CN103903304A - Axle wire generating method of random quadrangular curved surface for product three-dimensional data processing - Google Patents

Abstract

The invention discloses a method for generating the central axis of an arbitrary quadrilateral curved surface for three-dimensional data processing of a product, and proposes a simple and fast central axis generating algorithm by using a trajectory method combined with a moving Frenet frame. The B-rep three-dimensional solid model is widely used in engineering problems. This model has a clear boundary definition. The information of the vertices, edges and faces of the model is clearly stored in the model file. Using these clear information can simplify The traditional method of generating the central axis by the trajectory method reduces the amount of data required for the generation of the central axis. The invention uses the definite boundary information in the B-rep model as the condition of the central axis generation method, and utilizes the characteristics of the bisector to eliminate the calculation of the branch points and reduce the factors affecting the accuracy of the central axis. In order to ensure the accuracy of the central axis, the density of the insertion point can be used for control.

Description

For the arbitrary quadrilateral curved surface axis generation method of product three-dimensional data processing

Technical field

The present invention relates to a kind of arbitrary quadrilateral curved surface axis generation method for the processing of product three-dimensional data, be particularly useful for the CAD/CAE technology such as model simplification, model reconstruction, finite element grid generation, solid modelling.Relating to Patent classificating number G06 calculates; Calculate; The 3D modeling of G06T17/00 for computerized mapping processed or produced to the view data that counting G06T is general.

Background technology

Model skeleton or center line are also in axis, conventionally be described as follows: in two-dimensional closed region, there is the disk of a radius variable at this intra-zone, keep this disk and zone boundary to have two point of contacts at least, then mobile this disk, after path that disk is walked covers All Ranges, the line that the track that the center of circle of disk stays forms is called axis, is called as middle axial plane in three-dimensional model.Axis is in each key link extensive application of CAD/CAE system, such as model reconstruction, model analysis, computer vision, solid modelling and in feature extraction of geometry designs etc.But axis is also restricted in some applications, because do not have at present fast a kind of and calculate accurately the algorithm of axis, particularly under curved boundary conditions, generate axis.

Blum proposed the concept of axis for the first time in 1967, the advantageous characteristic such as that axis has is unique, reversible, symmetrical, homeomorphic.It is very large that axis is affected by the continuity on border, and be also difficult at that time extract curved boundary accurately.Before 10 years, main discovery is the introducing of bisector, and it is a special case of axis, but the method needs cut operator, has increased the complexity of processing procedure.In recent years, Teixeira and Zucker have studied the relation between bent limit and axis curvature of a curve, tangent line and normal line vector three, have provided the differential equation of the curve axis motion of being described by mean curvature.Although there is the algorithm much generating about axis, generating stable and accurate axis is not also a pipe course.

The axis generating algorithm of commonly using in CAD/CAE system at present is roughly divided into two classes: method of loci and Voronoi figure method.Method of loci is the definition according to axis, allows the circle of a radius variable remain tangent with border, at least two, this region, and the left track of this round heart is exactly axis.But this algorithm need to judge whether distance and this point on each each border of some arrival are take-off point, have two positions with incision superius.This has increased data volume virtually, will inevitably affect the efficiency of calculating.Voronoi figure method be first by the border of given area point set by its discretize, generate Voronoi figure according to this discrete point set, what the summit of Voronoi figure formed is exactly axis.But this algorithm is more suitable for two dimensional model, need to increases extra alternative condition and just can be extended to three-dimensional model.

Summary of the invention

The present invention is directed to the proposition of above problem, and a kind of arbitrary quadrilateral curved surface axis generation method for the processing of product three-dimensional data of development has following steps:

-model decomposition step: quadrilateral surface model is decomposed into four limits; Non-conterminous both sides in four described limits are divided into one group, and two groups of limits that obtain are respectively as playing initial line and stopping limit and two binding sides;

-interpolation procedure: obtain the merging vector of two binding sides by calculating the vector of two described binding sides, using the direction of this merging vector as interpolation direction, insert multiple insertion points by step-length δ;

Step is moved in-insertion point: in the time that described multiple insertion points are all positioned at model inside, calculate the distance of each insertion point and described two binding sides; Use the mode of iteration, move described insertion point, make the distance approximately equal of this insertion point and described two binding sides;

-axis determining step: repeat above-mentioned steps; In the time of the equal approximately equal of distance of described insertion point and two binding sides, utilize described each insertion point structure B-spline curve, this curve approximation is the axis of quadrilateral surface model.

Step-length δ of the present invention is two distances between insertion point, utilizes step-length δ can control the quantity of insertion point.Step-length δ is less, and insertion point is more, and the axis precision generating is higher; Insertion point is fewer, and the calculated amount that generates axis is less.

Further, consider due to the difference of having chosen initial line and termination limit, may cause the volume coordinate of insertion point to be positioned at outside quadrilateral model.Therefore insertion point of the present invention is moved and is at least also had insertion point in step and judge and mobile step:

The distance d of-calculating insertion point P and two binding sides ₁and d ₂, and and d ₁and d ₂intersection point q with two binding sides ₁and q ₂, q ₁and q ₂between distance be d, d=||q ₁q ₂||; Obtain the direction vector of insertion point P according to Frenet frame formula ${\stackrel{\→}{e}}_1=q_1{q}_2/\left|\right|q_1{q}_2{\left|\right|}_2;$

-relatively d ₁, d ₂relation with d: if d ₁>d or d ₂>d, insertion point P is in model area outside;

If-d ₁>d ₂, put P along

direction displacement Δ=| (d ₁-d ₂) |/2; If d ₁<d ₂, put P along reverse direction displacement Δ=| (d ₁-d ₂) |/2; Insertion point P is moved to behind model inside, upgrade this point coordinate;

If-d ₁<d and d ₂<d, shows that a P, in model inside, does not temporarily adjust.

By above-mentioned decision process, can ensure that each insertion point is positioned at the inside of quadrilateral model, has ensured the carrying out of algorithm.

Preferably, described " use alternative manner, move described insertion point, make the distance approximately equal of this insertion point and described two binding sides " specifically comprises following steps:

The distance d of-calculating insertion point P and two binding sides ₁and d ₂, determining that insertion point is in the time that quadrilateral model is inner: obtain according to Frenet frame formula the direction vector that P position is adjusted

If-d ₁>d ₂, some P along

direction displacement Δ ₁;

If-d ₁<d ₂, some P along

reverse direction displacement Δ ₁;

-repeat above-mentioned steps, until d ₁/ d ₂=ε, stops computing; Wherein 0.99< ε <1.01; Displacement Δ ₁value be | (d ₁-d ₂) |/2 ⁿ, wherein n value is twice inconsistent number of times of continuous moving direction.

In described interpolation procedure, described " obtaining the merging vector of two binding sides by calculating the vector of two described binding sides " is specially:

Set A ₁and A ₂for two described binding sides; Press initial line to the direction that stops limit, obtained two binding side A ₁and A ₂vector

utilize vector merging formula (1) to calculate the interpolation direction vector of insertion point

$\stackrel{\&RightArrow;}{e}={\stackrel{\&RightArrow;}{e}}_{a1}+{\stackrel{\&RightArrow;}{e}}_{a2}---\left(1\right)$

Described " inserting multiple insertion points by step-length δ " is specially:

On an initial line, get mid point P ₁as starting point, according to the direction vector of interpolation

with step-length δ, utilize formula (2) to calculate the coordinate P of next insertion point ₂(x, y)

$\left\{\begin{array}{c}{P}_2.x=P_1.x+\stackrel{\&RightArrow;}{e}.x*\mathrm{\&delta;}\\ P_2.y=P_1.y+\stackrel{\&RightArrow;}{e}.y*\mathrm{\&delta;}\end{array}\right.---\left(2\right)$

If P ₂be less than step-length δ to the distance d that stops limit, represent that new insertion point approaches and stops limit, no longer needs new insertion point, so P ₂it is exactly last terminating point; Otherwise, continue to insert new insertion point P*.

Owing to having adopted technique scheme, the arbitrary quadrilateral curved surface axis generation method for the processing of product three-dimensional data provided by the invention, adopts method of loci in conjunction with mobile Frenet frame, has proposed a kind of simple and quick axis generating algorithm.In engineering problem, applying more is B-rep three-dimensional entity model, this model has clear and definite boundary definition, the information of the top points, edges, faces of model is clear and definite being stored in model file all, utilize these clear and definite information just can simplify traditional method of loci axis generation method, generate needed data volume thereby reduced axis.The present invention utilizes the condition of boundary information clear and definite in B-rep model as axis generation method, and utilizes the feature of bisector to eliminate the calculating to take-off point, reduces the factor that affects axis degree of accuracy.In order to ensure the degree of accuracy of axis, can adopt the density of insertion point to control.

Brief description of the drawings

For the technical scheme of clearer explanation embodiments of the invention or prior art, introduce simply the accompanying drawing of required use in embodiment or description of the Prior Art being done to one below, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skill in the art, do not paying under the prerequisite of creative work, can also obtain according to these accompanying drawings other accompanying drawing.

Fig. 1 is axis generating algorithm process flow diagram;

Fig. 2 is the each limit grouping of quadrilateral schematic diagram;

Fig. 3 is insertion point algorithm schematic diagram;

Fig. 4 is for adjusting insertion point to model schematic internal view;

Fig. 5 is for adjusting position, insertion point schematic diagram on axis;

Fig. 6 is that four irregular curved tetragonal axis, limit generate design sketch, and Fig. 6 a is original quadrilateral model; Fig. 6 b is the axis schematic diagram obtaining by algorithm of the present invention;

Fig. 7 is Fillet Feature face decomposing schematic representation;

Fig. 8 is that mechanical component model fillet is simplified the rear model of design sketch a master pattern b Fillet Feature simplification;

Fig. 9 utilizes axis parted pattern generating mesh a initial model b model four c limit, limit ab, cd to intersect for the axis that constraint condition generates axis e both direction for constraint condition generates d limit, axis bc, da, forms grid.

Embodiment

For making object, technical scheme and the advantage of embodiments of the invention clearer, below in conjunction with the accompanying drawing in the embodiment of the present invention, the technical scheme in the embodiment of the present invention is known to complete description:

As Figure 1-Figure 5: a kind of arbitrary quadrilateral curved surface axis generation method for the processing of product three-dimensional data, mainly comprises following step:

In model decomposition step, divide into groups shown in schematic diagram 2 in the each limit of quadrilateral.Every limit of quadrilateral is divided into two groups, A by " limit and limit are non-conterminous " principle ₁and A ₂be one group as binding side, limit B ₁and B ₂for another group, respectively as playing initial line and stopping limit.Select the main consideration on initial line peace treaty bundle limit to want the trend of the axis of selecting.

The described interpolation procedure of content of the present invention is as shown in Figure 3: insertion point algorithm schematic diagram.Set B ₁for playing initial line, B ₂for stopping limit, A ₁and A ₂for binding side.In the time specifically having selected initial line and stopped limit, the main trend of considering to want the target axis obtaining, as binding side, remaining two limits can be used as initial limit and stop limit with two limits of target axis approximate parallel (moving towards consistent).

Press initial line to the direction that stops limit, obtained the vector of two binding sides

utilizing vector to merge formula (1) calculates

be the interpolation direction vector of insertion point.

$\stackrel{\&RightArrow;}{e}={\stackrel{\&RightArrow;}{e}}_{a1}+{\stackrel{\&RightArrow;}{e}}_{a2}---\left(1\right)$

On an initial line, get mid point P ₁as starting point, according to the direction vector of interpolation

with step-length δ, utilize formula (2) to calculate the coordinate P of next insertion point ₂(x, y).

$\left\{\begin{array}{c}{P}_2.x=P_1.x+\stackrel{\&RightArrow;}{e}.x*\mathrm{\&delta;}\\ P_2.y=P_1.y+\stackrel{\&RightArrow;}{e}.y*\mathrm{\&delta;}\end{array}\right.---\left(2\right)$

If P ₂be less than step-length δ to the distance d that stops limit, represent that new insertion point approaches and stops limit, no longer needs new insertion point, so P ₂it is exactly last terminating point.Otherwise, continue to insert new insertion point P ^*.

In the present embodiment step-length δ be in quadrilateral model minor face 1/5th.

Further, consider due to the difference of having chosen initial line and termination limit, may cause the volume coordinate of insertion point to be positioned at outside quadrilateral model.Therefore insertion point of the present invention move in step, at least also have that insertion point is judged and mobile step as shown in Figure 4.Adjust insertion point to model schematic internal view.To acquired insertion point P, calculate the bee-line d of two binding sides of this some arrival ₁and d ₂, and the some q corresponding with bee-line ₁and q ₂(be positioned at binding side A ₁and A ₂on point), calculate direction vector according to Frenet frame formula.

${\stackrel{\&RightArrow;}{e}}_1=q_1{q}_2/\left|\right|q_1{q}_2{\left|\right|}_2;$

Judge that whether the position of insertion point P is in model area inside, if not in model inside, will adjust to model inside, judgement and method of adjustment are as follows:

According to a P to binding side A ₁and A ₂the corresponding point q of bee-line ₁and q ₂, obtain apart from d=||q ₁q ₂||.Comparison point P is to the bee-line d of binding side ₁, d ₂and the relation between d.

If d ₁>d or d ₂>d, puts P in model area outside.If d ₁>d ₂, put P along

direction move a certain distance Δ=| (d ₁-d ₂) |/2; If d ₁<d ₂, put P along

reverse direction move a certain distance Δ=| (d ₁-d ₂) |/2.Insertion point P is moved to behind model inside, upgrade this point coordinate;

If d ₁<d and d ₂<d, shows that a P is in model inside, does not temporarily need to adjust.

As Fig. 5: adjust position, each insertion point and make it drop on schematic diagram on axis.To each insertion point, adjust by following rule:

(a) if d ₁>d ₂, some P along

direction moves a certain distance Δ ₁;

(b) if d ₁<d ₂, some P along

reverse direction moves a certain distance Δ ₁;

(c) if d ₁=d ₂(approximately equal), upgrades some P coordinate figure.

Displacement Δ in the present invention ₁initial value is made as | (d ₁-d ₂) |/2.In the time that twice continuous moving direction is inconsistent, secondary displacement Δ ₁be reduced into | (d ₁-d ₂) |/4, adopt iterative algorithm repeatedly to adjust position a little until two constraint back gauges of some P arrival are equal.Displacement Δ ₁value be | (d ₁-d ₂) |/2 ⁿ, wherein n value is twice inconsistent number of times of continuous moving direction.Adopt alternative manner repeatedly to adjust position a little, the point inserting if make arrives binding side distance and equates there is certain difficulty completely, so approximately equal, i.e. d ₁/ d ₂=ε, ε approaches 1.For high-precision axis, it is 1 better that ε more approaches, but operation efficiency can reduce.

In the present invention, each model instance ε value is 0.99< ε <1.01.

As Fig. 6: four irregular curved tetragonal axis, limit generate design sketchs.In commercial production cad model, be regular compared with multi-model, but also have some irregular models.Adopt the high irregular boundary model of complexity to test axis generating algorithm.As shown in Figure 6 a, adopt the quadrilateral of the four irregular bendings in limit.Import after this model, application axis generating algorithm, effect is as shown in Figure 6 b.Although bilateral bending makes the direction of insertion point not be equal to the direction vector of binding side, through the adjustment to insertion point, insertion point can be dropped on axis.Recycling has obtained the some structure B-spline curve on axis, and this curve is the axis of bending quadrilateral curved surface, has met the symmetry of axis.

Utilizing axis to realize Fillet Feature simplifies.Patent " the Fillet Feature short-cut method based on fillet axis " (201310578463.1) discloses utilizes axis to cut apart Fillet Feature curved surface, the divided curved surface of reconstruct and corresponding seating surface, thus realize the method that Fillet Feature is simplified.Most of Fillet Feature are made up of four edges, two smooth limits and two common limits.With smooth limit, as constraint condition, common limit, as playing initial line and stopping limit, meets the axis generating algorithm that this patent proposes, just therefore can generate axis on this Fillet Feature curved surface.As shown in Figure 7, with smooth limit e ₄and e ₇for binding side, common limit e ₅and e ₆for playing initial line and stopping limit, generate axis e ₈fillet Feature curved surface is divided into two face F3 and F4; Two end points of axis are by limit e simultaneously ₅be divided into two sections of a ₁and a ₂, limit e ₆be divided into b ₁and b ₂two sections.Fig. 8 is mechanical component, is made up of 20 faces, wherein comprises multiple Fillet Feature, utilizes axis to cut apart reconstruct fillet surface, simplifies Fillet Feature, and the face number of model reduces to 11, and the model after simplification as shown in Figure 8 b.

Utilizing axis parted pattern to realize grid generates.In axis generation method, the constraint condition border having been generated as axis.Utilize axis parted pattern, the border of each subregion is also subject to the constraint on border, makes all subregion boundary similarity.The Jiang Gai region, axis generating in given area is divided into reciprocity two parts, if these two regions are continued to divide, just can obtain some approximately equalised subregions.Exchange the relation that binding side has been followed initial line, stopped limit, again carry out same division, obtained a series of sequential cells like this are the grid of generation.

Given closed region as shown in Fig. 9 a, is divided into quadrilateral: limit ab, bc, cd, da composition.Generate axis ef using limit ab, cd as constraint condition, by this Region Decomposition, form two regions, as shown in Fig. 9 b.Then judge decision condition: in each region, minor face is less than or equal to initial mesh border setting value.Dissatisfied decision condition is divided again, and as shown in Fig. 9 c, respectively using limit ae, fd and limit eb, cf as two axis of constraint condition regeneration, former region is divided into four sub regions, now meets decision condition and stops dividing.Exchange the relation that master mould binding side has been followed initial line, stopped limit, again carry out same division using limit bc, da as constraint condition, generate as three of Fig. 9 d axis.Twice different directions divided six axis that generate and intersected, and crossed node is grid node, and a series of continuous quadrilateral being formed by connecting by grid node is grid cell, as shown in Fig. 9 e.

The above; it is only preferably embodiment of the present invention; but protection scope of the present invention is not limited to this; any be familiar with those skilled in the art the present invention disclose technical scope in; be equal to replacement or changed according to technical scheme of the present invention and inventive concept thereof, within all should being encompassed in protection scope of the present invention.

Claims (5)

1. A method for generating the central axis of an arbitrary quadrilateral surface for three-dimensional data processing of a product, comprising the following steps: —Model decomposition step: decompose the quadrilateral surface model into four sides; divide the non-adjacent two sides in the four sides into one group, and obtain the two groups of sides as the starting side, the termination side and the two constraint sides respectively; - Interpolation step: obtain the merging vector of the two constrained sides by calculating the vectors of the two constrained sides, take the direction of the merging vector as the interpolation direction, and insert a plurality of insertion points according to the step size δ; - Insertion point moving step: when the plurality of insertion points are located inside the model, calculate the distance between each insertion point and the two constraint sides; use an iterative method to move the insertion point so that the insertion point and the two constraint sides The distances between the two constraining sides are approximately equal; - Central axis determination step: repeat the above steps; when the distances between the insertion points and the two constraining sides are approximately equal, use each of the insertion points to construct a B-spline curve, which is approximately the central axis of the quadrilateral surface model . 2. The method for generating the central axis of any quadrilateral curved surface for product three-dimensional data processing according to claim 1, further characterized in that the insertion point moving step also has at least an insertion point judgment and a moving step: - Calculate the distance d ₁ and d ₂ between the insertion point P and the two constrained sides, and the intersection points q ₁ and q ₂ with d ₁ and d ₂ and the two constrained sides, the distance between q ₁ and q ₂ is d, and d= ||q ₁ q ₂ ||; According to the Frenet frame formula, the direction vector of P position adjustment is obtained ${\stackrel{\&Right\; Arrow;}{e}}_1=q_1{q}_2/\left|\right|q_1{q}_2{\left|\right|}_2;$ —Comparing the relationship between d ₁ , d ₂ and d: if d ₁ >d or d ₂ >d, the insertion point P is outside the model area; — If d ₁ >d ₂ , the point P is along

Direction moving distance Δ=|(d ₁ -d ₂ )|/2; if d ₁ <d ₂ , then point P along

The moving distance in the opposite direction of Δ=|(d ₁ -d ₂ )|/2; after moving the insertion point P into the model, update the coordinates of the point; —If d ₁ <d and d ₂ <d, it indicates that the point P is inside the model and no adjustment will be made temporarily. 3. The method for generating the central axis of any quadrilateral surface for processing three-dimensional product data according to claim 1 or 2, further characterized in that said "use an iterative method to move the insertion point so that the insertion point is consistent with the The distance between the above two constraint sides is approximately equal" specifically includes the following steps: —Calculate the distances d ₁ and d ₂ between the insertion point P and the two constraining sides, when it is determined that the insertion point is inside the quadrilateral model: get the direction vector for adjusting the position of P according to the Frenet frame formula

—If d ₁ >d ₂ , point P along

direction movement distance Δ ₁ ; —If d ₁ <d ₂ , point P along

Move distance Δ ₁ in the opposite direction; —Repeat the above steps until d ₁ /d ₂ =ε, stop the operation; where 0.99<ε<1.01; the value of the moving distance Δ ₁ is |(d ₁ -d ₂ )|/2 ⁿ , where n is The number of times that two consecutive movement directions did not agree. 4. The method for generating the central axis of any quadrilateral surface for three-dimensional data processing of products according to claim 1, further characterized in that in the interpolation step, the "obtained by calculating the vectors of the two constrained sides" The merging vector of two constrained sides" is specifically: Set A ₁ and A ₂ as the two constrained sides; get the vectors of the two constrained sides A ₁ and A ₂ according to the direction from the start side to the end side

Use the vector combination formula (1) to calculate the interpolation direction vector of the insertion point $\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; =</span>{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}}_{\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ 5. The method for generating the central axis of any quadrilateral curved surface for product three-dimensional data processing according to claim 4, further characterized in that said "insert a plurality of insertion points according to the step size δ" is specifically: Take the midpoint P ₁ on the starting edge as the starting point, according to the interpolated direction vector and step size δ, use formula (2) to calculate the coordinates P ₂ (x, y) of the next insertion point $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; +</span>\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; +</span>\stackrel{<span\; class="notranslate">\; \&Right\; Arrow;</span>}{\mathrm{<span\; class="notranslate">\; e</span>}}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; \&delta;</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$ If the distance d from P ₂ to the termination edge is less than the step length δ, it means that the new insertion point is close to the termination edge and no new insertion point is needed, so P ₂ is the final termination point; otherwise, continue to insert a new insertion point P* .

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