CN101923502B - Ring-cutting subdivision method of simple polygon in computer graphics - Google Patents

Abstract

The invention discloses a ring-cutting subdivision method of simple polygon in computer graphics, comprising: 1) connecting the first vertex of the polygon with four vertexes of an outsourcing diamond of the polygon, wherein for the n<th> vertex of the polygon, n=1; 2) setting n to be n+1, if the n<th> vertex and the first vertex are coincident and the (n-1)<th> vertex and the first vertex are connected into an edge, performing 3); otherwise, performing 2.1),i.e. taking the directed edge from the (n-1)<th> vertex to the n<th> vertex as a probe, if the probe and certain additional edge are overlapped, deleting the overlapped additional edge, carrying out collineation processing, and returning to 2); otherwise, performing 2.2), and if the probe passes through certain additional edge, respectively processing three edges on two sides of the probe, deleting the additional edge, and repeating 2.2); if the n<th> vertex and the first vertex are coincident, respectively processing three edges on two sides of the probe, and performing 3); if the probe is provided with a front collineation vertex, carrying out collineation processing, and otherwise carrying out conventional processing; returning to 2); and 3), correcting the direction of the equilateral polygon.

Description

Ring-cutting subdivision method of simple polygon in the computer graphical

Technical field:

The present invention is applied to computer graphical processing, pattern-recognition, curved surface approaches and the data processing of aspect such as mesh of finite element generation.

Background technology:

The polygon decomposition problem all has very important practical value in a lot of fields.The subdivision method that is proposed at present roughly has following several kinds: triangulation, quadrilateral subdivision (are seen " computational geometry-algorithm and application ", publishing house of Tsing-Hua University, 2005; The 68th page) or protruding subdivision (see " computational geometry in the C language " " Computational geometry in C ", China Machine Press, 2005; The 58th page); What though the optimal algorithm of at present known triangulation can be for linear session, but unusual complicacy, and be off-line.Complexity working time of other subdivision methods all is higher than O (N).And protruding subdivision then possibly need to add extra steiner point and extra limit (seeing " computational geometry-algorithm and application ", publishing house of Tsing-Hua University, 2005, the 232 pages).

Therefore, seek one more easy, particularly online, the linear session subdivision method be always the computational geometry field an open difficult problem (The Open Problems Project, Http:// maven.smith.edu/～orourke/TOPP/P10.html#Problem.10).

Beginning in 2007; I take up the research of this problem; Through the exploration more than 2 years; Finally invented ring-cutting subdivision method of simple polygon in a kind of online computer graphical; Be a series of bulge loops (each summit can be connected with every other summit straight line on the ring or inside) and scrobicular ring (only having a summit can or encircle inside on ring is connected with every other summit straight line) with the simple polygon subdivision exactly, form polygonal a kind of bifurcation subdivision (dimorphic partition), be O's (N) the working time under its worst case.And polygonal ring-cutting subdivision is in case completion can directly be converted into triangulation with scrobicular ring and bulge loop in linear session.

Summary of the invention:

Technical matters: the purpose of this invention is to provide ring-cutting subdivision method of simple polygon in a kind of computer graphical, along with the embedding one by one of polygon vertex, online, accomplish polygonal subdivision with linear session.

Technical scheme: the technical scheme that ring-cutting subdivision method of simple polygon adopts in the computer graphical of the present invention is following:

Step 1.1: set up the XOY plane coordinate, can contain polygonal rhombus one of the polygonal outer setting of treating subdivision, the coordinate on 4 structure summits of rhombus is respectively (a; 0), (0, a), (a, 0) and (0;-a); A is greater than the horizontal scale and the vertical real number of yardstick sum, and horizontal scale is the greater of absolute value of left side or the right side X coordinate of the minimum outsourcing rectangle of polygon, and vertically yardstick is the greater of absolute value of upside or the downside Y coordinate of the minimum outsourcing rectangle of polygon; Rhombus constitutes a ring, and ring is a downstream direction with the counter clockwise direction; , one by one the 1st summit is connected with 4 structure summits of rhombus as the 1st summit with arbitrary summit on the polygon, rhombus is decomposed into 4 than circlet, as initial subdivision set; With in two adjacent on polygon summits with the 1st summit any one as the 2nd summit, suppose that the trend with the 1st summit and the 2nd summit order is polygonal downstream direction;

Step 1.2: one by one the n summit is embedded the subdivision set along polygonal downstream direction, n=2,3,4 ..., N, N+1, N are polygonal number of vertices, the N+1 summit overlaps with the 1st summit;

Step 1.3: if according to the 1st summit, the 2nd summit, the 3rd summit ..., the polygonal trend that forms of the order on 1 summit, N summit to the is clockwise; Then polygonal each summit travel direction is reversed and handle; The upper reaches adjacent vertex that is about to former setting is modified to the downstream adjacent vertex, and the downstream adjacent vertex of former setting is modified to upper reaches adjacent vertex;

The above-mentioned method that the n summit is embedded the subdivision set is:

Step 1.2.1: with the directed edge from n summit, n-1 summit to the is probe, different with the 1st summit as if the n summit, then execution in step 1.2.2; Otherwise; If with the n-1 summit is that then polygon is closed, and this additional side is updated to polygonal limit when having the terminal point of an additional side to be the 1st summit in all additional sides of starting point; Embed and finish, described additional side is by any two directed edges that non-conterminous summit forms; Otherwise,

Step 1.2.2: article one directed edge that is positioned at the probe right side with from the n-1 summit is limit, the 1st right, is the 1st first-in-chain(FIC) with the downstream side on limit, the 1st right, is the 1st last-of-chain with the upstream edge of probe; With i as follow-up counter; I=1, if article one be positioned at limit, the 1st right the left side the additional side with the probe overlapping, then the downstream side with this additional side is the conllinear limit; And delete overlapping additional side, execution in step 1.2.6; Otherwise,

Step 1.2.3: respectively as the beginning and the ending of i segment, described segment is the directed edge of a series of tandem arrays with i first-in-chain(FIC), i last-of-chain; If the i segment is monolateral shape, protruding chain or recessed chain segment, then the i segment is carried out the segment pinching and confirm the alignment probe limit, and hit the limit as i, then execution in step 1.2.5 with the alignment edge that is obtained; As if the i segment is left spill, then execution in step 1.2.4; Otherwise the i segment is right spill, if probe is crossing with the upstream edge of i last-of-chain, then hits the limit with said upstream edge as i; Execution in step 1.2.5; Otherwise,, then hit the limit as i with alignment edge if there is alignment edge to being that the upstream edge of first-in-chain(FIC), i last-of-chain is that the segment of last-of-chain carries out the segment pinching and confirms the alignment probe limit with the i first-in-chain(FIC); Otherwise the upstream edge with the i last-of-chain is that i hits the limit, then execution in step 1.2.5; The last-of-chain of above-mentioned monolateral shape segment is the downstream side of first-in-chain(FIC); Angle＜the π of all intermediate vertexs of above-mentioned protruding chain segment; The intermediate vertex of said segment is all summits between the starting point of the starting point of first-in-chain(FIC) and last-of-chain on the segment, and the angle on said summit is to be the angle in two adjacent directed edges left sides of linking with this summit on the segment; Angle >=the π of all intermediate vertexs of above-mentioned recessed chain segment; Above-mentioned left spill segment removes the angle＜π of the terminal point of first-in-chain(FIC), the angle >=π of all the other intermediate vertexs on the segment; Above-mentioned right spill segment is except that the angle＜π of the starting point of the upstream edge of last-of-chain, the angle >=π of all the other intermediate vertexs on the segment;

Step 1.2.4: if probe intersects with the i first-in-chain(FIC), be that i hits the limit then with the i first-in-chain(FIC), execution in step 1.2.5; Otherwise, be that first-in-chain(FIC), i last-of-chain are that the segment of last-of-chain carries out the segment pinching and confirms the alignment probe limit to downstream side with the i first-in-chain(FIC), if there is alignment edge, then hits the limit as i, otherwise hit the limit as i with the i first-in-chain(FIC) with alignment edge,

Step 1.2.5:, then hit the limit as the conllinear limit, execution in step 1.2.6 with i if i hits the starting point and probe conllinear on limit; Otherwise; Hit the limit as playing initial line, medial side and termination limit with limit, the i right, i first-in-chain(FIC) and i respectively; With the unbounded mode the three limits chain of rings is carried out on the right side of probe; The downstream side, i last-of-chain and the probe that hit the limit with i respectively again carry out the three limits chain of rings with the unbounded mode to the left side of probe as playing initial line, medial side and a termination limit; The new downstream side that hits the limit with the i behind the three limit chains of rings is the i+1 last-of-chain; If hitting the limit with i, probe intersects; The downstream side that then hits the reverse edge on limit with i is the i+1 first-in-chain(FIC), is limit, the i+1 right with the downstream side of the reverse edge of probe, deletes i and hits the limit; As new i, return to step 1.2.3 with i+1; Otherwise; To the reverse edge with probe is that to hit the limit be that the segment of last-of-chain carries out the segment correction for first-in-chain(FIC), i; With newly connect from the additional side on n summit as the right flank limit; Again to being that first-in-chain(FIC), right flank limit are that the segment of last-of-chain carries out the segment correction with the i+1 last-of-chain, with newly connect from the additional side of the starting point of i+1 last-of-chain as the left wing limit; Respectively with the upstream edge on downstream side, left wing limit and the right flank limit on right flank limit as playing initial line, medial side and a termination limit, thereby with the unbounded mode the place ahead of probe is carried out that three limits are a chain of guarantees that the place ahead of popping one's head in is a triangle, embed and finish; The method of above-mentioned segment correction is: if segment is protruding chain; Be that ring limit, last-of-chain are that the bilateral chain of rings is carried out on the stop ring limit then with first-in-chain(FIC); Otherwise if segment is left spill; Be new first-in-chain(FIC) to the downstream side with first-in-chain(FIC) then, last-of-chain is that new segment that new last-of-chain constituted carries out postponing from the recessed chain of upper type and makes it to become monolateral shape or recessed chain, otherwise if segment is right spill; Then to being new first-in-chain(FIC) with first-in-chain(FIC), the upstream edge of last-of-chain is that the new segment of new last-of-chain carries out postponing from the recessed chain of mode down and makes it to become monolateral shape or recessed chain; Finish the segment correction; The method of the above-mentioned bilateral chain of rings is: with the upstream edge on stop ring limit is the ring inner edge, has connected the starting point of encircling the limit and the starting point on stop ring limit, with formed additional side as sealing the limit; If play the ring limit is an additional side and a ring limit long enough; And two rings that work the both sides of encircling the limit then can have been deleted the ring limit playing bigger scrobicular ring of formation under the prerequisite that the terminal point that can form a bigger bulge loop under the situation of being deleted on the ring limit or play the ring limit is not the 1st summit; In like manner; If the ring inner edge also is additional side and ring inner edge long enough; And two rings of the both sides of ring inner edge then can be deleted the ring inner edge forming a bigger bulge loop under the ring inner edge situation of being deleted or not being to form a bigger scrobicular ring under the prerequisite on the 1st summit in the starting point of ring inner edge; Bilateral a chain of the end; Above-mentioned rise ring limit long enough be meant the length on ring limit be at least the ring limit terminal point to the starting point on stop ring limit distance half the or to work the length of encircling the limit be half of length of sealing the limit at least; Above-mentioned ring inner edge long enough is that the length of finger ring inner edge has been that the length half the or the ring inner edge of distance of starting point to the starting point of encircling inner edge on ring limit is length half that seals the limit at least at least; Above-mentioned bulge loop refers to the angle＜π on all summits in ring; Above-mentioned scrobicular ring refers in ring except the angle＜π on three adjacent summits only being arranged, the angle >=π on all the other all summits;

Step 1.2.6: respectively with limit, the i right, i first-in-chain(FIC) and conllinear limit as playing initial line, medial side and a termination limit; With the bounded mode the three limits chain of rings is carried out on the right side of probe; Again respectively with conllinear limit, i last-of-chain and probe as initial line, medial side and a termination limit; It is a chain of with the bounded mode three limits to be carried out in the left side of probe, if there is the new additional side of downstream direction on the conllinear summit, then with this additional side as follow-up conllinear limit; If the n summit overlaps with the starting point that i hits the limit, then n returns on the summit to the 1st summit and polygon is closed, and the position of coordinating probe and conllinear limit concerns the embedding end; Otherwise; To the downstream side with the probe reverse edge is that first-in-chain(FIC), conllinear limit are that the segment of last-of-chain obtains the limit, middle part and as limit in the right side; Reverse edge with probe has been that the limit is that the bilateral chain of rings is carried out on the stop ring limit in ring limit, the right side then; And connecting n summit and conllinear summit as continuing the limit, the reverse edge of above-mentioned directed edge is end points and limit that direction with it opposite identical with said directed edge; To being that first-in-chain(FIC), probe are that the segment of last-of-chain obtains the limit, middle part and as limit in the left side, be that ring limit, the limit of continuing are that the bilateral chain of rings is carried out on the stop ring limit with limit in the left side with the conllinear limit, the embedding end; The above-mentioned method that segment is obtained the limit, middle part is: if segment is left spill, then the upstream edge with last-of-chain is the limit, middle part, and with first-in-chain(FIC) be encircle the limit, the limit, middle part is that the bilateral chain of rings is carried out on the stop ring limit, obtains end; Otherwise if the downstream side of first-in-chain(FIC) is the upstream edge of last-of-chain, then the downstream side with first-in-chain(FIC) is the limit, middle part, obtains end; Otherwise the downstream side with first-in-chain(FIC) has been that ring limit, last-of-chain are that the bilateral chain of rings is carried out on the stop ring limit, serves as the limit, middle part with the bilateral a chain of new downstream side of first-in-chain(FIC) afterwards, obtains end;

The a chain of method in three above-mentioned limits is:

Step 1.2.5.1: to being that first-in-chain(FIC), medial side are that the segment of last-of-chain carries out the segment correction, playing the start position of initial line new additional side arranged if revise the back to play initial line, then with new additional side as new initial line, otherwise be new initial line with an initial line; Again to be first-in-chain(FIC) with the medial side, to stop the limit be that the segment of last-of-chain carries out the segment correction, if revise the additional side that the start position of back medial side has new downstream direction, then with said additional side as limit in new, otherwise be newly middle limit with the medial side; If an initial line is identical with medial side or medial side is identical with the termination limit or the angle >=π of new middle limit starting point then finishes; As if twice correction result all is recessed chain, then execution in step 1.2.5.2; Otherwise,, then finish if a chain of mode in three limits is the bounded mode; Otherwise being the unbounded mode, if twice correction result all is monolateral shape, has been that ring limit, termination limit are that the bilateral chain of rings is carried out on the stop ring limit with new initial line then, finishes; If a preceding correction result is monolateral shape, a back correction result is recessed chain, to be first-in-chain(FIC) with limit in new, to stop the limit be that the segment of last-of-chain carries out postponing from the recessed chain of upper type, finish; Otherwise a preceding correction result is recessed chain, and a back correction result is monolateral shape, to new initial line be first-in-chain(FIC), new in the limit be that the segment of last-of-chain carries out postponing from the recessed chain of mode down, finish;

Step 1.2.5.2: with limit in new is the 1st strange flanging, and the starting point of new initial line is a upper extreme point, and the starting point that stops the limit is a lower extreme point, and j is a cycle counter, and j=1 carries out the processing of alternately postponing of two recessed chains;

Step 1.2.5.3: if the three limit chains of rings are unbounded modes, and the starting point that j takes turns strange flanging is angle >=π that lower extreme point or j take turns strange flanging starting point, finishes; If the starting point that the three limit chains of rings are bounded mode and j takes turns the upstream edge of strange flanging is that upper extreme point then finishes; Otherwise to take turns strange flanging with j is that first-in-chain(FIC), termination limit are the segment of last-of-chain, postpones to carry out recessed chain from upper type, and takes turns even flanging with the additional side of new connection as j;

Step 1.2.5.4: if the three limit chains of rings are unbounded modes, and the starting point that j takes turns even flanging is angle >=π that upper extreme point or j take turns even flanging starting point, finishes; If the starting point that the three limit chains of rings are bounded mode and j takes turns the downstream side of even flanging is that lower extreme point then finishes; Otherwise to being that first-in-chain(FIC), j take turns the segment that even flanging is a last-of-chain with new initial line, postpone, and take turns strange flanging as j+1 with the downstream side of the additional side of new connection to descend mode to carry out recessed chain certainly; With the j of j+1 as a new round, execution in step 1.2.5.3; The method that above-mentioned recessed chain is postponed is: if from the postponing of upper type, then the upstream edge with first-in-chain(FIC) serves as to observe the limit, otherwise serves as to observe the limit with the downstream side of last-of-chain; Starting point to observe the limit is an observation point, carries out the visible range that observation point is confirmed in the segment pinching, and cuts edge as upper reaches class with the directed edge of the upper reaches reference position of visible range, and the downstream side of the directed edge of the downstream final position of visible range is cut edge as the downstream class; For from the postponing of upper type, be that the bilateral chain of rings is carried out on the stop ring limit to observe that the limit be the ring limit, the downstream class is cut edge, for descend postponing of mode certainly, then with the side cut of upper reaches class be encircle the limit, the observation limit is that the bilateral chain of rings is carried out on the stop ring limit; Behind the bilateral chain of rings, if the starting point of the upstream edge on stop ring limit is identical with the starting point that plays the ring limit, then the new segment that constitutes of the starting point from starting point to the stop ring limit of encircling the limit is monolateral shape, otherwise is recessed chain;

Above-mentioned segment pinching confirms that the method on alignment probe limit is:

Step 2.1: the upstream edge that with the first-in-chain(FIC) is order the 1st limit, last-of-chain is backward the 1st limit, and k is a cycle counter, k=1,

Step 2.2: if the terminal point on order k limit is positioned at the probe left side starting point on order k limit be positioned at the probe right side or with the probe conllinear; Be alignment edge then with order k limit; Finish; Otherwise if the terminal point on backward k limit is positioned at the probe left side starting point on backward k limit be positioned at the probe right side or with the probe conllinear, be alignment edge then with backward k limit, finish; Otherwise the starting point on backward k limit is positioned at the probe right side if the terminal point on backward k limit is with the probe conllinear, and then the downstream side with backward k limit is an alignment edge; Otherwise if order k limit is identical with backward k limit or order k limit is the upstream edge on backward k limit, then seeking all over whole sequence does not have alignment edge yet, finishes; Otherwise if segment is recessed chain, and the starting point on the order k limit terminal point that is positioned at left side or the backward k limit of probe is positioned at the right side of probe, then alignment edge can not be arranged, and finishes; Otherwise with the downstream side on order k limit serves as that the upstream edge on order k+1 limit, backward k limit is backward k+1 limit, with k+1 as new k, execution in step 2.2;

Above-mentioned segment pinching confirms that the method for the visible range of observation point is:

Step 3.1: the upstream edge that with the first-in-chain(FIC) is order the 1st limit, last-of-chain is backward the 1st limit; If observation point is in left side, order the 1st limit, the directed edge of the upper reaches reference position of visible range is order the 1st limit; If observation point is in left side, backward the 1st limit, the directed edge of the downstream final position of visible range is backward the 1st limit; If the upper and lower position of visible range is all definite, then finish; Otherwise with k is cycle counter, k=1,

Step 3.2: with the downstream side on order k limit serves as that the upstream edge on order k+1 limit, backward k limit is backward k+1 limit, with k+1 as follow-up k; Not in left side, order k limit, then the directed edge of downstream final position is order k-1 limit, finishes if observation point is positioned at left side, order the 1st limit; Otherwise if observation point is positioned at left side, order the 1st limit and left side, backward k limit simultaneously, then the directed edge of downstream final position is backward k limit, finishes; Otherwise if observation point is positioned at left side, order k limit and left side, backward the 1st limit simultaneously, then the directed edge of upper reaches reference position is order k limit, finishes; Not in left side, backward k limit, then the directed edge of upper reaches reference position is backward k-1 limit, finishes otherwise if observation point is positioned at left side, backward the 1st limit; Otherwise execution in step 3.2.

Beneficial effect:

Ring-cutting subdivision method of simple polygon is online disposal route in 1 computer graphical

The method of handling is one by one with polygonal the 1st summit V ₁, the 2nd summit V ₂..., N summit V _NThe set of embedding subdivision, termination condition is n summit V _nReturn to the 1st summit V ₁As long as, obtain the 1st summit and just begin the subdivision process, need not to obtain by the time after all summits.

The time complexity of ring-cutting subdivision method of simple polygon is O (N) in 2 computer graphicals

In whole subdivision process, the processing of cycle repeats has:

The polygon vertex of the embedding one by one V of the 1st level _nMajor cycle loopMain, comprise:

The probe E of the 2nd level _pWith hit limit H _UiLimit cutting circulation loopCut when intersecting further comprises:

The probe E of the 3rd level _pOn segment, seek and hit limit H _UiAsk and hand over circulation loopHit,

The probe E of the 3rd level _pThe connection circulation loopLink of the consistent protruding or consistent recess reason of both sides;

The polygonal correction for direction circulation loopClockwise of the 1st level;

More than various circulations need different working times separately:

2.1 outermost major cycle loopMain is under the situation of not considering the subordinate level, working time, complexity was O (N);

2.2 the time complexity of the limit of the 2nd level cutting circulation loopCut is a more complicated, for different summits, the quantity of the additional side that probe is passed through is many more, and working time is long more; Total time complexity equals the summation of each summit cutting number of times; According to method of the present invention, additional side cutting (Figure 24 a～24i), suppose and go up summit V all will be carried out in summit of every under specific circumstances embedding _uM is arranged _uBar additional side, following summit V _dM is arranged _dThe bar additional side, then worst case is m _u=m _d=m (if m _u≠ m _d, for the additional side that has more, the cutting of an additional side of every completion need embed two summits at least, makes time complexity reduce); Set summit V _dBe the n summit, V then on the n+1 summit _N+1Embed after the subdivision set accomplishes, the quantity of the new additional side that possibly passed through by the formed probe in next summit that produces is m/2+1 (seeing Figure 13 a, 13b, the method for 14a, the bilateral chain of rings of 14b) at most, so:

N+1 summit V _N+1Embedding the total cutting number of times in back is m time;

N+2 summit V _N+2Embedding the total cutting number of times in back is m+m time;

N+3 summit V _N+3Embedding the total cutting number of times in back is 2m+m/2+1 time;

N+4 summit V _N+4Embedding the total cutting number of times in back is 2m+m/2+1+m/2+1 time;

N+5 summit V _N+5Embedding the total cutting number of times in back is 2m+m+2+m/4+1 time;

N+6 summit V _N+6Embedding the total cutting number of times in back is 2m+m+2+m/4+1+m/4+1 time;

……

N+2m summit V _N+2mEmbedding the total cutting number of times in back does $2\underset{i=0}{\overset{m}{\mathrm{\&Sigma;}}}\frac{m}{2^i}+2m-2=2(2-2^{-m})m+2m-2<6m$ Inferior;

Follow-up more summit embeds and is cut the influence that m is counted on the limit before the total cutting number of times in back is then no longer gone into;

The total number of vertex of ∵ is N>=[(2m+1)+(2m+2)-1]+2m=6m+2, m≤(N-2)/and 6＜N/6,6m＜N;

The time complexity of ∴ limit cutting circulation loopCut is O (6m)=O (N) under worst case;

2.3 the asking of the 3rd level hands over circulation loopHit to adopt the segment pinching to confirm the method on alignment probe limit (Figure 18 a～18c); Each operation time is by the number decision on summit on the segment; Do not taking place under the situation of cutting process; Total time complexity is the adding up of number on summit on all rings that are cut open, and promptly is O (N); But under the situation of cutting repeatedly; When a big ring is broken down into two less rings, different as if the number of vertex of the little ring that is decomposed out, then spent operation times only with both in the less ring of number of vertex relevant; And be the twice (the every circulation primary of segment pinching need be carried out order respectively and judged that with backward each once) of its number of vertices; And have nothing to do with the more ring of number of vertex, therefore, worst case is that two little rings respectively are half of original big ring number of vertex; When the summit of new embedding gets into the little ring of a certain side, ask and hand over circulation loopHit to search for once again little ring, produced additional operation time; If the number of vertices on the big ring is n, the number of vertices on the little ring is n/2, and the cap number of times on the then big ring is at most n, and the cap number of times on the little ring is at most n/2; In like manner; If little ring is by the decomposition of going forward one by one once again; Then newly-increased operation times is at most n/4 ..., on ring, be less than 4 summits and all become triangle; If summit after this is positioned within the little ring that decomposes that gone forward one by one, then ask the influence of handing over circulation loopHit no longer to receive big before ring number of vertices n; So for the little ring of probe either side, the friendship cycle index summation of asking that adds up of the little ring of all decomposition of going forward one by one is:

$\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{n}{2^i}=n(1-\frac{1}{2^n})<n,$

Itself and asking separately of big ring are handed over cycle index n addition, then ask and hand over cycle index summation＜2n;

∵ will encircle greatly to go forward one by one and decompose in the time of cannot decomposing again, and needing an other n summit, then total number of vertex at least should be n summit sum, i.e. 2n≤N of n the summit on the big ring and the decomposition of going forward one by one at least;

The time complexity of asking friendship circulation loopHit of ∴ the 3rd level also is O (2n)=O (N) under worst case;

2.4 the time complexity of the connection circulation loopLink of the 3rd level; With the ring shape and whether connected relevant repeatedly; For the connection that forms bulge loop, only need the time of constant at every turn, and for the connection that forms scrobicular ring; Then possibly be the time of constant, also possibly be equivalent to the number of vertices that forms scrobicular ring; When scrobicular ring decomposition situation does not take place; Recessed chain for from upper type is postponed, and is positioned at last-of-chain (Figure 16 b) if the downstream class is cut edge, and then needs the constant time; Be positioned at segment middle part (Figure 17 b) if the downstream class is cut edge, then the number of times of the location under the worst case is the number of vertices of segment; Same principle also is suitable for from the following recessed chain of mode postpone (Figure 16 c, 17d); Therefore, total connection round-robin time complexity is O (N); And be cut and cause under the situation that ring decomposes in that the limit has taken place; Can be divided into tie-time and repetition tie-time first total working time; Tie-time is O (N) first; Repeat then repetitive operation (this repetitive operation has comprised two situation that recessed chain is alternately postponed, Figure 15 a, 15b) of tie-time because big scrobicular ring is broken down into little scrobicular ring; Ask the analysis of handing in the circulation according to above-mentioned, when big ring is broken down into the little ring that two number of vertices equate, need the number of times of computing maximum; So, under worst case, connect required total operation times once again and be: $\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{N}{2^i}=N(1-\frac{1}{2^N})<N$

The time complexity that ∴ connects circulation loopLink for first with repetition tie-time sum: O (N)+O (N)=O (N);

2.5 outermost polygon correction for direction circulation loopClockwise, complexity working time that its direction reverses is O (N);

Though more than various circulations be in different circulating levels; But various round-robin number of times are by the decision of the geometric relationship of polygon self; Not by the decision of round-robin level, and the summation of various round-robin time complexities is exactly the time complexity of ring-cutting subdivision method; Because every kind of round-robin time complexity all is linear, therefore, they be linear also, promptly the time complexity of ring-cutting subdivision method is O (N).

Ring-cutting subdivision method of simple polygon is simpler than other known linear session subdivision methods in 3 computer graphicals

1991; By " the linear session subdivision of simple polygon " Triangulating a SimplePolygon in Linear Time method of Bernard Chazelle invention, be published in authoritative academic journal " discrete and computational geometry " (Discrete&Computational Geometry, 6 (5) of computational geometry circle; The 485th～524 page; 1991), this method is not only quite complicated, indigestion, and is off-line;

Ring-cutting subdivision Faxian of the present invention so is easily understood than the method for Bernard Chazelle, realizes easily;

Ring-cutting subdivision method of simple polygon can be handled the polygon that apex coordinate is a random value in real time in 4 computer graphicals

When the summit of simple polygon clocklike constitutes recessed chain or the protruding chain of big section continuously, might accumulate a large amount of additional sides at some position, make follow-up some summits need spend considerable time when embedding and carry out cutting process; If apex coordinate is random random value, then cutting process is spread out, and this graphics process for routine is very favorable, thereby can realize real-time processing.

Description of drawings

Fig. 1 is polygonal outsourcing rhombus Q ₀The setting synoptic diagram.

Fig. 2 is by the 1st summit V ₁Formed initial subdivision set.

Fig. 3 is clockwise polygonal instance.

Fig. 4 is probe E _pThe instance overlapping with the additional side.

Fig. 5 is monolateral shape segment C _C1And starting point is n-1 summit V _N-1Whole directed edge G _eThe formation instance of (3 or 4).

Fig. 6 a, 6b, 6c and 6d are respectively the synoptic diagram of protruding chain, recessed chain, left spill and right spill segment.

Fig. 7 a, 7b are respectively that right spill segment obtains and hits limit H _UiThe time the both sides synoptic diagram that decomposites of original state and right flank.

Fig. 8 a, 8b are respectively that left spill segment obtains and hits limit H _UiThe time the both sides synoptic diagram that decomposites of original state and left wing.

Fig. 9 is probe E _pWith hit limit H _UiThe instance that intersects.

Figure 10 is to probe E _pThe place ahead carry out a chain of instance that constitutes 2 additional sides in three limits.

Figure 11 is at probe E _pThere is collinear point V in the place ahead _cThe time constitute the instance of 3 additional sides.

Figure 12 is n summit V _nReturn to the 1st summit V ₁And hit limit H _UiStarting point be the 1st summit V ₁Instance.

Figure 13 a is respectively when the bilateral chain of rings, plays ring limit R _uCan be formed a bigger bulge loop instance by deletion;

Figure 13 b is when the bilateral chain of rings, plays ring limit R _uWith ring inner edge R _iAll can be formed the instance of a bigger bulge loop by deletion;

Figure 14 a is when the bilateral chain of rings, plays ring limit R _uCan be formed the instance of a bigger scrobicular ring by deletion;

Figure 14 b is when the bilateral chain of rings, ring inner edge R _iCan be formed the instance of a bigger scrobicular ring by deletion.

Figure 15 a, 15b are on three limits when a chain of, the instance of postponing and handling that replaces about carrying out with bounded, unbounded mode respectively.

Figure 15 c is on three limits of bounded mode when a chain of, and twice segment correction result is the instance of two monolateral shapes.

Figure 15 d is on three limits of unbounded mode when a chain of, and twice segment correction result is the instance of two monolateral shapes.

Figure 15 e is on three limits of unbounded mode when a chain of, and twice segment correction result is respectively the instance of monolateral shape, recessed chain.

Figure 15 f is on three limits of unbounded mode when a chain of, and twice segment correction result is respectively the instance of recessed chain, monolateral shape.

Figure 16 a, 16b, 16c are respectively the instances of the segment correction of protruding chain segment, left spill segment, right spill segment.

Figure 16 d, 16e are respectively in the segment makeover process, the instance that does not deal with of monolateral segment segment, recessed chain segment.

Figure 17 a, 17b postpone from the recessed chain of upper type for being the result of new recessed chain.

Figure 17 c, 17d postpone from the recessed chain of following mode for being the result of new monolateral shape.

Figure 18 a, 18b implement the instance that the alignment probe limit is confirmed in the segment pinching to protruding chain segment, recessed chain segment.

Figure 18 c is when implementing between the definite probe of segment pinching target area to recessed chain segment, does not have the instance of alignment edge.

Figure 19 a is segment pinching when confirming the visible range of observation point, directly obtains the instance of visible range;

Figure 19 b is segment pinching when confirming the visible range of observation point, and an end becomes visible instance by invisible.

Figure 19 c is segment pinching when confirming the visible range of observation point, and an end becomes sightless instance by visible.

Figure 20 a, 20b are the initial setting up of the complete ring-cutting subdivision instance of polygon polygon;

Figure 20 c～20f is the 2nd summit V of polygon polygon ₂～the 3 summit V ₃Embed the synoptic diagram of subdivision set;

Figure 21 a～21f is the 4th summit V of polygon polygon ₄～the 10 summit V ₁₀Embed the synoptic diagram of subdivision set;

Figure 22 a～22f is the 11st summit V of polygon polygon ₁₁～the 13 summit V ₁₃Embed the synoptic diagram of subdivision set;

Figure 23 a～23d is the 14th summit V of polygon polygon ₁₄～the 15 summit V ₁₅Embed the synoptic diagram of subdivision set;

Figure 23 e, 23f are the closed results of polygon polygon.

Figure 24 a～24i is the number change synoptic diagram that is cut the limit number under the worst case.

Annotate: the trend of the reference character of each directed edge order is consistent with the direction of the directed edge that is marked among the figure.

Embodiment

Embodiment 1

With reference to Fig. 1, in step 1.1, by a＞horizontal scale X _Max+ vertical yardstick Y _Max, confirm the numerical value of a, set up the rhombus Q outside the polygon ₀Note during actual program; Although a can be infinity in theory, employed computerese has qualification to effective numerical range, in addition; Half of data structure is used in the directed edge suggestion, when carrying out direction distinguishing, need consider the influence of numerical evaluation round-off error;

With reference to Fig. 2, in step 1.1, selected the 1st summit V ₁, the 2nd summit V ₂, connect the 1st summit V ₁Each summit with rhombus;

With reference to Fig. 3, with n summit V _nWhen embedding the subdivision set, because of n summit V _nWith the 1st summit V ₁Overlap and with N summit V _NAdditional side E for starting point _NTerminal point be the 1st summit V ₁, N summit E then _NBe polygonal limit, embed and finish; When execution in step 1.3 because polygon be clockwise (can be by crossing rhombus Q ₀The additional side on a certain structure summit obtain polygonal peripheral summit, the upstream and downstream adjacent vertex by described peripheral summit comes decision diamond Q again ₀Above-mentioned structure summit whether in polygonal left side, rhombus Q for example ₀The 2nd structure summit P was arranged ₂Additional side E _2N, said additional side E _2NTerminal point be N summit V _N, and the 2nd structure summit P ₂Be positioned at V from the N summit _NUpper reaches adjacent vertex V _N-1Through N summit V _NThe V to the N summit _NDownstream adjacent vertex V ₁Within the angle of formed left side), handle so need travel direction to reverse, with N summit V _NDemarcation is the 1st summit V ₁The downstream adjacent vertex, with the 1st summit V ₁Demarcation is the 2nd summit V ₂The downstream adjacent vertex, with the 2nd summit V ₂Demarcation is the 3rd summit V ₃The downstream adjacent vertex, with the 3rd summit V ₃Demarcation is the 4th summit V ₄The downstream adjacent vertex, And with the 1st summit V ₁Demarcation is N summit V _NUpper reaches adjacent vertex, with the 2nd summit V ₂Demarcation is the 1st summit V ₁Upper reaches adjacent vertex, with the 3rd summit V ₃Demarcation is the 2nd summit V ₂Upper reaches adjacent vertex, with the 4th summit V ₄Demarcation is the 3rd summit V ₃Upper reaches adjacent vertex ...

With reference to Fig. 4, with n summit V _nWhen embedding the subdivision set, from n-1 summit V _N-1Selecting article one directed edge that is positioned at the probe right side in the directed edge for starting point is the 1st right limit E _R1, with the 1st right limit E _R1Downstream side be the 1st first-in-chain(FIC) E _H1, be the 1st last-of-chain E with the upstream edge of probe _T1, as follow-up counts, i=1 is because be positioned at the 1st right limit E with i _R1Article one additional side E in left side _cWith probe E _pOverlapping, so with overlapping limit E _cDownstream side be conllinear limit H _a, delete overlapping limit E then _c, carry out collineation processing by step 1.2.6 again.

With reference to Fig. 5, Fig. 6 a～6d, when execution in step 1.2.1, from all with n-1 summit V _N-1Directed edge G for starting point _eIn select be positioned at the probe E _pArticle one limit on right side is the 1st right limit E _R1, with the 1st right limit E _R1Downstream side be the 1st first-in-chain(FIC) E _H1, with probe E _pUpstream edge be the 1st last-of-chain E _T1, the segment of having created a monolateral shape (Fig. 5) is the 1st segment C _C1The segment of other shape have protruding chain (Fig. 6 a), recessed chain (Fig. 6 b), left spill (Fig. 6 c), right spill (Fig. 6 d).

With reference to Fig. 7 a, Fig. 7 b, carry out ring cutting at execution in step 1.2.3 and cut when handling i segment C _CiBe right spill and probe E _pWith i last-of-chain E _TiUpstream edge non-intersect, so to i first-in-chain(FIC) E _HiBe first-in-chain(FIC), i last-of-chain E _TiUpstream edge be that the segment of last-of-chain carries out the segment pinching and confirms between the probe target area, and it is hit limit H as i _Ui

With reference to Fig. 8 a, Fig. 8 b, when execution in step 1.2.4, i segment C _CiBe left spill and probe E _pWith i first-in-chain(FIC) E _HiNon-intersect, so to i first-in-chain(FIC) E _HiBe first-in-chain(FIC), i last-of-chain E _TiUpstream edge be that the segment of last-of-chain carries out the segment pinching and confirms between the probe target area, but alignment edge do not exist, so with i first-in-chain(FIC) E _HiHit limit H as i _Ui

With reference to Fig. 9, when execution in step 1.2.5, hit limit H with i _UiDownstream side H _DiBe i+1 last-of-chain E _Ti+1, because probe E _pHit limit H with i _UiIntersect, so hit limit H with i _UiReverse edge E _HrDownstream side E _HrdBe i+1 first-in-chain(FIC) E _Hi+1, with probe E _pReverse edge E _PrDownstream side E _PrdBe i+1 right limit E _Ri+1, and delete i and hit limit H _Ui, as new i, return to step 1.2.3 and carry out the next round ring cutting and cut processing with i+1.

With reference to Figure 10, when execution in step 1.2.5, because probe E _pHit limit H with i _UiNon-intersect, thus to the probe E _pReverse edge E _PrFor first-in-chain(FIC), i hit limit H _UiFor the segment of last-of-chain carries out the segment correction, with newly connect from the additional side on n summit as right flank limit W _r, again to i+1 last-of-chain E _Ti+1Be first-in-chain(FIC), right flank limit W _rFor the segment of last-of-chain carries out the segment correction, with newly connect from the additional side of the starting point of i+1 last-of-chain as the limit W of left wing _lRespectively with right flank limit W _rDownstream side W _Rd, the limit W of left wing _lWith right flank limit W _rUpstream edge W _RuAs playing initial line, medial side and termination limit, the three limits chain of rings is carried out in the place ahead of probe with the unbounded mode.

With reference to Figure 11, carry out in the collineation processing process at execution in step 1.2.6, earlier to probe E _pTo carry out three limits of bounded mode respectively a chain of in both sides, then to i right limit E _RiBe first-in-chain(FIC), conllinear limit H _aFor the segment of last-of-chain obtains the limit, middle part, because of this segment is left spill, so with last-of-chain H _aUpstream edge H _AuBe the limit, middle part, and with first-in-chain(FIC) E _RiFor carrying out the bilateral chain of rings for the stop ring limit in a ring limit, limit, middle part; Be limit M in the right side with the limit, middle part then _u, probe E _pReverse edge E _PrFor playing limit M in ring limit, the right side _uFor carrying out the bilateral chain of rings in the stop ring limit, and connect n summit V _nWith conllinear summit V _cAs the limit E that continues _xTo with conllinear limit H _aBe first-in-chain(FIC), probe E _pFor the segment of last-of-chain obtains the limit, middle part, not the downstream side H of left spill and first-in-chain(FIC) because of this segment _AdNot the upstream edge of last-of-chain, so with the downstream side H of first-in-chain(FIC) _AdFor a ring limit, last-of-chain carry out the bilateral chain of rings for the stop ring limit, the downstream side with the first-in-chain(FIC) after the bilateral chain of rings is the limit, middle part then; Then with the limit, middle part as limit M in the left side _d, with limit M in the left side _dFor a ring limit, limit E continues _xFor carrying out the bilateral chain of rings in the stop ring limit.

With reference to Figure 12, when execution in step 1.2.6, n summit V _nBe that i hits limit H _UiStarting point (also must be the 1st summit V ₁), polygon is closed.

With reference to Figure 13 a, when the bilateral chain of rings, with stop ring limit R _dUpstream edge be ring inner edge R _i, to have connected ring limit R _uStarting point and stop ring limit R _dThe additional side that forms of starting point as sealing limit R _r, because play ring limit R _uBe an additional side and a ring limit R _uA long enough and a ring limit R _uTwo rings of both sides playing a ring limit R _uCan be formed a bigger bulge loop by under the situation of deleting, so can have been deleted ring limit R _uBecause of ring inner edge R _iNot the additional side, so ring inner edge R _iCannot delete.

With reference to Figure 13 b, when the bilateral chain of rings, different with Figure 14 a is ring inner edge R _iAlso be additional side and ring inner edge R _iLong enough and ring inner edge R _iTwo rings of both sides at ring inner edge R _iCan be formed a bigger bulge loop by under the situation of deleting, so ring inner edge R _iAlso can be deleted.

With reference to Figure 14 a, when the bilateral chain of rings, with stop ring limit R _dUpstream edge be ring inner edge R _i, to have connected ring limit R _uStarting point and stop ring limit R _dThe additional side that forms of starting point as sealing limit R _r, because play ring limit R _uBe an additional side and a ring limit R _uA long enough and a ring limit R _uTerminal point be not the 1st summit V ₁And play ring limit R _uTwo rings of both sides playing a ring limit R _uCan be formed a bigger scrobicular ring by under the situation of deleting, so can have been deleted ring limit R _uBecause of ring inner edge R _iNot the additional side, so ring inner edge R _iCannot delete.

With reference to Figure 14 b, when the bilateral chain of rings, play ring limit R _uNot the additional side, so play ring limit R _uCannot delete; Because of ring inner edge R _iBe additional side and ring inner edge R _iLong enough and ring inner edge R _iTerminal point be not the 1st summit V ₁And ring inner edge R _iTwo rings of both sides at ring inner edge R _iCan be formed a bigger scrobicular ring by under the situation of deleting, so ring inner edge R _iAlso can be deleted.

With reference to Figure 15 a, in a chain of process in three limits of bounded mode, to play an initial line S _uBe first-in-chain(FIC), medial side S _mThe segment of creating for last-of-chain carries out the segment correction, to play initial line S _uBe new initial line S _NuAgain to medial side S _mBe first-in-chain(FIC), termination limit S _dThe segment of creating for last-of-chain carries out the segment correction, with medial side S _mThe additional side of new downstream direction of start position as limit S in new _NmBecause play initial line S _uNot medial side S _mAnd medial side S _mNot to stop limit S _dAnd limit S in new _NmAngle＜the π of starting point and twice correction result all are recessed chains, and execution in step 1.2.5.2 is with limit S in new _NmBe the 1st to take turns strange flanging O ₁, new initial line S _NuStarting point be upper extreme point P _u, stop limit S _dStarting point be lower extreme point P _d, execution in step 1.2.5.3 takes turns strange flanging O with the 1st ₁Be recessed first-in-chain(FIC), termination limit S _dBe recessed last-of-chain, postpone, and take turns even flanging E as the 1st with the additional side of new connection to carry out recessed chain from upper type ₁Execution in step 1.2.5.4 is with new initial line S _NuFor recessed first-in-chain(FIC), the 1st is taken turns even flanging E ₁Be recessed last-of-chain, postpone to descend mode to carry out recessed chain certainly, and with the downstream side of the additional side of new connection as the 2nd strange flanging O that takes turns ₂Take turns execution in step 1.2.5.3 with the 2nd, because the strange flanging O that the three limit chains of rings are bounded modes and the 2nd takes turns ₂The starting point of upstream edge be upper extreme point P _uSo, finish.

With reference to Figure 15 b, in a chain of process in three limits of unbounded mode, to play an initial line S _uBe first-in-chain(FIC), medial side S _mThe segment of creating for last-of-chain carries out the segment correction, to play initial line S _uBe new initial line S _NuAgain to medial side S _mBe first-in-chain(FIC), termination limit S _dThe segment of creating for last-of-chain carries out the segment correction, because medial side S _mStart position the new updrift side rather than the additional side of downstream direction are arranged, so with medial side S _mAs limit S in new _NmBecause play initial line S _uNot medial side S _mAnd medial side S _mNot to stop limit S _dAnd limit S in new _NmAngle＜the π of starting point and twice correction result all are recessed chains, and execution in step 1.2.5.2 is with limit S in new _NmBe the 1st to take turns strange flanging O ₁, new initial line S _NuStarting point be upper extreme point P _u, stop limit S _dStarting point be lower extreme point P _d, execution in step 1.2.5.3 takes turns strange flanging O with the 1st ₁Be recessed first-in-chain(FIC), termination limit S _dBe recessed last-of-chain, postpone, and take turns even flanging E as the 1st with the additional side of new connection to carry out recessed chain from upper type ₁Execution in step 1.2.5.4 is with new initial line S _NuFor recessed first-in-chain(FIC), the 1st is taken turns even flanging E ₁Be recessed last-of-chain, postpone, and take turns strange flanging O as the 2nd with the downstream side of the additional side of new connection to descend mode to carry out recessed chain certainly ₂Take turns execution in step 1.2.5.3 with the 2nd, take turns strange flanging O with the 2nd ₂Be recessed first-in-chain(FIC), termination limit S _dBe recessed last-of-chain, postpone, and take turns even flanging E as the 2nd with the additional side of new connection to carry out recessed chain from upper type ₂Execution in step 1.2.5.4 takes turns even flanging E because the three limit chains of rings are unbounded modes and the 2nd ₂Starting point be upper extreme point P _uSo, finish.

With reference to Figure 15 c, in a chain of process in three limits of bounded mode,, obtain new initial line S through twice segment correction _Nu, new middle limit S _NmThough play initial line S _uNot medial side S _mAnd medial side S _mNot to stop limit S _dAnd limit S in new _NmAngle＜the π of starting point, but twice segment correction result is not two recessed chains.

With reference to Figure 15 d, in a chain of process in three limits of unbounded mode,, obtain new initial line S through twice segment correction _Nu, new middle limit S _NmBecause play initial line S _uNot medial side S _mAnd medial side S _mNot to stop limit S _dAnd limit S in new _NmAngle＜the π of starting point, and twice correction result all be monolateral shape, so with new initial line S _NuFor playing the ring limit, stopping limit S _dFor carrying out the bilateral chain of rings in the stop ring limit.

With reference to Figure 15 e, in a chain of process in three limits of unbounded mode, the situation of similar Figure 15 d, but correction result is monolateral shape for the first time, correction result is recessed chain for the second time, so respectively with limit S in new _NmBe recessed first-in-chain(FIC), termination limit S _dBe recessed last-of-chain, postpone to carry out recessed chain from upper type.

With reference to Figure 15 f, in a chain of process in three limits of unbounded mode, the situation of similar Figure 15 d, but correction result is recessed chain for the first time, correction result is monolateral shape for the second time, so respectively with new initial line S _NuBe recessed first-in-chain(FIC), new middle limit S _NmBe recessed last-of-chain, postpone to descend mode to carry out recessed chain certainly.

With reference to Figure 16 a, in the segment makeover process, segment is protruding chain, with first-in-chain(FIC) E _hFor playing ring limit, last-of-chain E _tFor carrying out the bilateral chain of rings (correction result is monolateral shape) in the stop ring limit.

With reference to Figure 16 b, in the segment makeover process, segment is left spill, with first-in-chain(FIC) E _hDownstream side be recessed first-in-chain(FIC), last-of-chain E _tBe recessed last-of-chain, to carry out recessed chain postpone (correction result is monolateral shape) from upper type.

With reference to Figure 16 c, in the segment makeover process, segment is right spill, with first-in-chain(FIC) E _hBe recessed first-in-chain(FIC), last-of-chain E _tUpstream edge be recessed last-of-chain, carry out recessed chain postpone (correction result is recessed chain) to descend mode certainly.

With reference to Figure 16 d, Figure 16 e, in the segment makeover process, segment is monolateral shape or recessed chain, does not deal with.

With reference to Figure 17 a, Figure 17 b, in the recessed chain of upper type is postponed process, with first-in-chain(FIC) E _hUpstream edge for observing limit W, and to first-in-chain(FIC) E _h, last-of-chain E _tThe visible range that observation point is confirmed in the segment pinching is carried out on two limits forming, is upper reaches class side cut T with the directed edge of the upper reaches reference position of visible range _u, the downstream side of the directed edge of the downstream final position of visible range is downstream class side cut T _dBe ring limit R to observe limit W then _u, downstream class side cut T _dBe stop ring limit R _dCarry out the bilateral chain of rings (result that postpones is recessed chain).

With reference to Figure 17 c, Figure 17 d, in the recessed chain that descends mode is certainly postponed process, with last-of-chain E _tDownstream side for observing limit W, and to first-in-chain(FIC) E _h, last-of-chain E _tThe visible range that observation point is confirmed in the segment pinching is carried out on two limits forming, is upper reaches class side cut T with the directed edge of the upper reaches reference position of visible range _u, the downstream side of the directed edge of the downstream final position of visible range is downstream class side cut T _dThen with upper reaches class side cut T _uFor playing ring limit R _u, to observe limit W be stop ring limit R _dCarry out the bilateral chain of rings (result that postpones is monolateral shape).

With reference to Figure 18 a, confirm in the process on alignment probe limit in the segment pinching, with the first limit E on both sides _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Through the circulation of step 2.2, can be in the hope of probe E _pAlignment edge be backward the 3rd limit E _B3

With reference to Figure 18 b, confirm in the process on alignment probe limit in the segment pinching, with the first limit E on both sides _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Through the circulation of step 2.2, can be in the hope of probe E _pAlignment edge be the order the 5th limit E _F5

With reference to Figure 18 c, confirm in the process on alignment probe limit in the segment pinching, with the first limit E on both sides _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Through the circulation of step 2.2, the angle>=π of the starting point on backward the 3rd limit and probe E _pStarting point be positioned at simultaneously the order the 3rd limit E _F3, backward the 3rd limit E _B3The left side, and E _F3Terminal point be positioned at the probe E _pThe right side, alignment edge can not be arranged.

With reference to Figure 19 a, confirm in the process of visible range of observation point, with the first limit E on both sides in the segment pinching _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Observation point V _EyeBe positioned at order the 1st limit E simultaneously _F1Left side and backward the 1st limit E _B1The left side, then the directed edge of upper reaches reference position is order the 1st limit E _F1, the directed edge of the downstream final position of visible range is backward the 1st limit E _B1

With reference to Figure 19 b, confirm in the process of visible range of observation point, with the first limit E on both sides in the segment pinching _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Observation point V _EyeThe E in left side, backward the 1st limit _B1, the directed edge of the downstream final position of visible range is backward the 1st limit E _B1Through the circulation of step 3.2, observation point V _EyeBe positioned at order the 4th limit E simultaneously _F4Left side and backward the 1st limit E _B1The left side, then the directed edge of upper reaches reference position is order the 4th limit E _F4

With reference to Figure 19 c, confirm in the process of visible range of observation point, with the first limit E on both sides in the segment pinching _Nip1Be order the 1st limit E _F1, both sides the second limit E _Nip2Upstream edge be backward the 1st limit E _B1Observation point V _EyeThe left side E on order the 1st limit _F1, the directed edge of the upper reaches final position of visible range is E _F1Through the circulation of step 3.2, observation point V _EyeBe positioned at order the 1st limit left side but left side E not on order the 4th limit _F4, then the directed edge of downstream final position is order the 4th limit E _F4Upstream edge E _F3

Embodiment 2

Initialization: with reference to Figure 20 a, 20b, 15 apex coordinates of polygon Polygon are respectively: V ₁(1.72,10.72), V ₂(2.0,6.0), V ₃(7.0,4.0), V ₄(11.6,3.2), V ₅(5.0 ,-2.0), V ₆(3.88 ,-3.32), V ₇(3.68 ,-4.48), V ₈(3.14 ,-3.64), V ₉(2.0 ,-3.52), V ₁₀(0.0 ,-4.0), V ₁₁(4.0 ,-2.4), V ₁₂(1.71,2.8), V ₁₃(4.25 ,-0.81), V ₁₄(5.92,3.05), V ₁₅(3.61,0.56), the 16th summit V ₁₆It is the reproduction on the 1st summit; By a＞horizontal scale X _Max+ vertical yardstick Y _Max=|-11.6|+10.72=22.32, get a=30, it is respectively P that 4 structure summits are set outside polygon Polygon ₁(30,0), P ₂(0,30), P ₃(30,0), P ₄The rhombus Q of (0 ,-30) ₀The 1st summit V ₁Respectively with rhombus Q ₀4 the structure summits be connected;

With the 2nd summit V ₂Embedding subdivision set: with reference to Figure 20 c, 20d, by step 1.2.1, from the 1st summit V ₁To the 2nd summit V ₂Constitute probe E _p, because of the 2nd summit V ₂With the 1st summit V ₁Overlap, so select the E that pops one's head in that is positioned at from the 2nd summit _pArticle one directed edge on right side is the 1st right limit E _R1, and with the 1st right limit E _R1Downstream side be the 1st first-in-chain(FIC) E _H1, with probe E _pUpstream edge be the 1st last-of-chain E _T1, because do not have and probe E _pOverlapping additional side, thus execution in step 1.2.3, to the 1st first-in-chain(FIC) E _H1, the 1st last-of-chain E _T1Monolateral shape the 1st segment C that is constituted _C1Carry out the segment pinching and confirm the alignment probe limit, the alignment edge that is obtained is hit limit H as the 1st _U1, execution in step 1.2.5 then is because the 2nd summit V ₂With the 1st summit V ₁Do not overlap, and the 1st hits limit H _U1Two end points all not with the probe E _pSo conllinear is respectively with the 1st right limit E _R1, the 1st first-in-chain(FIC) E _H1With the 1st hit limit H _U1As playing initial line, medial side and a termination limit, with the unbounded mode to probe E _pTo carry out three limits a chain of on the right side owing to be that first-in-chain(FIC), medial side are that the segment of last-of-chain is monolateral shape to play initial line, and medial side is identical with the termination limit, so the three limit chains of rings do not change; Hit limit H with the 1st respectively again _U1Downstream side, the 1st last-of-chain E _T1With probe E _pAs playing initial line, medial side and termination limit; With the unbounded mode the three limits chain of rings is carried out in the left side of probe; Because it is identical with medial side to play initial line, and be first-in-chain(FIC) with the medial side, segment that the termination limit is last-of-chain is monolateral shape, this time the three limit chains of rings also do not change; Hit limit H with the 1st _U1Downstream side be the 2nd last-of-chain E _T2, because of probe E _pDo not hit limit H with the 1st _U1Intersect, so to the E that pops one's head in _pReverse edge E _PrFor first-in-chain(FIC), the 1st hits limit H _U1For the protruding chain segment of last-of-chain carries out the segment correction, be that ring limit, last-of-chain are that bilateral the 1st right limit E that deleted when a chain of is carried out on the stop ring limit with first-in-chain(FIC) _R1, with newly connect from the additional side on the 2nd summit as right flank limit W _rAgain to the 2nd last-of-chain E _T2Be first-in-chain(FIC), right flank limit W _rFor the protruding chain segment of last-of-chain carries out the segment correction, equally at bilateral the 2nd last-of-chain E that deleted when a chain of _T2, with newly connect from the 2nd last-of-chain E _T2The additional side of starting point as the limit W of left wing _lRespectively with right flank limit W _rDownstream side, the limit W of left wing _lWith right flank limit W _rUpstream edge as playing initial line, medial side and a termination limit, it is a chain of with the unbounded mode three limits to be carried out in the place ahead of probe, this time three limit chains of rings do not change;

With the 3rd summit V ₃Embedding subdivision set: with reference to Figure 20 e, 20f, with above-mentioned with the 2nd summit V ₂It is similar to embed the subdivision aggregation process, has connected respectively from the 3rd summit V ₃To the 3rd structure summit P ₃Right flank limit W _rWith from the 4th structure summit P ₄To the 3rd summit V ₃The limit W of left wing _l, difference is when carrying out the bilateral chain of rings, the 2nd last-of-chain E _T2Can not be deleted;

With the 4th summit V ₄Embedding subdivision set: with reference to Figure 21 a, 21b, with the 1st right limit E _R1Downstream side be the 1st first-in-chain(FIC) E _H1, with probe E _pUpstream edge be the 1st last-of-chain E _T1, its special feature is that article one is positioned at the 1st right limit E _R1The additional side E in left side _cWith probe E _pOverlapping, so with this additional side E _cDownstream side be conllinear limit H _a, and delete overlapping additional side E _c, execution in step 1.2.6 then is respectively with the 1st right limit E _R1, the 1st first-in-chain(FIC) E _H1With conllinear limit H _aAs playing initial line, medial side and termination limit, with the bounded mode the three limits chain of rings is carried out on the right side of probe, cause at the 2nd summit V ₂With the 3rd structure summit P ₃Between connected the additional side; Again respectively with conllinear limit H _a, the 1st last-of-chain E _T1With probe E _pAs playing initial line, medial side and termination limit, with the bounded mode the three limits chain of rings is carried out in the left side of probe, do not change; To with the probe E _pReverse edge E _PrDownstream side E _PrdBe first-in-chain(FIC), conllinear limit H _aFor the segment of last-of-chain obtains the limit, middle part and as limit M in the right side _u, then with probe E _pReverse edge E _PrFor playing limit M in ring limit, the right side _uFor carrying out the bilateral chain of rings in the stop ring limit, and connect the 4th summit V ₄With conllinear summit P ₃As the limit E that continues _xTo with conllinear limit H _aBe first-in-chain(FIC), probe E _pFor the segment of last-of-chain obtains the limit, middle part and as limit M in the left side _d, then with limit M in the left side _dFor a ring limit, limit E continues _xFor carrying out the bilateral chain of rings in the stop ring limit;

With the 5th summit V ₅The set of embedding subdivision: with reference to Figure 21 c, 21d, its special feature is when execution in step 1.2.5, probe E _pHit limit H at probe with the 1st _U1Intersect, thereby need deletion the 1st to hit limit H _U1, twice execution in step 1.2.3; Embedding subdivision aggregation process and the 2nd summit V with the 6th～9 summit ₂, the 3rd summit V ₃Similar, connect two new additional sides at every turn, and possibly delete one to two additional side;

With the 10th summit V ₁₀Embedding subdivision set: with reference to Figure 21 e, 21f, with the 5th summit V ₅Embed the subdivision collection class seemingly, hit limit H in deletion the 1st _U1Afterwards, execution in step 1.2.3 for the second time, thereby with the 10th summit V ₁₀Respectively with the 1st structure summit P ₁, the 4th structure summit P ₄Link;

With the 11st summit V ₁₁Embedding subdivision set:, take turns the 1st respectively and deleted the 1st and hit limit H with reference to Figure 22 a, 22b _U1, the 2nd takes turns and has deleted the 2nd and hit limit H _U2Afterwards, the 3rd take turns hit limit H _U3Starting point V ₅With probe E _pConllinear, when execution in step 1.2.6, need be with the bounded mode to probe E _pTo carry out three limits respectively a chain of in both sides, wherein the three limit chains of rings to the left side have caused alternately postponing of two recessed chains; Similar the 4th summit V of subsequent step ₄Embed the process of subdivision set;

Embedding subdivision set with the 12nd～15 summit: with reference to Figure 22 c～22f, 23a～23d, with the 2nd summit V ₂, the 3rd summit V ₃Embedding subdivision aggregation process similar;

With the 16th summit V ₁₆Embedding subdivision set: with reference to Figure 19 e, 19f, though the 16th summit V ₁₆With the 1st summit V ₁Overlap, but the 15th summit V ₁₅To the 1st summit V ₁There is not the additional side, therefore need be with probe E _pAs probe E _pExecution in step 1.2.3～step 1.2.5 takes turns through the 1st respectively and has deleted the 1st and hit limit H _U1, the 2nd takes turns and has deleted the 2nd and hit limit H _U2Afterwards, the 3rd take turns hit limit H _U3Starting point be the 1st summit V ₁, polygon is closed, respectively with the bounded mode to probe E _pTo carry out three limits respectively a chain of in both sides, accomplish the embedding subdivision aggregation process on all summits;

Polygonal correction for direction: because according to the 1st summit, the 2nd summit, the 3rd summit ..., the polygonal trend that forms of the order on 1 summit, the 15th summit to the is anticlockwise, need not to revise the upstream and downstream neighbouring relations;

Above step has been accomplished the ring-cutting subdivision of the polygon Polygon of embodiment 2.

Claims (3)

1. ring-cutting subdivision method of simple polygon in the computer graphical is characterized in that:

Step 1.1: set up the XOY plane coordinate, can contain polygonal rhombus one of the polygonal outer setting of treating subdivision, the coordinate on 4 structure summits of rhombus is respectively (a; 0), (0; A), (a, 0) with (0 ,-a); A is a real number; A is greater than horizontal scale and vertical yardstick sum, and horizontal scale is the greater of absolute value of left side or the right side X coordinate of polygonal minimum outsourcing rectangle, and vertically yardstick is the greater of absolute value of upside or the downside Y coordinate of polygonal minimum outsourcing rectangle; Rhombus constitutes a ring, and ring is a downstream direction with the counter clockwise direction; As the 1st summit, 4 structure summits with the 1st summit and rhombus are connected one by one, and rhombus is decomposed into 4 than circlet with arbitrary summit on the polygon, gather as initial subdivision than circlet with said 4; With in two adjacent on polygon summits with the 1st summit any one as the 2nd summit, be polygonal downstream direction with the trend of the 1st summit and the 2nd summit order;

Step 1.2: one by one the n summit is embedded the subdivision set along polygonal downstream direction, n=2,3,4 ..., N, N+1, N are polygonal number of vertices, the N+1 summit overlaps with the 1st summit;

Step 1.3: if according to the 1st summit, the 2nd summit, the 3rd summit ..., the polygonal trend that forms of the order on 1 summit, N summit to the is clockwise; Then polygonal each summit travel direction is reversed and handle; The upper reaches adjacent vertex that is about to former setting is modified to the downstream adjacent vertex, and the downstream adjacent vertex of former setting is modified to upper reaches adjacent vertex;

The above-mentioned method that the n summit is embedded the subdivision set is:

Step 1.2.1: with the directed edge from n summit, n-1 summit to the is probe, different with the 1st summit as if the n summit, then execution in step 1.2.2; Otherwise; If with the n-1 summit is that then polygon is closed, and this additional side is updated to polygonal limit when having the terminal point of an additional side to be the 1st summit in all additional sides of starting point; Embed and finish, described additional side is by any two directed edges that non-conterminous summit forms; Otherwise,

Step 1.2.2: article one directed edge that is positioned at the probe right side with from the n-1 summit is limit, the 1st right, is the 1st first-in-chain(FIC) with the downstream side on limit, the 1st right, is the 1st last-of-chain with the upstream edge of probe; With i as follow-up counter; I=1, if article one be positioned at limit, the 1st right the left side the additional side with the probe overlapping, then the downstream side with this additional side is the conllinear limit; And delete overlapping additional side, execution in step 1.2.6; Otherwise,

Step 1.2.3: respectively as the beginning and the ending of i segment, described segment is the directed edge of a series of tandem arrays with i first-in-chain(FIC), i last-of-chain; If the i segment is monolateral shape, protruding chain or recessed chain segment, then the i segment is carried out the segment pinching and confirm the alignment probe limit, and hit the limit as i, then execution in step 1.2.5 with the alignment edge that is obtained; As if the i segment is left spill, then execution in step 1.2.4; Otherwise the i segment is right spill, if probe is crossing with the upstream edge of i last-of-chain, then hits the limit with said upstream edge as i; Execution in step 1.2.5; Otherwise,, then hit the limit as i with alignment edge if there is alignment edge to being that the upstream edge of first-in-chain(FIC), i last-of-chain is that the segment of last-of-chain carries out the segment pinching and confirms the alignment probe limit with the i first-in-chain(FIC); Otherwise the upstream edge with the i last-of-chain is that i hits the limit, then execution in step 1.2.5; The last-of-chain of above-mentioned monolateral shape segment is the downstream side of first-in-chain(FIC); Angle＜the π of all intermediate vertexs of above-mentioned protruding chain segment; The intermediate vertex of said segment is all summits between the starting point of the starting point of first-in-chain(FIC) and last-of-chain on the segment, and the angle on said summit is to be the angle in two adjacent directed edges left sides of linking with this summit on the segment; Angle >=the π of all intermediate vertexs of above-mentioned recessed chain segment; Above-mentioned left spill segment removes the angle＜π of the terminal point of first-in-chain(FIC), the angle >=π of all the other intermediate vertexs on the segment; Above-mentioned right spill segment is except that the angle＜π of the starting point of the upstream edge of last-of-chain, the angle >=π of all the other intermediate vertexs on the segment;

Step 1.2.4: if probe intersects with the i first-in-chain(FIC), be that i hits the limit then with the i first-in-chain(FIC), execution in step 1.2.5; Otherwise, be that first-in-chain(FIC), i last-of-chain are that the segment of last-of-chain carries out the segment pinching and confirms the alignment probe limit to downstream side with the i first-in-chain(FIC), if there is alignment edge, then hits the limit as i, otherwise hit the limit as i with the i first-in-chain(FIC) with alignment edge,

Step 1.2.5:, then hit the limit as the conllinear limit, execution in step 1.2.6 with i if i hits the starting point and probe conllinear on limit; Otherwise; Hit the limit as playing initial line, medial side and termination limit with limit, the i right, i first-in-chain(FIC) and i respectively; With the unbounded mode the three limits chain of rings is carried out on the right side of probe; The downstream side, i last-of-chain and the probe that hit the limit with i respectively again carry out the three limits chain of rings with the unbounded mode to the left side of probe as playing initial line, medial side and a termination limit; The new downstream side that hits the limit with the i behind the three limit chains of rings is the i+1 last-of-chain; If hitting the limit with i, probe intersects; The downstream side that then hits the reverse edge on limit with i is the i+1 first-in-chain(FIC), is limit, the i+1 right with the downstream side of the reverse edge of probe, deletes i and hits the limit; As new i, return to step 1.2.3 with i+1; Otherwise; To the reverse edge with probe is that to hit the limit be that the segment of last-of-chain carries out the segment correction for first-in-chain(FIC), i; With newly connect from the additional side on n summit as the right flank limit; Again to being that first-in-chain(FIC), right flank limit are that the segment of last-of-chain carries out the segment correction with the i+1 last-of-chain, with newly connect from the additional side of the starting point of i+1 last-of-chain as the left wing limit; Respectively with the upstream edge on downstream side, left wing limit and the right flank limit on right flank limit as playing initial line, medial side and a termination limit, thereby with the unbounded mode the place ahead of probe is carried out that three limits are a chain of guarantees that the place ahead of popping one's head in is a triangle, embed and finish; The method of above-mentioned segment correction is: if segment is protruding chain; Be that ring limit, last-of-chain are that the bilateral chain of rings is carried out on the stop ring limit then with first-in-chain(FIC); Otherwise if segment is left spill; Be new first-in-chain(FIC) to the downstream side with first-in-chain(FIC) then, last-of-chain is that new segment that new last-of-chain constituted carries out postponing from the recessed chain of upper type and makes it to become monolateral shape or recessed chain, otherwise if segment is right spill; Then to being new first-in-chain(FIC) with first-in-chain(FIC), the upstream edge of last-of-chain is that the new segment of new last-of-chain carries out postponing from the recessed chain of mode down and makes it to become monolateral shape or recessed chain; Finish the segment correction; The method of the above-mentioned bilateral chain of rings is: with the upstream edge on stop ring limit is the ring inner edge, has connected the starting point of encircling the limit and the starting point on stop ring limit, with formed additional side as sealing the limit; If play the ring limit is an additional side and a ring limit long enough; And two rings that work the both sides of encircling the limit then can have been deleted the ring limit playing bigger scrobicular ring of formation under the prerequisite that the terminal point that can form a bigger bulge loop under the situation of being deleted on the ring limit or play the ring limit is not the 1st summit; In like manner; If the ring inner edge also is additional side and ring inner edge long enough; And two rings of the both sides of ring inner edge then can be deleted the ring inner edge forming a bigger bulge loop under the ring inner edge situation of being deleted or not being to form a bigger scrobicular ring under the prerequisite on the 1st summit in the starting point of ring inner edge; Bilateral a chain of the end; Above-mentioned rise ring limit long enough be meant the length on ring limit be at least the ring limit terminal point to the starting point on stop ring limit distance half the or to work the length of encircling the limit be half of length of sealing the limit at least; Above-mentioned ring inner edge long enough is that the length of finger ring inner edge has been that the length half the or the ring inner edge of distance of starting point to the starting point of encircling inner edge on ring limit is length half that seals the limit at least at least; Above-mentioned bulge loop refers to the angle＜π on all summits in ring; Above-mentioned scrobicular ring refers in ring except the angle＜π on three adjacent summits only being arranged, the angle >=π on all the other all summits;

Step 1.2.6: respectively with limit, the i right, i first-in-chain(FIC) and conllinear limit as playing initial line, medial side and a termination limit; With the bounded mode the three limits chain of rings is carried out on the right side of probe; Again respectively with conllinear limit, i last-of-chain and probe as initial line, medial side and a termination limit; It is a chain of with the bounded mode three limits to be carried out in the left side of probe, if there is the new additional side of downstream direction on the conllinear summit, then with this additional side as follow-up conllinear limit; If the n summit overlaps with the starting point that i hits the limit, then n returns on the summit to the 1st summit and polygon is closed, and the position of coordinating probe and conllinear limit concerns the embedding end; Otherwise; To the downstream side with the probe reverse edge is that first-in-chain(FIC), conllinear limit are that the segment of last-of-chain obtains the limit, middle part and as limit in the right side; Reverse edge with probe has been that the limit is that the bilateral chain of rings is carried out on the stop ring limit in ring limit, the right side then; And connecting n summit and conllinear summit as continuing the limit, the reverse edge of above-mentioned directed edge is end points and limit that direction with it opposite identical with said directed edge; To being that first-in-chain(FIC), probe are that the segment of last-of-chain obtains the limit, middle part and as limit in the left side, be that ring limit, the limit of continuing are that the bilateral chain of rings is carried out on the stop ring limit with limit in the left side with the conllinear limit, the embedding end; The above-mentioned method that segment is obtained the limit, middle part is: if segment is left spill, then the upstream edge with last-of-chain is the limit, middle part, and with first-in-chain(FIC) be encircle the limit, the limit, middle part is that the bilateral chain of rings is carried out on the stop ring limit, obtains end; Otherwise if the downstream side of first-in-chain(FIC) is the upstream edge of last-of-chain, then the downstream side with first-in-chain(FIC) is the limit, middle part, obtains end; Otherwise the downstream side with first-in-chain(FIC) has been that ring limit, last-of-chain are that the bilateral chain of rings is carried out on the stop ring limit, serves as the limit, middle part with the bilateral a chain of new downstream side of first-in-chain(FIC) afterwards, obtains end;

The a chain of method in three above-mentioned limits is:

Step 1.2.5.1: to being that first-in-chain(FIC), medial side are that the segment of last-of-chain carries out the segment correction, playing the start position of initial line new additional side arranged if revise the back to play initial line, then with new additional side as new initial line, otherwise be new initial line with an initial line; Again to be first-in-chain(FIC) with the medial side, to stop the limit be that the segment of last-of-chain carries out the segment correction, if revise the additional side that the start position of back medial side has new downstream direction, then with said additional side as limit in new, otherwise be newly middle limit with the medial side; If an initial line is identical with medial side or medial side is identical with the termination limit or the angle >=π of new middle limit starting point then finishes; As if twice correction result all is recessed chain, then execution in step 1.2.5.2; Otherwise,, then finish if a chain of mode in three limits is the bounded mode; Otherwise being the unbounded mode, if twice correction result all is monolateral shape, has been that ring limit, termination limit are that the bilateral chain of rings is carried out on the stop ring limit with new initial line then, finishes; If a preceding correction result is monolateral shape, a back correction result is recessed chain, to be first-in-chain(FIC) with limit in new, to stop the limit be that the segment of last-of-chain carries out postponing from the recessed chain of upper type, finish; Otherwise a preceding correction result is recessed chain, and a back correction result is monolateral shape, to new initial line be first-in-chain(FIC), new in the limit be that the segment of last-of-chain carries out postponing from the recessed chain of mode down, finish;

Step 1.2.5.2: with limit in new is the 1st strange flanging, and the starting point of new initial line is a upper extreme point, and the starting point that stops the limit is a lower extreme point, and j is a cycle counter, and j=1 carries out the processing of alternately postponing of two recessed chains;

Step 1.2.5.3: if the three limit chains of rings are unbounded modes, and the starting point that j takes turns strange flanging is angle >=π that lower extreme point or j take turns strange flanging starting point, finishes; If the starting point that the three limit chains of rings are bounded mode and j takes turns the upstream edge of strange flanging is that upper extreme point then finishes; Otherwise to take turns strange flanging with j is that first-in-chain(FIC), termination limit are the segment of last-of-chain, postpones to carry out recessed chain from upper type, and takes turns even flanging with the additional side of new connection as j;

Step 1.2.5.4: if the three limit chains of rings are unbounded modes, and the starting point that j takes turns even flanging is angle >=π that upper extreme point or j take turns even flanging starting point, finishes; If the starting point that the three limit chains of rings are bounded mode and j takes turns the downstream side of even flanging is that lower extreme point then finishes; Otherwise to being that first-in-chain(FIC), j take turns the segment that even flanging is a last-of-chain with new initial line, postpone, and take turns strange flanging as j+1 with the downstream side of the additional side of new connection to descend mode to carry out recessed chain certainly; With the j of j+1 as a new round, execution in step 1.2.5.3; The method that above-mentioned recessed chain is postponed is: if from the postponing of upper type, then the upstream edge with first-in-chain(FIC) serves as to observe the limit, otherwise serves as to observe the limit with the downstream side of last-of-chain; Starting point to observe the limit is an observation point, carries out the visible range that observation point is confirmed in the segment pinching, and cuts edge as upper reaches class with the directed edge of the upper reaches reference position of visible range, and the downstream side of the directed edge of the downstream final position of visible range is cut edge as the downstream class; For from the postponing of upper type, be that the bilateral chain of rings is carried out on the stop ring limit to observe that the limit be the ring limit, the downstream class is cut edge, for descend postponing of mode certainly, then with the side cut of upper reaches class be encircle the limit, the observation limit is that the bilateral chain of rings is carried out on the stop ring limit; Behind the bilateral chain of rings, if the starting point of the upstream edge on stop ring limit is identical with the starting point that plays the ring limit, then the new segment that constitutes of the starting point from starting point to the stop ring limit of encircling the limit is monolateral shape, otherwise is recessed chain.

2. ring-cutting subdivision method of simple polygon in the computer graphical as claimed in claim 1 is characterized in that, the segment pinching confirms that the method on alignment probe limit is:

Step 2.1: the upstream edge that with the first-in-chain(FIC) is order the 1st limit, last-of-chain is backward the 1st limit, and k is a cycle counter, k=1,

Step 2.2: if the terminal point on order k limit is positioned at the probe left side starting point on order k limit be positioned at the probe right side or with the probe conllinear; Be alignment edge then with order k limit; Finish; Otherwise if the terminal point on backward k limit is positioned at the probe left side starting point on backward k limit be positioned at the probe right side or with the probe conllinear, be alignment edge then with backward k limit, finish; Otherwise the starting point on backward k limit is positioned at the probe right side if the terminal point on backward k limit is with the probe conllinear, and then the downstream side with backward k limit is an alignment edge; Otherwise if order k limit is identical with backward k limit or order k limit is the upstream edge on backward k limit, then seeking all over whole sequence also can not have alignment edge, finishes; Otherwise if segment is recessed chain, and the starting point on the order k limit terminal point that is positioned at left side or the backward k limit of probe is positioned at the right side of probe, then alignment edge can not be arranged, and finishes; Otherwise with the downstream side on order k limit serves as that the upstream edge on order k+1 limit, backward k limit is backward k+1 limit, with k+1 as new k, execution in step 2.2.

3. ring-cutting subdivision method of simple polygon in the computer graphical as claimed in claim 1 is characterized in that, the segment pinching confirms that the method for the visible range of observation point is:

Step 3.1: the upstream edge that with the first-in-chain(FIC) is order the 1st limit, last-of-chain is backward the 1st limit; If observation point is in left side, order the 1st limit, the directed edge of the upper reaches reference position of visible range is order the 1st limit; If observation point is in left side, backward the 1st limit, the directed edge of the downstream final position of visible range is backward the 1st limit; If the upper and lower position of visible range is all definite, then finish; Otherwise with k is cycle counter, k=1,

Step 3.2: with the downstream side on order k limit serves as that the upstream edge on order k+1 limit, backward k limit is backward k+1 limit, with k+1 as follow-up k; Not in left side, order k limit, then the directed edge of downstream final position is order k-1 limit, finishes if observation point is positioned at left side, order the 1st limit; Otherwise if observation point is positioned at left side, order the 1st limit and left side, backward k limit simultaneously, then the directed edge of downstream final position is backward k limit, finishes; Otherwise if observation point is positioned at left side, order k limit and left side, backward the 1st limit simultaneously, then the directed edge of upper reaches reference position is order k limit, finishes; Not in left side, backward k limit, then the directed edge of upper reaches reference position is backward k-1 limit, finishes otherwise if observation point is positioned at left side, backward the 1st limit; Otherwise execution in step 3.2.

Images