ}{a}_i{z}^i,$

Described form S uniform convergence, then meet:

$\underset{i\∈z}{\Σ}{a}_{3i+j}=1,j=0,1,2;$

Initial control vertex inserts summit with with initial vertax with between produce trisection point, described insertion top displacement amount is set for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Described insertion summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Fixing all initial vertaxs by described insertion summit with move to reposition with the distance of setting movement is respectively with right with be defined as:

${V^{\′}}_{3i+d}^0={\mathrm{\αV}}_{i+d-1}^0+{\mathrm{\βV}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p^{\′}}_{3i}^{k+1}=p_i^k\\ {p^{\′}}_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)+{\mathrm{\αV}}_i^k+{\mathrm{\βV}}_{i+1}^k\\ {p^{\′}}_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)+{\mathrm{\βV}}_i^k+{\mathrm{\αV}}_{i+1}^k\end{array}\right.;$

Introduce parameter ω to merge and approach and being subdivided into of interpolation:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\ωV}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\ωV}}_i^k+(1-\mathrm{\ω})({\mathrm{\αV}}_i^k+{\mathrm{\βV}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\ωV}}_{i+1}^k+(1-\mathrm{\ω})({\mathrm{\βV}}_i^k+{\mathrm{\αV}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\ω}\\ a_1=1-\frac{8}{27}\mathrm{\ω}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\ω})(\mathrm{\α}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\ω})(2\mathrm{\α}+2-\mathrm{\β})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\ω})(2\mathrm{\β}-1-\mathrm{\α})\\ b_3=-\frac{1}{27}(1-\mathrm{\ω})\mathrm{\β}\end{array}\right.;$

Wherein, $\mathrm{\α}=\frac{3}{2}(1+\mathrm{\μ}),\mathrm{\β}=\frac{3}{2}(1-\mathrm{\μ}).$

Preferably, the described parameter to described Curve Subdivision pattern is modified, and the curve second order under described Curve Subdivision pattern is led continuously, comprising:

Parameter μ, ω meet curve second order under described Curve Subdivision pattern is led continuously.

Preferably, described leading in continuous print parameter area at described curve second order adjusts parameter, determines that the step of girth error in threshold range before and after segmenting comprises:

Obtain the girth that described second order leads continuous print curve;

The girth that the girth of described non-Subdivision Curves and described second order lead continuous print curve is subtracted each other, obtains and subtract each other result;

By described subtract each other result and the girth of described non-Subdivision Curves obtain segment before and after compared with the threshold value of girth error that allows, acquisition comparative result;

According to described comparative result, the parameter that described second order leads continuous print curve is adjusted.

In addition, the invention allows for a kind of Curve Subdivision system of protecting girth, described system comprises:

Data obtaining module: for obtaining the information of non-Subdivision Curves;

Processing module: approaching and 4 triple Subdividing Processing of interpolation for merging the described curve do not segmented, obtaining Curve Subdivision pattern;

Parameter area determination module: for modifying to the parameter of described Curve Subdivision pattern, makes the curve second order under described Curve Subdivision pattern lead continuously;

Parameter adjustment module: adjust parameter for leading in continuous print parameter area at described curve second order, determines that the girth error segmenting front and back is in threshold range;

Output module: approach the Subdivision Curves with guarantor's girth of interpolation for exporting to merge.

Preferably, the information of described non-Subdivision Curves comprises the threshold value of the girth error allowed before and after the girth acquisition segmentation of the shape of non-Subdivision Curves, the girth of non-Subdivision Curves, the control vertex of non-Subdivision Curves and described non-Subdivision Curves.

Preferably, described processing module comprises:

Form definition unit: for the control vertex according to described non-Subdivision Curves after k+1 segmentation, obtain new control vertex the form of described triple segmentation is defined as:

$p_i^{k+1}=\underset{j\∈z}{\Σ}{a}_{i-3j}{p}_j^k,i\∈z;$

Polynomial generation unit: for arranging coefficient a={a _i∈ R, i ∈ z}; Described form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\∈z}{\Σ}{a}_i{z}^i,$

Described form S uniform convergence, then meet:

$\underset{i\∈z}{\Σ}{a}_{3i+j}=1,j=0,1,2;$

Approach segmentation definition unit: for inserting summit on initial control vertex with with initial vertax with between produce trisection point, described insertion top displacement amount is set for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Described insertion summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Interpolation subdividing definition unit: for fixing all initial vertaxs by described insertion summit with move to reposition with the distance of setting movement is respectively with right with be defined as:

${V^{\′}}_{3i+d}^0={\mathrm{\αV}}_{i+d-1}^0+{\mathrm{\βV}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p^{\′}}_{3i}^{k+1}=p_i^k\\ {p^{\′}}_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)+{\mathrm{\αV}}_i^k+{\mathrm{\βV}}_{i+1}^k\\ {p^{\′}}_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)+{\mathrm{\βV}}_i^k+{\mathrm{\αV}}_{i+1}^k\end{array}\right.;$

Merge and approach and interpolation subdividing unit: merge approach and being subdivided into of interpolation for introducing parameter ω:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\ωV}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\ωV}}_i^k+(1-\mathrm{\ω})({\mathrm{\αV}}_i^k+{\mathrm{\βV}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\ωV}}_{i+1}^k+(1-\mathrm{\ω})({\mathrm{\βV}}_i^k+{\mathrm{\αV}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\ω}\\ a_1=1-\frac{8}{27}\mathrm{\ω}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\ω})(\mathrm{\α}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\ω})(2\mathrm{\α}+2-\mathrm{\β})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\ω})(2\mathrm{\β}-1-\mathrm{\α})\\ b_3=-\frac{1}{27}(1-\mathrm{\ω})\mathrm{\β}\end{array}\right.;$

Wherein, $\mathrm{\α}=\frac{3}{2}(1+\mathrm{\μ}),\mathrm{\β}=\frac{3}{2}(1-\mathrm{\μ}).$

Preferably, the described parameter to described Curve Subdivision pattern is modified, and the curve second order under described Curve Subdivision pattern is led continuously, comprising:

Parameter μ, ω meet $\frac{1}{5}<\mathrm{\&mu;}<\frac{1}{3},0\&le;\mathrm{\&omega;}\&le;1,$ Curve second order under described Curve Subdivision pattern is led continuously.

Preferably, described parameter adjustment module comprises:

Girth acquiring unit: the girth of leading continuous print curve for obtaining described second order;

Difference acquiring unit: the girth for the girth of described non-Subdivision Curves and described second order being led continuous print curve subtracts each other, obtains and subtracts each other result;

Contrast unit: for by described subtract each other result and the girth of described non-Subdivision Curves obtains segment the girth error that front and back allow threshold value compared with, acquisition comparative result;

Parameter adjustment unit: for adjusting the parameter that described second order leads continuous print curve according to described comparative result.

Implement the embodiment of the present invention, can merge and approach and when interpolation subdividing, ensure that the girth of curve does not change, the smoothness of curve can also be ensured simultaneously, greatly improve the experience sense of user.

Accompanying drawing explanation

In order to be illustrated more clearly in the embodiment of the present invention or technical scheme of the prior art, be briefly described to the accompanying drawing used required in embodiment or description of the prior art 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 prerequisite not paying creative work, other accompanying drawing can also be obtained according to these accompanying drawings.

Fig. 1 is the method flow schematic diagram of the Curve Subdivision method of guarantor's girth of the embodiment of the present invention;

Fig. 2 be the embodiment of the present invention lead at curve second order the method flow schematic diagram adjusting parameter in continuous print parameter area;

Fig. 3 is the structure composition schematic diagram of the Curve Subdivision system of guarantor's girth of the embodiment of the present invention.

Embodiment

Below in conjunction with the accompanying drawing in the embodiment of the present invention, be clearly and completely described the technical scheme in the embodiment of the present invention, obviously, described embodiment is only the present invention's part embodiment, instead of whole embodiments.Based on the embodiment in the present invention, those of ordinary skill in the art, not making the every other embodiment obtained under creative work prerequisite, belong to the scope of protection of the invention.

Fig. 1 is the method flow schematic diagram of the Curve Subdivision method of guarantor's girth of the embodiment of the present invention, and as shown in Figure 1, the method comprises:

S11: the information obtaining non-Subdivision Curves;

S12: the curve do not segmented is merged and approaches and 4 triple Subdividing Processing of interpolation, obtain Curve Subdivision pattern;

S13: modify to the parameter of Curve Subdivision pattern, makes the curve second order under Curve Subdivision pattern lead continuously;

S14: lead in continuous print parameter area at curve second order and adjust parameter, determines that the girth error segmenting front and back is in threshold range;

S15: export to merge and approach the Subdivision Curves with guarantor's girth of interpolation.

S11 is described further:

The information of non-Subdivision Curves comprises the shape of non-Subdivision Curves, the perimeter L of non-Subdivision Curves, the control vertex of non-Subdivision Curves the threshold xi of the girth error allowed before and after non-Subdivision Curves segmentation.

S12 is described further:

According to the control vertex of non-Subdivision Curves after k+1 segmentation, obtain new control vertex the form of triple segmentation is defined as:

$p_i^{k+1}=\underset{j\&Element;z}{\&Sigma;}{a}_{i-3j}{p}_j^k,i\&Element;z;$

Coefficient a={a is set _i∈ R, i ∈ z}; Form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\&Element;z}{\&Sigma;}{a}_i{z}^i,$

Form S uniform convergence, then meet:

$\underset{i\&Element;z}{\&Sigma;}{a}_{3i+j}=1,j=0,1,2;$

Initial control vertex inserts summit with with initial vertax with between produce trisection point, arrange and insert top displacement amount for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Insert summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Fixing all initial vertaxs summit will be inserted with move to reposition with the distance of setting movement is respectively with right with be defined as:

${V^{\&prime;}}_{3i+d}^0={\mathrm{\&alpha;V}}_{i+d-1}^0+{\mathrm{\&beta;V}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p^{\&prime;}}_{3i}^{k+1}=p_i^k\\ {p^{\&prime;}}_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)+{\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k\\ {p^{\&prime;}}_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)+{\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k\end{array}\right.;$

Introduce parameter ω to merge and approach and being subdivided into of interpolation:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\&omega;V}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_i^k+(1-\mathrm{\&omega;})({\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_{i+1}^k+(1-\mathrm{\&omega;})({\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\&omega;}\\ a_1=1-\frac{8}{27}\mathrm{\&omega;}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\&omega;})(\mathrm{\&alpha;}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&alpha;}+2-\mathrm{\&beta;})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&beta;}-1-\mathrm{\&alpha;})\\ b_3=-\frac{1}{27}(1-\mathrm{\&omega;})\mathrm{\&beta;}\end{array}\right.;$

Wherein, $\mathrm{\&alpha;}=\frac{3}{2}(1+\mathrm{\&mu;}),\mathrm{\&beta;}=\frac{3}{2}(1-\mathrm{\&mu;}).$

According to above-mentioned steps, Curve Subdivision pattern can be obtained.

S13 is described further:

Carry out derivative to the Curve Subdivision pattern of above-mentioned acquisition to ask for, ask for their second derivative, obtain the second derivative of Curve Subdivision pattern; Again according to the sufficient and necessary condition of continuous second derivative under Curve Subdivision pattern is asked for parameter μ, ω according to the continuous print sufficient and necessary condition of function, and the scope of get parms μ, ω exists middlely the curve second order under Curve Subdivision pattern can be made to lead continuously.

Composition graphs 2 couples of S14 are described further:

As shown in Figure 2:

S141: obtain the girth that second order leads continuous print curve model;

S142: the girth of non-Subdivision Curves and girth are subtracted each other, obtains and subtracts each other result;

S143: by subtracting each other result compared with the threshold value of the girth error allowed before and after non-Subdivision Curves segments, obtain comparative result;

S144: the parameter that second order leads continuous print curve is adjusted according to comparative result

Wherein, S141 is described further:

The acquisition methods that second order leads the girth of continuous print curve model is obtained by following formula:

$L^{k+1}=\underset{i=0}{\overset{3^{k+1}(n-1)}{\&Sigma;}}\left|p_i^{k+1}{p}_{i\mathrm{+1}}^{k+1}\right|$

Specifically expand into:

$L^{k+1}=\underset{i=0}{\overset{3^k(n-1)}{\&Sigma;}}\left|p_{3i}^{k+1}{p}_{3i\mathrm{+1}}^{k+1}\right|+\underset{i=0}{\overset{3^k(n-1)}{\&Sigma;}}\left|p_{3i+1}^{k+1}{p}_{3i+2}^{k+1}\right|+\underset{i=0}{\overset{3^k(n-1)}{\&Sigma;}}\left|p_{3i+2}^{k+1}{p}_{3i+3}^{k+1}\right|;$

The girth having above-mentioned formula to obtain curve is L ^k+1.

S142 is described further:

By the perimeter L obtaining non-Subdivision Curves in above-mentioned S11, the perimeter L of non-Subdivision Curves is adopted to deduct the perimeter L of curve ^k+1, that is:

|L-L ^k+1|＝η；

Wherein, η is for subtracting each other result.

S143 is described further:

By the threshold xi obtaining the girth error allowed before and after the segmentation of non-Subdivision Curves in above-mentioned S11, adopt ξ and η to compare, obtain comparative result.

S144 is described further:

If comparative result is ξ >=η, then selected parameter (leading the Selecting parameter of continuous print curve at second order) meets the parameter request protecting girth segmentation; If comparative result is ξ < η, then selected parameter does not meet the parameter request protecting girth segmentation, needs to change parameter and repeats S14 step.

In addition, first embodiment of the invention additionally provides a kind of Curve Subdivision system of protecting girth, and as shown in Figure 3, this system comprises:

Data obtaining module 11: for obtaining the information of non-Subdivision Curves;

Processing module 12: approaching and 4 triple Subdividing Processing of interpolation for merging the curve do not segmented, obtaining Curve Subdivision pattern;

Parameter area determination module 13: for modifying to the parameter of Curve Subdivision pattern, makes the curve second order under Curve Subdivision pattern lead continuously;

Parameter adjustment module 14: adjust parameter for leading in continuous print parameter area at curve second order, determines that the girth error segmenting front and back is in threshold range;

Output module 15: approach the Subdivision Curves with guarantor's girth of interpolation for exporting to merge.

Preferably, the information of non-Subdivision Curves comprises the threshold value of the girth error allowed before and after the girth acquisition segmentation of the shape of non-Subdivision Curves, the girth of non-Subdivision Curves, the control vertex of non-Subdivision Curves and non-Subdivision Curves.

Preferably, processing module 12 comprises:

Form definition unit: for the control vertex according to non-Subdivision Curves after k+1 segmentation, obtain new control vertex $p^{k+1}=\{p_i^{k+1},i\&Element;z\},$ The form of triple segmentation is defined as:

$p_i^{k+1}=\underset{j\&Element;z}{\&Sigma;}{a}_{i-3j}{p}_j^k,i\&Element;z;$

Polynomial generation unit: for arranging coefficient a={a _i∈ R, i ∈ z}; Form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\&Element;z}{\&Sigma;}{a}_i{z}^i,$

Form S uniform convergence, then meet:

$\underset{i\&Element;z}{\&Sigma;}{a}_{3i+j}=1,j=0,1,2;$

Approach segmentation definition unit: for inserting summit on initial control vertex with with initial vertax with between produce trisection point, arrange and insert top displacement amount for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Insert summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Interpolation subdividing definition unit: for fixing all initial vertaxs summit will be inserted with move to reposition with the distance of setting movement is respectively with right with be defined as:

${V^{\&prime;}}_{3i+d}^0={\mathrm{\&alpha;V}}_{i+d-1}^0+{\mathrm{\&beta;V}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p^{\&prime;}}_{3i}^{k+1}=p_i^k\\ {p^{\&prime;}}_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)+{\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k\\ {p^{\&prime;}}_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)+{\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k\end{array}\right.;$

Merge and approach and interpolation subdividing unit: merge approach and being subdivided into of interpolation for introducing parameter ω:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\&omega;V}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_i^k+(1-\mathrm{\&omega;})({\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_{i+1}^k+(1-\mathrm{\&omega;})({\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\&omega;}\\ a_1=1-\frac{8}{27}\mathrm{\&omega;}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\&omega;})(\mathrm{\&alpha;}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&alpha;}+2-\mathrm{\&beta;})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&beta;}-1-\mathrm{\&alpha;})\\ b_3=-\frac{1}{27}(1-\mathrm{\&omega;})\mathrm{\&beta;}\end{array}\right.;$

Wherein, $\mathrm{\&alpha;}=\frac{3}{2}(1+\mathrm{\&mu;}),\mathrm{\&beta;}=\frac{3}{2}(1-\mathrm{\&mu;}).$

Preferably, the parameter of Curve Subdivision pattern is modified, the curve second order under Curve Subdivision pattern is led continuously, comprising:

Parameter μ, ω meet curve second order under Curve Subdivision pattern is led continuously.

Preferably, parameter adjustment module 14 comprises:

Girth acquiring unit: the girth of leading continuous print curve for obtaining second order;

Difference acquiring unit: the girth for the girth of non-Subdivision Curves and second order being led continuous print curve subtracts each other, obtains and subtracts each other result;

Contrast unit: for by subtract each other result and the girth of non-Subdivision Curves obtains segment the girth error that front and back allow threshold value compared with, acquisition comparative result;

Parameter adjustment unit: for adjusting the parameter that second order leads continuous print curve according to comparative result.

Particularly, the principle of work of the system related functions module of the embodiment of the present invention see the associated description of embodiment of the method, can repeat no more here.

Implement the embodiment of the present invention, can merge and approach and when interpolation subdividing, ensure that the girth of curve does not change, the smoothness of curve can also be ensured simultaneously, greatly improve the experience sense of user.

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

In addition, above a kind of Curve Subdivision method and system of protecting girth that the embodiment of the present invention provides are described in detail, apply specific case herein to set forth principle of the present invention and embodiment, the explanation of above embodiment just understands method of the present invention and core concept thereof for helping; Meanwhile, for one of ordinary skill in the art, according to thought of the present invention, all will change in specific embodiments and applications, in sum, this description should not be construed as limitation of the present invention.

Claims (10)

1. protect a Curve Subdivision method for girth, it is characterized in that, described method comprises:

Obtain the information of non-Subdivision Curves;

The described curve do not segmented is merged and approaches and 4 triple Subdividing Processing of interpolation, obtain Curve Subdivision pattern;

The parameter of described Curve Subdivision pattern is modified, makes the curve second order under described Curve Subdivision pattern lead continuously;

Lead in continuous print parameter area at described curve second order and adjust parameter, determine that the girth error segmenting front and back is in threshold range;

Export to merge and approach the Subdivision Curves with guarantor's girth of interpolation.

2. the Curve Subdivision method of guarantor's girth according to claim 1, it is characterized in that, the information of described non-Subdivision Curves comprises the threshold value of the girth error allowed before and after the segmentation of the shape of non-Subdivision Curves, the girth of non-Subdivision Curves, the control vertex of non-Subdivision Curves and non-Subdivision Curves.

3. the Curve Subdivision method of guarantor's girth according to claim 1, is characterized in that, described fusion the described curve do not segmented approaches and 4 triple Subdividing Processing steps of interpolation comprise:

According to the control vertex of described non-Subdivision Curves after k+1 segmentation, obtain new control vertex the form of described triple segmentation is defined as:

$p_i^{k+1}=\underset{j\&Element;z}{\&Sigma;}{a}_{i-3j}{p}_j^k,i\&Element;z;$

Coefficient a={a is set _i∈ R, i ∈ z}; Described form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\&Element;z}{\&Sigma;}{a}_i{z}^i,$

Described form S uniform convergence, then meet:

$\underset{i\&Element;z}{\&Sigma;}{a}_{3i+j}=1,j=0,1,2;$

Initial control vertex inserts summit with with initial vertax with between produce trisection point, described insertion top displacement amount is set for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Described insertion summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Fixing all initial vertaxs by described insertion summit with move to reposition with the distance of setting movement is respectively with right with be defined as:

$V_{3i+d}^{\&prime;0}={\mathrm{\&alpha;V}}_{i+d-1}^0+{\mathrm{\&beta;V}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p}_{3i}^{\&prime;k+1}=p_i^k\\ p_{3i+1}^{\&prime;k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)+{\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k\\ p_{3i+2}^{\&prime;k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)+{\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k\end{array}\right.;$

Introduce parameter ω to merge and approach and being subdivided into of interpolation:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\&omega;V}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_i^k+(1-\mathrm{\&omega;})({\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_{i+1}^k+(1-\mathrm{\&omega;})({\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\&omega;}\\ a_1=1-\frac{8}{27}\mathrm{\&omega;}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\&omega;})(\mathrm{\&alpha;}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&alpha;}+2-\mathrm{\&beta;})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&beta;}-1-\mathrm{\&alpha;})\\ b_3=-\frac{1}{27}(1-\mathrm{\&omega;})\mathrm{\&beta;}\end{array}\right.;$

Wherein, $\mathrm{\&alpha;}=\frac{3}{2}(1+\mathrm{\&mu;}),\mathrm{\&beta;}=\frac{3}{2}(1-\mathrm{\&mu;}).$

4. the Curve Subdivision method of guarantor's girth according to claim 1, is characterized in that, the described parameter to described Curve Subdivision pattern is modified, and the curve second order under described Curve Subdivision pattern is led continuously, comprising:

Parameter μ, ω meet curve second order under described Curve Subdivision pattern is led continuously.

5. the Curve Subdivision method of guarantor's girth according to claim 1, is characterized in that, described leading in continuous print parameter area at described curve second order adjusts parameter, determines that the step of girth error in threshold range before and after segmenting comprises:

Obtain the girth that described second order leads continuous print curve model;

The girth of described non-Subdivision Curves and described girth are subtracted each other, obtains and subtract each other result;

By described result of subtracting each other compared with the threshold value of the girth error allowed before and after described non-Subdivision Curves segments, obtain comparative result;

According to described comparative result, the parameter that described second order leads continuous print curve is adjusted.

6. protect a Curve Subdivision system for girth, it is characterized in that, described system comprises:

Data obtaining module: for obtaining the information of non-Subdivision Curves;

Processing module: approaching and 4 triple Subdividing Processing of interpolation for merging the described curve do not segmented, obtaining Curve Subdivision pattern;

Parameter area determination module: for modifying to the parameter of described Curve Subdivision pattern, makes the curve second order under described Curve Subdivision pattern lead continuously;

Parameter adjustment module: adjust parameter for leading in continuous print parameter area at described curve second order, determines that the girth error segmenting front and back is in threshold range;

Output module: approach the Subdivision Curves with guarantor's girth of interpolation for exporting to merge.

7. the Curve Subdivision system of guarantor's girth according to claim 6, it is characterized in that, the information of described non-Subdivision Curves comprises the threshold value of the girth error allowed before and after the girth acquisition segmentation of the shape of non-Subdivision Curves, the girth of non-Subdivision Curves, the control vertex of non-Subdivision Curves and described non-Subdivision Curves.

8. the Curve Subdivision system of guarantor's girth according to claim 6, is characterized in that, described processing module comprises:

Form definition unit: for the control vertex according to described non-Subdivision Curves after k+1 segmentation, obtain new control vertex the form of described triple segmentation is defined as:

$p_i^{k+1}=\underset{j\&Element;z}{\&Sigma;}{a}_{i-3j}{p}_j^k,i\&Element;z;$

Polynomial generation unit: for arranging coefficient a={a _i∈ R, i ∈ z}; Described form is designated as S, then generator polynomial is:

$a\left(z\right)=\underset{i\&Element;z}{\&Sigma;}{a}_i{z}^i,$

Described form S uniform convergence, then meet:

$\underset{i\&Element;z}{\&Sigma;}{a}_{3i+j}=1,j=0,1,2;$

Approach segmentation definition unit: for inserting summit on initial control vertex with with initial vertax with between produce trisection point, described insertion top displacement amount is set for:

$V_i^0=-\frac{1}{27}{p}_{i-1}^0+\frac{2}{27}{p}_{-1}^0-\frac{1}{27}{p}_{i+1}^0,$

Described insertion summit with move to new position with adopt the mode of approaching to segment to be defined as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4V_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-V_i^k\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-V_{i+1}^k\end{array}\right.;$

Interpolation subdividing definition unit: for fixing all initial vertaxs by described insertion summit with move to reposition with the distance of setting movement is respectively with right with be defined as:

$V_{3i+d}^{\&prime;0}={\mathrm{\&alpha;V}}_{i+d-1}^0+{\mathrm{\&beta;V}}_{i+2-d}^0,d=1,2,$

Introduce variable α, β, wherein, then the mode of interpolation is subdivided into:

$\left\{\begin{array}{c}{p}_{3i}^{\&prime;k+1}=p_i^k\\ p_{3i+1}^{\&prime;k+1}=\left(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k\right)+{\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k\\ p_{3i+2}^{\&prime;k+1}=\left(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k\right)+{\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k\end{array}\right.;$

Merge and approach and interpolation subdividing unit: merge approach and being subdivided into of interpolation for introducing parameter ω:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=p_i^k-4{\mathrm{\&omega;V}}_i^k\\ p_{3i+1}^{k+1}=(\frac{2}{3}{p}_i^k+\frac{1}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_i^k+(1-\mathrm{\&omega;})({\mathrm{\&alpha;V}}_i^k+{\mathrm{\&beta;V}}_{i+1}^k)\\ p_{3i+2}^{k+1}=(\frac{1}{3}{p}_i^k+\frac{2}{3}{p}_{i+1}^k)-{\mathrm{\&omega;V}}_{i+1}^k+(1-\mathrm{\&omega;})({\mathrm{\&beta;V}}_i^k+{\mathrm{\&alpha;V}}_{i+1}^k)\end{array}\right.,$

Be standardized as:

$\left\{\begin{array}{c}{p}_{3i}^{k+1}=a_0{p}_{i-1}^k+a_1{p}_i^k+a_0{p}_{i+1}^k\\ p_{3i+1}^{k+1}=b_0{p}_{i-1}^k+b_1{p}_i^k+b_2{p}_{i+1}^k+b_3{p}_{i+2}^k\\ p_{3i+2}^{k+1}=b_3{p}_{i-1}^k+b_2{p}_i^k+b_1{p}_{i+1}^k+b_0{p}_{i+2}^k\end{array}\right.,$

Wherein:

$\left\{\begin{array}{c}{a}_0=\frac{4}{27}\mathrm{\&omega;}\\ a_1=1-\frac{8}{27}\mathrm{\&omega;}\\ b_0=\frac{1}{27}-\frac{1}{27}(1-\mathrm{\&omega;})(\mathrm{\&alpha;}+1)\\ b_1=\frac{16}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&alpha;}+2-\mathrm{\&beta;})\\ b_2=\frac{10}{27}+\frac{1}{27}(1-\mathrm{\&omega;})(2\mathrm{\&beta;}-1-\mathrm{\&alpha;})\\ b_3=-\frac{1}{27}(1-\mathrm{\&omega;})\mathrm{\&beta;}\end{array}\right.;$

Wherein, $\mathrm{\&alpha;}=\frac{3}{2}(1+\mathrm{\&mu;}),\mathrm{\&beta;}=\frac{3}{2}(1-\mathrm{\&mu;}).$

9. the Curve Subdivision system of guarantor's girth according to claim 6, is characterized in that, the described parameter to described Curve Subdivision pattern is modified, and the curve second order under described Curve Subdivision pattern is led continuously, comprising:

Parameter μ, ω meet curve second order under described Curve Subdivision pattern is led continuously.

10. the Curve Subdivision system of guarantor's girth according to claim 6, is characterized in that, described parameter adjustment module comprises:

Girth acquiring unit: the girth of leading continuous print curve for obtaining described second order;

Difference acquiring unit: the girth for the girth of described non-Subdivision Curves and described second order being led continuous print curve subtracts each other, obtains and subtracts each other result;

Contrast unit: for by described subtract each other result and the girth of described non-Subdivision Curves obtains segment the girth error that front and back allow threshold value compared with, acquisition comparative result;

Parameter adjustment unit: for adjusting the parameter that described second order leads continuous print curve according to described comparative result.