CN1205546C - Automatic design simulating system for plane figure of highway in mountain area - Google Patents

Abstract

The present invention relates to a linear automatic design simulation system for highway planes in mountain areas, which comprises the following functional modules: a contour drafting module, a deterministic point value module, a curve fitting curvature map transformation module, a curve fitting curvature map drawing module and a linear element design algorithm module. With the establishment of the plane linear design simulation system in the present invention, repeated labor of manual determination of slope points on a large-scale topographical map is eliminated, and manual point fixing is converted to computer automatic identification. Through the linear dynamic adjustment by definite edge restraint conditions, the linear adjustment is visual and simple; manual operation is reduced as much as possible, no leading wire is needed to be arranged, and linear arrangement and laying are fully automatized; thus, the plane linear design is fast and simple, lots of manpower and material resources are saved, and map drawing period is shortened.

Description

Plane figure of highway in mountain area automatic simulation method for designing

Technical field

The present invention relates to Correction in Road Alignment Design and optimization method, be specifically related to plane figure of highway in mountain area automatic simulation method for designing, belong to the highway layout technical field.

Background technology

In China's plane figure of highway in mountain area design, always be [1] at the enterprising line operate of large scale topographical map at present, the method for employing is the linear pattern method for designing (claiming traverse method again) that tradition is put sloping alignment.At first determine the integral layout of route, then according to average gradient i _{On average}(5.0% ~ 5.5%, decide on relative relief) makes grade point (or claiming data point) successively on level line, adjust some unsuitable backs and connect these grade points formation leading lines.Leading line remains the bar broken line, so requirement according to the horizontal alignment standard, in conjunction with the horizontal wall inscription situation of change, determine the point that must pass through, the point and the negligible point of appropriate preferential treatment, take with a line, in the mode of line intersection point, make the plane and revise leading line, and measure corner, on figure, apply out curve.Require to do the adjustment of line position according to horizontal alignment at last and make center line again.Paper location of line is the process of a repeated examinations.Examination line that must be by repeatedly could obtain desirable suitable linear.

Traditional horizontal alignment method for designing has following limitation:

1. grade point all be by the deviser with bisecting compasses successively on each bar level line section go to determine.Put the slope when having arrived terminal point, do not reach the terminal point elevation if put the elevation that the slope obtains, redefine grade point according to the high extent of residue again, this need of work repeatedly just can be finished.Simultaneously,, dodge arduous engineering or serious unfavorable geology location, need to redefine average gradient again, adjust grade point, carry out so repeatedly in order to make full use of favorable terrain.These work repeat, loaded down with trivial details, fallibility.

2. revising leading line, take with a line, when making plane " examination line " in the mode of line intersection point, which point must pass through, which point is to want appropriate preferential treatment, which point is again can be irrespective, and definite standard can reference for this, relies on experience and self abundant space imagining ability of deviser's accumulation at ordinary times fully, and often have certain blindness and randomness, also can't determine suitable linear through repeatedly trying line sometimes.

3. after lead is drawn, begin to lay plane curve.Owing to be subjected to the constraint of landform and geologic condition, obtain more rational circular curve radius, length of transition curve, make that the route engineering amount that designs is few, combine well with landform, do not destroy linear whole ride comfort again, this is that the deviser must consider and is comparison stubborn problem [2].

4. because the conventional wires method is as main linear with straight line, curve (circular curve, adjustment curve) is just as auxiliary linear, therefore the linear alpine terrain environment coordination of designing by the conventional wires method that generally is difficult to and wriggles fluctuating, cause digging greatly greatly filling out, destroy natural landscape, caused the visual effect of highway relatively poor.

5. the conventional wires method is difficult to handle geometry linear complicated and changeable.

And present shaped form method for designing [3], [4], only be the approach that removes to seek to solve the horizontal alignment design problem from some aspects, do not realize the full-automation design of mountain area horizontal alignment from the angle of systems such as landform, linear, method for designing, computer program and emulation technology.Linear unit and the automatic description of constraint condition and the problem of finding the solution of unresolved linear laying process of while.Thereby practicality is relatively poor, often is difficult to apply.

Summary of the invention

The objective of the invention is to overcome the deficiency of above-mentioned prior art, a kind of plane figure of highway in mountain area automatic simulation method for designing that classic method exists the artificial duplication of labour, fallibility, defective such as loaded down with trivial details that solves is provided.

Realize above-mentioned purpose, to the effect that of the present invention: a kind of plane figure of highway in mountain area automatic simulation method for designing mainly comprises the steps:

(1) set up digital terrain model, draw out hypsographic map on this basis, its concrete steps are: a) terrain data collection; B) terrain data pre-service; C) terrain data ordering and retrieval; D) the elevation interpolation of digital terrain model adopts rectangle graticule mesh interpolation method; E) fracture line are handled; (2) adopt the method for rectangle graticule mesh cubic polynomial elevation interpolation to obtain grade point; (3) looking these grade points is data point, utilizes the local coordinate cubic spline curve to carry out match, and the plane fitting that generates route is linear; (4), generate the linear curvature chart of match to the linear curvature of calculating at a certain distance of the match of route; (5) the linear curvature chart of match is discerned conversion module automatically, and transform the linear curvature chart of generation conventional planar with least square method; (6) utilize Tinkertoy approach to lay horizontal alignment, calculate the line position of going forward side by side.

Described terrain data ordering comprises following method: sort all known numbers (1) to the x direction; (2) according to the density of raw data, determine graticule mesh width dc automatically, obtain total columns of x direction, thereby can determine first and every row x direction true origin; (3) from x _{0 (i)}＜x＜x _{0 (i)}In+dc the zone, determine the known topographic(al) point of every row; (4) to the known terrain data of these row, the y direction is arranged; (5) determine this row graticule mesh number according to this distribution range that is listed as known terrain data, and the local coordinate initial point y that should be listed as first lattice _{0 (i)}(6) from y _{0 (i)}+ jdc＜x＜y _{0 (i)}+ (j+1) in the dc zone, this column data is write down the known point first point sequence number in each graticule mesh and always counts;

Repeat (3) ~ (6), all known points in digital-to-analogue zone back that disposes is formed the digital-to-analogue data, on this basis, as long as provide the planimetric coordinates of any point to be located, just can retrieve this fast and put graticule mesh of living in, thus known topographic(al) point in this graticule mesh of rapid extraction.

Described least square method transforms and generates the linear curvature chart design of conventional planar, and further method is: (1) determines that matched curve passes through the topography and geomorphology parameter γ in area; (2) determine the parameters for shape characteristic λ of curvature chart according to the sample curvature chart, go to judge turning to of route according to continuous 0 or 1 number among the vectorial λ, thus the sign of the curvature chart mean curvature k that settles the standard; (3) structure standard curvature chart: at first according to the combinations of values of parameter γ, as rough sledding appears, then increase and consider convex, C type, compound these three kinds special linear array modes, otherwise only consider these three kinds of linear combination modes commonly used of basic model, S type and ovum type; According to the mileage length of known road quality classification and sample curvature chart, determine l then _s, R _AveAnd l _sAnd l _yOptimum ratio, construct the standard curvature chart; (4) to the vectorial value of the curvature K vector sum mileage L of standard curvature chart and sample curvature chart, and curvature K vector is carried out " normalization " handle, calculate the C value of each curvature chart; (5) the C value is relatively reached a conclusion.

Described Tinkertoy approach is that the plane figure of highway that combination is complicated " breaks the whole up into parts ", resolve into several linear unit, if origin information such as coordinate, tangent line or normal direction and the radius-of-curvature of known route plane curve, then begin to be provided with arbitrary unit from starting point, extend along any direction, the information of this element terminal point such as coordinate, tangent line or normal line direction angle and radius-of-curvature can be calculated, and simultaneously its identical information as the next unit starting point are used; The unit down calculates so one by one, and as playing with building blocks, each unit connects from beginning to end, constitutes the plane figure of highway of a continuous whole; Wherein linear unit is that line element can be respectively straight line, adjustment curve and circular curve, also can be the linear unit that these three kinds of line elements combine.

Can make various functional modules as stated above and form system, adopt the modularized program method for designing that following advantage is arranged:

1. the program structure that adopts modular approach to obtain is clear, well arranged;

2. adopt modular approach can improve procedure development efficient.Each module can compile separately, and program module is by can be shared by many programs after well-designed;

3. adopt modular approach can improve the feasibility of program, and make program be convenient to safeguard and revise;

4. adopt modular approach, each module can test or verify independently, baroque large program of this comparison test or verify to be easy to many.

Major advantage of the present invention also is: by setting up the analogue system of horizontal alignment design, eliminate the artificial duplication of labour of determining grade point on large scale topographical map, make artificial fixed point transfer Computer Automatic Recognition to; Utilize certain edge-restraint condition dynamically to adjust linear, it is directly perceived, easy that linear adjustment is accomplished; Reduce manually-operated as much as possible, no longer need to be provided with leading line, linear layout with lay full automation, make horizontal alignment design quick, simple and direct, save great amount of manpower and material resources, shorten drafting period.According to said method the automatic simulation system of the plane figure of highway in mountain area design of design frees the deviser from the work that repeats, and more energy is put in the optimization work of horizontal alignment and is removed raising designing quality and design level.

Description of drawings

Fig. 1 is a contour mapping process flow diagram of the present invention.

Fig. 2 is a program flow chart of determining data point in the analogue system of the present invention.

Fig. 3 is the local coordinate curve map.

Fig. 4 is matched curve curvature chart conversion process figure.

Fig. 5 is the ultimate principle figure of Tinkertoy approach.

Fig. 6 is a line element algorithm for design process flow diagram.

Fig. 7 is that process flow diagram is calculated at the position angle at Point of Inflection place.

Fig. 8 is the highway alignment of grade point curve fitting.

Fig. 9 is the curvature chart of matched curve.

Figure 10 is the curvature chart after the match.

Figure 11 is traditional linear after transforming.

Embodiment

The invention will be further described below in conjunction with accompanying drawing and concrete embodiment.Adopt the inventive method:

One, the foundation of digital terrain model

Topographic(al) data is the important foundation data of highway design, in traditional design, generally represents landform with topomap or sectional drawing.Utilize computing machine to carry out the highway design, will allow and calculate function understanding and handle topographic(al) data, for this reason, must become receptible information one numeral of computing machine to topographic(al) data, digital terrain model is introduced into the highway design field under this background.

Digital terrain model (Digital Terrain Model is called for short digital-to-analogue, DTM) is meant the numerical value describing mode of expressing terrain feature according to certain mathematical model.It is made up of topographic(al) point three-dimensional coordinate x, y, the z of many rules or random arrangement, and the topographic(al) data that is digitizing is stored in the product of computing machine.

Set up digital terrain model, generally will pass through the main processes such as elevation interpolation of the ordering of terrain data collection, data pre-service, raw data and retrieval, point to be located.In order to improve the precision of digital terrain model, strengthen ability to the landform surface description, must effectively handle atural object, landform fracture line in the digital-to-analogue.

(1) terrain data collection

The source of digital terrain model raw data has multiple, and that mainly adopts in practicality at present has wherein three kinds: obtain terrain data by aerial survey instrument from aerophoto; From existing topomap, import terrain data by digitizer; Obtain terrain data by instruments such as the electronic theodolite that can write down metric data, full standing posture tacheometers from the field actual measurement.

Obtain the digital-to-analogue raw data with airborne survey method and mainly contain two kinds of approach: the one, by the side's of elephant mapping thought, utilize stereocomparator directly to measure image point coordinate x, y, the p of photo; Another is by object space mapping thought, utilizes stereoscopic plotter to set up stereoscopic model, rating model coordinate x, y, z.

Existing topomap is carried out digitizing can use large format digitizer or large format graphics scanner.When utilizing digitizer that topomap is sampled, generally be to obtain the planimetric coordinates of topographic(al) point by digitizer along level line, this isocontour height value is then by manually reading; To small-scale topographical map (1: 10000 ~ 1: 50000), if level line is intensive, certain width is moved towards on road along the line, can duplicate amplification, is beneficial to sample along level line; When data such as landform independent point and atural object, fracture line are measured, can be earlier to the Various types of data coding convention.

At sheet data and the topomap of not navigating, in the time of can not gathering terrain data with said method, utilize the full standing posture tacheometer of tape recording device to survey terrain data in the open air, this also is a kind of practicable digital-to-analogue collecting method.Because open-air actual measurement labour intensity is big, poor working environment, efficient is low, be not suitable for gathering too much topographic(al) point, thereby open-air actual measurement collection terrain data is set up digital-to-analogue, general regional area or the local construction site survey that only is adapted to route than weak point, and be used for known counting required less triangulation network digital-to-analogue program.

(2) terrain data pre-service

The pretreated purpose of data is the different initial landform data of data layout that various device is collected, convert the terrain data file of unified coordinate system, uniform data format and coding form to, and carry out debugging, correct mistakes and the processing of data compression.Then, roughly move towards according to route, consider that belt-like zone sets up the needs of digital terrain model and the edge fit between the digital-to-analogue, with whole landform raw data through rotation, move in the unified completely digital-to-analogue coordinate system, call continuously for the banded digital terrain model of setting up along route trend.

The computing velocity of logarithmic mode, its data structure is particularly important, and the data structure quality of digital-to-analogue directly has influence on the efficient of digital-to-analogue, also is to be related to the key that can digital-to-analogue be applied to actual engineering.At the data pretreatment stage, determine the data structure that the digital-to-analogue data are adopted, also be one of its main task.When research digital-to-analogue data structure and data organization, mainly should consider two factors: the one, the digital-to-analogue of building should be occupied minimum storage space, reduces data redundancy; The 2nd, on the basis of minimum storage space, digital-to-analogue should provide complete data message, in order to retrieval fast, interpolation and various application.

(3) terrain data ordering and retrieval

With regard to sort algorithm, the ordering of insertion, Shell sorting, selection ordering, heapsort, quicksort, merge sort, the ordering of B+ method, the ordering of jump dichotomy, radix sorting or the like are arranged.To the selection of sort method, mainly to make full use of the information in each sequencer procedure as much as possible, improve the complicacy of algorithm effectively, reduce number of comparisons, and reduce the interim parking space that data are occupied, with this speed of accelerating to sort.

From the analysis of sort algorithm as can be known, faster to the ordering of one-dimension array ordering comparison two-dimensional array, thus the ordering of digital-to-analogue data two dimension can be converted into the one-dimension array sequencing problem, respectively to x, the ordering of y direction.The sort algorithm thought of logarithmic mode data is:

1. all given datas are sorted to the x direction;

2. according to the density of raw data, determine graticule mesh width dc automatically, obtain total columns of x direction, thereby can determine first and every row x direction true origin x _{0 (i)}

3. determine that the known topographic(al) point of every row is (from x _{0 (i)}＜x＜x _{0 (i)}Extract in+dc the zone);

4. to the known terrain data of these row, the y direction is sorted;

5. determine this row graticule mesh number according to this distribution range that is listed as known terrain data, and the local coordinate initial point y that should be listed as first lattice _{0 (i)}

6. to this column data write down in each graticule mesh known point first point sequence number and always count (from y _{0 (i)}+ jdc＜x＜y _{0 (i)}+ (j+1) extract in the dc zone).

Repeat 3. ~ 6., all known points in digital-to-analogue zone back that disposes is formed the digital-to-analogue data, on this basis, as long as provide the planimetric coordinates of any point to be located, just can retrieve this fast and put graticule mesh of living in, thus known topographic(al) point in this graticule mesh of rapid extraction.Digital-to-analogue all must have to known point retrieve fast, the ability of rapid extraction, could satisfy practical needs.

(4) the elevation interpolation of digital terrain model

To the landform raw data that collects, carry out interpolation with certain mathematical method and encrypt, this is one of key problem of setting up digital-to-analogue.The method of elevation interpolation has multiple: mobile surface fitting method, weighted mean value method, curved surface summation method, simultaneous piecemeal polynomial method, rectangle graticule mesh interpolation method, triangular mesh interpolation method.This problem adopts rectangle graticule mesh interpolation method, therefore introduces, and all the other methods see list of references [7] for details.

Rectangle graticule mesh interpolation method is under the known situation of rectangle graticule mesh node elevation, the method of point to be located elevation in each graticule mesh scope of interpolation, these graticule mesh node elevation can directly measure from photogrammetric model and obtain, and also can be tried to achieve by certain above-mentioned interpolating method.In each unit of graticule mesh, set up the ground surface that a Local Polynomial is represented this unit during interpolation, determine the elevation of point to be located in this graticule mesh scope then by this polynomial surface.Since in be inserted in the single graticule mesh and carry out, the elevation of point to be located mainly is subjected to the influence of four node elevation of graticule mesh, so graticule mesh will have enough density, so as in each grid unit available one with respect to each planimetric coordinates once, the independent polynomial expression of secondary or three times carries out interpolation.

Interpolation in the rectangle graticule mesh, the most frequently used is in the bicubic polynomial expression several or all.The polynomial expression formula of bicubic is fully

$Z_p=\left[\begin{array}{cccc}1& x& x^2& x^3\end{array}\right]\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& a_{33}& a_{34}\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right]\left[\begin{array}{c}1\\ y\\ y^2\\ y^3\end{array}\right]=X^T\mathrm{AY}---\left(1\right)$

The bicubic polynomial expression to be determined as can be known from (1) formula, 16 coefficients in the factor arrays will be determined.For this reason, the elevation in the unit on four graticule mesh nodes, also need the known cubic surface slope on X, Y direction respectively

And the mixing slope of representing the curved surface distortion at graticule mesh node place These slope value can be calculated by the altimeter of node on the adjacent graticule mesh, and concrete grammar is according to approximate treatment principle in the mathematics, represent single order local derviation and second order mixing local derviation respectively with first order difference and second order difference.

Because these boundary conditions have been arranged, can guarantee that the whole interpolation surface that is formed by each Local Polynomial is continuous in smooth.So it is the most perfect in these class methods adopting 16 complete bicubic polynomial interpolation.

(5) fracture line are handled

Fracture line are significant datas of setting up digital terrain model, and the elevation interpolation precision of its logarithmic mode has considerable influence.Can handle fracture line effectively, be the key that can digital-to-analogue be used for engineering reality.The interpolating method difference that digital-to-analogue adopts is also different to the disposal route of fracture line.When adopting the interpolating method of surface fitting one class,, generally be to approach surface to be inserted along the fracture line burst as some digital-to-analogue, the square grid digital-to-analogue etc. of loosing.For this reason in the interpolation zone, differentiate which point at first automatically and intersect with fracture line with the straight line that point to be inserted is linked to be, determine the topographical data points that no longer participates in interpolation with to be inserted some heteropleural, and intersection point and the interpolation of obtaining they and fracture line go out this point height.For triangulation network digital-to-analogue, should network as leg-of-mutton border with each bar fracture line.Near fracture line, best leg-of-mutton condition does not exist, for this reason, when network forming, to the triangle of each new formation, all judgements that will whether intersect with the straight line that its fracture line point is linked to be, and, confirm new triangular apex according to the mutual relationship of triangle and fracture.

Digital terrain model gets final product topographic mapping after setting up.Generally be on rectangle square grid or triangulation network grid limit, to make the elevation interpolation to obtain contour point value, connect each contour point by certain judgement direction then.Its process flow diagram as shown in Figure 1.

Two, grade point (data point) determines

From the vertical section of route, the trend of route is nothing more than these three kinds of situations: 1. route go up a slope (i＞0); 2. route descending (i＜0); 3. route level (i=0.3%) (smooth and easy for making draining, as to prevent that ponding from infiltrating roadbed and influencing its stability) so the longitudinal gradient of reservation 0.3%.If this situation of route level (i=0.3%) aside from, just have only two kinds of situations so when determining grade point: route goes up a slope or the route descending.

The deviser goes up design plane when linear at large scale topographical map (1: 500 ~ 1: 2000), and according to the terminus absolute altitude of route and the elevation at reference mark, which direction this walks and at this upward slope of which highway section, at this descending of which highway section to judge route.Then with bisecting compasses aperture L=Δ h/i _{On average}% (engineer's scale is identical with topomap), along route trend rotate bisecting compasses successively on level line section advance, the point that is drawn is original grade point.

Therefore, can discern the trend of route automatically, make the linear deviser's of meeting that computer fitting comes out intention, just need on the hypsographic map that generates by digital terrain model (DTM), determine the reference mark of control route trend in order to make computing machine.These reference mark can be the reference mark that route must pass through, and also can be the economic control points of subgrade cross section equalization of embankments and cuttings, also can be pure virtual controlling points.Determined after these reference mark, when the establishment simulated program, a variable gradepoint (x=0 is set, y=0), whenever obtain a grade point, just the x of this grade point, y coordinate assignment being given x, the y coordinate of variable gradepoint, is the center of circle with a gradepoint then, radius R=Δ h/i _{On average}% makes circular arc.Simultaneously, (method that the elevation interpolation adopts is a rectangle graticule mesh interpolation to carry out the elevation interpolation every a minute angle Δ a (as Δ a=0.05 radian), see 5.1 joints for details), spot elevation (the x that interpolation is come out, y, if z) satisfy the condition of following two aspects: the difference of the known level line elevation that elevation z that 1. obtains and route working direction are the most contiguous is less than given arbitrarily small positive number ε (as ε=0.001); 2. the spot elevation planimetric coordinates of obtaining (x, y) to the most contiguous reference mark planimetric coordinates of route working direction (x, distance y) is the shortest, then this point is the grade point of being asked.Concrete program flow chart as shown in Figure 2.

Three, the foundation of local coordinate system cubic spline curve mathematical model

After a series of data point is determined, can what kind of function of this usefulness carries out the effect that interpolation or match draw just reach best? very naturally, can expect cubic spline function or B-spline function, but will notice that cubic spline function or B-spline function all have a constraint condition: horizontal ordinate must be dull (the dull rising or dull decline).For highway, improve, the horizontal ordinate of highway alignment can not be dull, so cubic spline function or B-spline function just are not suitable for being used for carrying out interpolation or match.So cubic spline function mathematical model under the employing local coordinate.

Under the local coordinate cubic spline function be a kind ofly have geometric invariance, applicable to the SPL of large deflection curve interpolation.Its segmentation is made of (small deflection) cubic polynomial curve, and mathematic(al) representation is simply unified, calculates easy to be reliable, integral body reaches Second Order Continuous, and the line style fairing has conformal performance preferably, generally unnecessary flex point can not occur, be a kind of desirable large deflection interpolation instrument.It has the following advantages:

1. adopt the form of local coordinate, meet the objective condition of road alignment design, the match design effort still can be used under the situation of coordinate as usual not having, enlarged the scope of its application, and made the work of laying of curve also become very easy;

2. its mathematic(al) representation is simple, and unity of form helps computing machine and carries out numerical evaluation, and easy to use;

3. curve has whole large deflection, the characteristics of local small deflection, thereby curve to depart from the distance of the string of a musical instrument less, be convenient to control and the adjustment of designer to route allocation.

4. mathematic(al) representation is simple, and unity of form helps computing machine and carries out numerical evaluation, and easy to use;

(1) derivation of equation of local coordinate system cubic spline function

Curve T _sBe orderly point range { P on the plane _j(j=1,2 ..., local cubic spline N) is meant that its satisfies two conditions: 1. in each local coordinate, curve is to be the cubic polynomial of independent variable and the two ends by the string of a musical instrument with the string of a musical instrument; 2. whole piece curve and tangent line thereof, curvature are continuous in the zone.

As shown in Figure 3, establish interior nodes P _jAdjacent local coordinate (x, y), (x, one in y), second derivative are designated as m respectively _j(-), M _j(-) and m _j(+), M _j(+), the coordinate corner is φ _j, according to tangent line and the continuous requirement of curvature (referring to Fig. 3) arranged then:

$\frac{m_j(+)-m_j(-)}{1+m_j(+)m_j(-)}\mathrm{tg}{\mathrm{\φ}}_j---\left(2\right)$

$\frac{M_j(-)}{{[1+m_j^2(-)]}^{3/2}}=\frac{M_j(+)}{{[1+m_j^2(+)]}^{3/2}}---\left(3\right)$

Cubic interpolation formula in each local coordinate can be write as:

$y\left(x\right)=m_{j-1}(+)\frac{x{(l_j-x)}^2}{l_j^2}-m_j(-)\frac{x^2(l_j-x)}{l_j^2}---\left(4\right)$

Or:

$y\left(x\right)=M_{j-1}(+)\frac{{(l_j-x)}^3}{6l_j}+M_j(-)\frac{x^3}{6l_j}\frac{M_{j-1}(+)}{6}{l}_j(l_j-x)-\frac{M_j}{6}{l}_{j}x---\left(4\right)$

In the formula: l _j-the string of a musical instrument

Length

So node one, second derivative has expression formula to make mark mutually:

$M_j(-)=\frac{{2m}_{j-1}(+)}{l_j}+\frac{{4m}_j(-)}{l_j},m_j(-)=\frac{l_j{M}_j(-)}{3}+\frac{l_j{M}_{j-1}(+)}{6}---\left(5\right)$

Wushu (2), formula (5) substitution formula (3), the abbreviation arrangement can obtain with { m _jBe node relationships formula-" m (+) relational expression " of parameter:

λ _jm _J-1+ 2m _j+ μ _jm _J+1=c _j+ F _j(7) in the formula:

${\mathrm{\λ}}_j=\frac{l_{j+1}}{l_j+l_{j+1}}$

${\mathrm{\μ}}_j=\frac{l_j}{l_j+l_{j+1}}$

c _j＝2λ _jtgφ _j+μ _jtgφ _j+1

F _j＝F _1j+F _2j+F _3j (8)

$F_{1j}=\frac{{\mathrm{\μ}}_j{m}_{j+1}{\mathrm{tg\φ}}_{j+1}[m_{j+1}-{\mathrm{tg\φ}}_{j+1}]}{1+m_{j+1}{\mathrm{tg\φ}}_{j+1}}$

$F_{2j}=\frac{{2\mathrm{\λ}}_j{m}_j{\mathrm{tg\φ}}_j[m_j-{\mathrm{tg\φ}}_j]}{1+m_j{\mathrm{tg\φ}}_j}{[{\cos \mathrm{\φ}}_j+m_j{\sin \mathrm{\φ}}_j]}^3$

F _3j＝λ _j[m _j-1+2(m _j-tgφ _j)]{1-[cosφ _j+m _jsinφ _j] ³}

(j=2 ... N-1) additional two end boundary constraints, formula (7) constitutes N+1 unknown parameter { m _jComplete Nonlinear System of Equations.Point out in passing, for border φ non-periodic _N≡ 0.

As make mark

Can obtain " m (-) relational expression " of node respectively through same derivation

λ _jm _J-1+ 2 m _j+ μ _jm _J+1=c _j+ F _j(10) in the formula:

c _j＝-2μ _jtgφ _j-λ _jtgφ _j-1

F _j＝ F _1j+ F _2j+ F _3j

F _1j＝-λ _jtgφ _j-1·m _j-1· m _j-1 (11)

F _2j＝-2μ _jtgφ _j·m _j· m _j·[cosφ _j- m _jsinφ _j] ³

F _3j=μ _j[2 m _j+ m _J+1+ 2tg φ _j] { 1-[cos φ _j-m _jSin φ _j] ³" M (+) relational expression " of node

μ _jM _J-1+ 2M _j+ λ _jM _J+1=d _j+ g _j(12) in the formula:

${\mathrm{\λ}}_j=\frac{l_{j+1}}{l_j+l_{j+1}}$

${\mathrm{\μ}}_j=\frac{l_j}{l_j+l_{j+1}}$

$d_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}---\left(13\right)$

$g_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}{m}_j{\stackrel{\‾}{m}}_j+{2\mathrm{\μ}}_j{M}_j\{1-{[{\cos \mathrm{\φ}}_j-{\stackrel{\‾}{m}}_j{\sin \mathrm{\φ}}_j]}^3\}$

$+{\mathrm{\λ}}_j{M}_{j+1}\{1-{[{\cos \mathrm{\φ}}_{j+1}-{\stackrel{\‾}{m}}_{j+1}{\sin \mathrm{\φ}}_{j+1}]}^3\}$

" M (-) relational expression " of node

μ _jM _J-1+ 2 M _j+ λ _jM _J+1=d _j+ g _j(14) in the formula:

$d_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}$

${\stackrel{\‾}{g}}_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}{m}_j{\stackrel{\‾}{m}}_j+{\mathrm{\μ}}_j{\stackrel{\‾}{M}}_{j-1}\{1-{[{\cos \mathrm{\φ}}_{j-1}+m_{j-1}{\sin \mathrm{\φ}}_{j-1}]}^3\}---\left(15\right)$

$+{2\mathrm{\λ}}_j{\stackrel{\‾}{M}}_j\{1-{[{\cos \mathrm{\φ}}_j+m_j{\sin \mathrm{\φ}}_j]}^3\}$

If with node curvature { ρ _jAs parameter, by interior nodes tangent line condition of continuity formula (2), the abbreviation arrangement can obtain " ρ relational expression ":

μ _jρ _j-1+2ρ _j+λ _jρ _j+1＝d _j+h _j (16)

In the formula:

$d_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}$

$h_j=-\frac{{6\mathrm{tg\φ}}_j}{l_j+l_{j+1}}{m}_j{\stackrel{\‾}{m}}_j+{\mathrm{\μ}}_j{\mathrm{\ρ}}_{j-1}\{1-{[1+{m^2}_{j-1}]}^{\frac{3}{2}}\}+{2\mathrm{\ρ}}_j\left\{{\mathrm{\μ}}_j\right[1-{(1+{{\stackrel{\‾}{m}}_j}^2)}^{\frac{3}{2}}]---\left(17\right)$

$+{\mathrm{\λ}}_j[1-{(1+{m_j}^2)}^{\frac{3}{2}}]\}+{\mathrm{\λ}}_j{\mathrm{\ρ}}_j\{1-{[1+{{\stackrel{\‾}{m}}_{j+1}}^2]}^{\frac{3}{2}}\}$

" m (+) relational expression " of node, " m (-) relational expression ", " M (+) relational expression ", " M (-) relational expression " and " ρ relational expression " these five groups of relational expressions are equivalent equivalences, can select for use as the case may be.

(2) the determining of cubic spline function boundary condition under the local coordinate

The same with general cubic spline, boundary condition also have the cycle and non-periodic condition branch, but, have only boundary condition non-periodic specific to highway alignment.Non-periodic, condition can consider that for " m (+) relational expression " general type is

2m ₁(+)+μ ₁m ₁(+)＝c ₁+F ₁(m ₁(+)，m ₂(+))

(18)

λ _Nm _N-1(+)+2m _N(-)=c _N+ F _N(m _{N-1 (+)}, m _N(1)) especially when two ends respectively when given tangent line, curvature, coefficient is shown in table 1, table 2 in the formula (18).

(3). cubic spline function finds the solution under the local coordinate

The node relationships formula adds upper boundary conditions, has promptly constituted the integrated equation group of node parameter.The author utilizes " m (+) relational expression " to find the solution cubic spline function under the local coordinate, the following describes whole interpolation solution procedure.

Suppose that boundary condition is: the tangent line of start, end is known, is tg α ₁=0, tg α _N=0 has μ ₁=0, c ₁=0, F ₁=0, λ _N=0, c _N=0, F _N=0.So the coefficient matrices A of formula (7) is

Though above-mentioned equation is a nonlinear equation under local coordinate system, this is a kind of special nonlinear equation, adopts simple iteration just can obtain separating of equation.The group of at first solving an equation Am ⁽⁰⁾=c obtains m ⁽⁰⁾, m ⁽⁰⁾Substitution F _1j, F _2j, F _3jExpression formula in draw F, solve an equation then the group Am ^(k)=c+F (m ^(k-1)) promptly separate three cornerwise alternative manners with chasing method, separate error ‖ m up to adjacent secondary ^(k)-m ^(k-1)In ‖＜ε (ε is given arbitrarily small positive number) time, meet the demands, and obtains m by formula (2) again _j(-), substitution formula (4) just obtains the concrete formula of each local coordinate system cubic spline function.

When drawing splines, be converted into unified coordinate system to the coordinate under the local coordinate by two-dimensional transformations at last, conversion formula is

In the formula: the horizontal ordinate under the x-unified coordinate system;

Ordinate under the y-unified coordinate system;

Horizontal ordinate under x '-local coordinate system;

Ordinate under y '-local coordinate system;

x ₀', y ₀True origin under the '-local coordinate system.

It must be noted that the author uses cubic spline function under the local coordinate system as matched curve,, be the calculating curvature of back, be converted into linear the providing convenience of tradition, and do not pay attention to its road characteristic especially just its transition instrument as a kind of calculating.

Four, the drafting of matched curve curvature chart

After the Cubic Spline Functions Fitting grade point under the local coordinate system, that can find out clearly that match is come out does not linearly have the linear straight line of tangible tradition, adjustment curve, a circular curve, and the route pile No. of matched curve inconvenience mark, linear index can't reflect visual pattern, is very inconvenient for the calculating and the construction lofting of concrete road center line.Therefore must transform, the carrier of conversion is exactly a curvature chart.Because the cubic spline function under the local coordinate system all has clear and definite calculation expression at each section, so the splines on each section is asked curvature every a slight distance (as 0.05m), corresponding arc length (being highway mileage length) all is to count from the off, and this point will add and note.The computing formula of curvature is:

$k=\frac{y^{\text{'}\text{'}}}{{(1+y^{{\text{'}\text{'}}^2})}^{\frac{3}{2}}}---\left(20\right)$

Five, the matched curve curvature chart is converted into the linear curvature chart of conventional planar

The linear linear combination form of highway conventional planar has six kinds: basic model, S type, ovum type, convex, C type, compound, these six kinds of linear combination modes corresponding respectively the difform curvature chart of six kinds of rules of combination.Not regular by matched curve as the linear curvature chart of tradition by the curvature chart that calculating curvature K draws out, can't be corresponding with the conventional planar alignment elements, therefore require the curvature chart of SPL is transformed, make it become a certain of the linear six kinds of curvature figures of conventional planar, thereby can reflect linear combination relation and alignment elements value between straight line, adjustment curve and the circular curve clearly.This problem is described as with mathematical linguistics: regard basic model, S type, ovum type, convex, C type, compound these six kinds of curvature charts as six classes in overall, the curvature chart of matched curve is regarded as a sample, go judgement sample which kind of should belong to then.By setting up discrimination standard, the minimum distance method of employing pattern recognition analysis goes to judge the classification of recognition sample to this paper in research process, and this is the core of whole simulation system.

Before concrete discussion minimum distance method, introduce two parameter lambda and γ earlier.Parameter lambda is defined as the parameters for shape characteristic of curvature chart, adopts the form of expression of vector, i.e. λ=(λ ₁, λ ₂..., λ _n), λ _iValue be:

In order to express the shape of curvature chart accurately and effectively, can save the memory space of computing machine again, reduce calculated amount, λ _iWhat show is the state of unique point on the curvature chart.So-called unique point is got adjacent three some a (L exactly on curvature chart _a, K _a), b (L _b, K _b), c (L _c, K _c) (K represents curvature, and L represents mileage), obtain the slope tg θ of a and 2 lines of b ₁=(K _b-K _a)/(L _b-L _a) and the slope tg θ of b and 2 lines of c ₂=(K _c-K _b)/(L _c-L _b), if tg is θ ₁With tg θ ₂The opposite in sign or the tg θ of value ₁With tg θ ₂Value produce sudden change, then the b point is unique point.

The purpose of introducing parameter lambda is because highway geometry has and the branch of right-hand rotation, be reflected in show as on the curvature chart curvature K on the occasion of and negative value.Judge turning to of route by continuous 0 or 1 number among the vectorial λ, thereby determine to provide foundation for the positive negative value of the standard curvature chart mean curvature K of structure.

Parameter γ is defined as the topography and geomorphology parameter, and the value of γ is:

So route just can be represented with the combinations of values of γ by the topography and geomorphology form in area.

The purpose of introducing parameter γ is: " specification of the highway route design " be middle regulation (JTJ011-94): convex linear combination mode only can adopt when the route strictness is subjected to landform, atural object restriction; Compound linear combination mode is (except the interchange) use when landform or the restriction of other cause specific only; C molded lines shape array mode only can be used under the extreme terrain condition.That is to say that generally speaking, commonly used in six kinds of linear array modes of highway conventional planar have only these three kinds of basic models, S type, ovum type, if adopt convex, C type or compound linear combination mode, also need be illustrated especially in design.After introducing topography and geomorphology parameter γ, have only when the combinations of values of γ reaches rough sledding, just increase and consider these three kinds special linear array modes, otherwise just do not consider.So just provide basis for estimation, also can reduce calculated amount and computing time simultaneously, improve program run efficient for participating in choosing of comparative standard curvature chart.

After introducing parameter lambda and γ, minimum distance method is discussed then.We know that the expression formula of adjustment curve is: rl=A ², A is the amplification or the coefficient of reduction of adjustment curve.Under the identical condition of length l, its radius r of different adjustment curves is inequality, by the comparison to the A value, can know the shape of adjustment curve.The author gains enlightenment thus: for the differentiation problem of curvature chart, under the prerequisite that route mileage length is determined, the six kinds of curvature charts that conventional planar is linear and the difference of sample curvature chart are curvature value.By curvature value being carried out standardization (adopt " normalization " method), and and the highway mileage value multiply each other, draw a certain nondimensional numerical value, this numerical value is compared the similarity degree size that can judge curvature chart.Therefore, might as well work out following computing formula:

K·L＝C (21)

In the formula: the row vector that the K-curvature value is formed.

K＝(k ₁，k ₂，……，k _n)

The column vector that L-route mileage value is formed.

L＝(l ₁，l ₂，……，l _n)T

The similarity coefficient of C-curvature chart, dimensionless.

Yet the linear curvature chart of conventional planar has just showed the syntagmatic between straight line, adjustment curve and the circular curve qualitatively, linear combination is not concerned quantification.And using formula (21) needs K value and the L value determined, just can draw concrete C value, and the similarity that just can carry out curvature chart relatively.So will carry out the quantification work of traditional wire shape curvature chart earlier, this step is called the standardisation process of the linear curvature chart of conventional planar for the time being, the curvature chart that constructs is called the standard curvature chart.So-called standard curvature chart is meant that the curvature chart that constructs has certain representativeness to this grade highway under the condition of known road grade and route mileage length.When the similarity of carrying out curvature chart compares, be that template and sample curvature chart compare just with the standard curvature chart.

Concrete definite curvature chart must be known three key elements: the long l of adjustment curve _s, the long l of circular curve radius R and circular curve _yBecause the l of each standard highway that " specification of the highway route design " (JTJ011-94) stipulated _sWith R desired value and inconsistent, it is impossible wanting to make the standard curvature chart that constructs that all standard highways all are suitable for, and therefore will make the standard curvature chart to each standard highway respectively.For fear of l _sWith the randomness of R value, the long l of regulation adjustment curve _sGet the minimum length of easement curve of each standard highway, circular curve radius R gets the general least radius R of circular curve of each standard highway _Ave, the long l of circular curve _yNeed determine l _sAnd l _yRatio could determine.L so _sAnd l _yRatio what get will decide on route mileage length in the sample curvature chart.

When structure standard curvature chart, require to satisfy following two requirements:

1. the mileage length of standard curvature chart will with the mileage equal in length of sample curvature chart;

2. Gou Zao the six kinds of linear standard curvature chart of conventional planar area requirements equate, if any error, error should be less than 10%, and purpose is to reduce the misjudgement probability of similarity.

The establishment of program for convenience, definition l _sAnd l _yThe poorest ratio be 1: 0.5, optimum ratio is 1: 2, under the given condition of mileage length, just can obtain the standard curvature chart like this in the sample curvature chart.

After the standard curvature chart constructs, just can carry out value to curvature K and mileage L.For the mileage L vector that makes standard curvature chart and sample curvature chart equates that the value mode that is defined in mileage L among both is:

L=(10,20,30 ..., l _n) l among the vectorial L _I+1-l _i=10m, l _nNumerical value be the mileage total length.The value mode of curvature K is: for the standard curvature chart, at circular curve segment, the k value is 1/R _Ave, in the adjustment curve section, the k value is tried to achieve by linear interpolation; For sample curvature chart, k _iBe l _iCorresponding curvature value.

Should be noted that, the vectorial K's of each standard curvature chart and sample curvature chart And unequal, at this moment to carry out standardization to curvature chart K vector, adopt " normalization " method, i.e. each component k in the curvature K vector _iAll divided by

The C that draws is skimble-skamble otherwise K and L multiply each other, and comparative result also is easy to misjudge.Be interpreted as from the angle of mathematics: curvature is regarded as power, with different power with weight vector multiply by an identical vector, the result who draws can't compare, and more also is skimble-skamble.

At last, using formula (21) calculates the C value of each standard curvature chart and the C value of sample curvature chart respectively, obtain the absolute value of the difference of sample curvature chart C value and standard curvature chart C value then, the corresponding standard curvature chart of number of absolute value minimum is the conversion template of sample curvature chart.

In sum, judge that with minimum distance method the step summary of curvature chart pattern is as follows:

1. determine the topography and geomorphology parameter γ of matched curve by the area;

2. determine the parameters for shape characteristic λ of curvature chart according to the sample curvature chart, go to judge turning to of route according to continuous 0 or 1 number among the vectorial λ, thus the sign of the curvature chart mean curvature k that settles the standard;

3. construct the standard curvature chart: at first according to the combinations of values of parameter γ, as rough sledding appears, then increase and consider convex, C type, compound these three kinds special linear array modes, otherwise only consider these three kinds of linear combination modes commonly used of basic model, S type and ovum type.According to the mileage length of known road quality classification and sample curvature chart, determine l, R then _AveAnd l _sAnd l _yOptimum ratio, construct the standard curvature chart;

4. carry out " normalization " processing to the vectorial value of the curvature K vector sum mileage L of standard curvature chart and sample curvature chart, and to curvature K vector, using formula (21) calculates the C value of each curvature chart;

5. the C value is relatively reached a conclusion.

The transformation mode of sample curvature chart can carry out the conversion of curvature chart after determining.The conversion of curvature chart must be followed three principles: 1. the highway mileage length requirement equates; 2. highway geometry turn to require identical; 3. the area of curvature chart (being tangent line position angle poor of highway geometry start, end) requires to equate.Also require to satisfy each desired value of the linear design that " specification of the highway route design " (JTJ011-94) stipulate simultaneously.The author adopts the least square fitting principle to carry out the match of curvature chart, and the principle of least square sees list of references [19] for details.The program flow diagram of the conversion process of the curvature chart of whole matched curve as shown in Figure 4.

Six, the linear generation of conventional planar

After the curvature chart match is finished, also need to be reduced into traditional straight line, adjustment curve and circular curve according to curvature chart, the method for employing is a Tinkertoy approach.Tinkertoy approach also claims the line element design method, and it is that the plane figure of highway that combination is complicated " breaks the whole up into parts ", resolves into several linear unit.If origin information such as coordinate, tangent line or normal direction and the radius-of-curvature of known route plane curve, then begin to be provided with arbitrary unit from starting point, extend along any direction, the information of this element terminal point such as coordinate, tangent line or normal line direction angle and radius-of-curvature can be calculated, and simultaneously its identical information as the next unit starting point are used.The unit down calculates so one by one, and as playing with building blocks, each unit connects from beginning to end, constitutes the plane figure of highway of a continuous whole.Wherein linear unit is that line element can be respectively straight line, adjustment curve and circular curve, also can be the linear unit that these three kinds of line elements combine in case of necessity.Be not difficult to find that Tinkertoy approach in most cases is used for the calculating of known highway primary design of curve element value and the horizontal alignment of construction documents design phase and lays from above-mentioned analysis.

With straight line, adjustment curve, circular curve is line element, and the origin information of known route such as coordinate, tangent line or normal line direction angle, radius-of-curvature etc. and the linear curve element value of whole piece.According to the definition and the basic ideas of Tinkertoy approach, the ultimate principle of Tinkertoy approach as shown in Figure 5, the algorithm of line element design is as shown in Figure 6.

(1). the position angle at Point of Inflection place

Lay horizontal alignment with Tinkertoy approach, the very important point is exactly on having laid must to know the tangent line position angle of next unit after the unit.Therefore will obtain azimuthal value at Point of Inflection place earlier.The process flow diagram of computer azimuth angle is shown in Fig. 5-7.The transition matrix that is transformed into whole coordinate from local coordinate when each line element lays is:

$A=\left[\begin{array}{cc}\cos (\frac{\mathrm{\π}}{2}-\mathrm{azimuthangle}\left(i\right))& -\sin (\frac{\mathrm{\π}}{2}-\mathrm{azimuthangle}\left(i\right))\\ \sin (\frac{\mathrm{\π}}{2}-\mathrm{azimuthangle}\left(i\right))& \cos (\frac{\mathrm{\π}}{2}-\mathrm{azimuthangle}\left(i\right))\end{array}\right]$

(2). the coordinate of each point under global coordinate system is referring to Fig. 5,6,7 on each line element.

Because known each pile No. stake of Point of Inflection place (i) is so the mileage length of line element i is

L (i)=stake (i+1)-stake (i) is an X-axis with Point of Inflection i place tangent line, normal is that Y-axis is set up local coordinate system, ask the coordinate of every bit under global coordinate system on the line element, now the coordinate in the hope of Point of Inflection i+1 place is an example, and the situation when connecing straight line, circular curve and adjustment curve is discussed respectively.

1. connect straight line:

The coordinate at Point of Inflection i+1 place is under the local coordinate system:

Then the coordinate of Point of Inflection i+1 is under the global coordinate system: $\left[\begin{array}{c}x(i+1)\\ y(i+1)\end{array}\right]=A\begin{array}{c}\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]\end{array}+\left[\begin{array}{c}x\left(i\right)\\ y\left(i\right)\end{array}\right]$

2. connect circular curve:

Circular curve radius: R=|1/k (i) |

Central angle: a=L (i)/R

The coordinate of Point of Inflection i+1 is under the local coordinate system:

Then the coordinate under the global coordinate system is:

$\left[\begin{array}{c}x(i+1)\\ y(i+1)\end{array}\right]=A\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]+\left[\begin{array}{c}x\left(i\right)\\ y\left(i\right)\end{array}\right]$

3. connect adjustment curve

1) complete adjustment curve

K (i)=0 o'clock, k (i+1) located radius and is: R=|1/k (i+1) | and the coordinate of its local coordinate system is:

The coordinate of global coordinate system is:

$\left[\begin{array}{c}x(i+1)\\ y(i+1)\end{array}\right]=A\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]+\left[\begin{array}{c}x\left(i\right)\\ y\left(i\right)\end{array}\right]$

When k (i)＜0 and k (i+1)=0, ask the coordinate of k (i+1) point under local coordinate x ' oy ' earlier:

Rotation angle: the rotateangle=pi/2-(3 pi/2s+| k (i) l/2|) transition matrix:

$A_1=\left[\begin{array}{cc}\cos \left(\mathrm{rotateangle}\right)& -\sin \left(\mathrm{rotateangle}\right)\\ \sin \left(\mathrm{rotateangle}\right)& \cos \left(\mathrm{rotateangle}\right)\end{array}\right]$

x ₀, y ₀Coordinate under x ' oy ' is:

$\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]=A_1\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$

Trying to achieve its coordinate under global coordinate system again is:

$\left[\begin{array}{c}x(i+1)\\ y(i+1)\end{array}\right]=A\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]+\left[\begin{array}{c}x\left(i\right)\\ y\left(i\right)\end{array}\right]$

In like manner can try to achieve the coordinate that k (i)＞0 and k (i+1)=0 o'clock k (i+1) locate.

2) non-complete adjustment curve

When k (i)＜0 and k (i+1)＜k (i), this line element is for connecting the two non-complete adjustment curves of circular curve in the same way.Locate coordinate in order to calculate k (i+1), earlier non-complete adjustment curve is reduced into complete adjustment curve, starting point is oo for the o terminal point ", as shown in Figure 8.

Then at the radius at o ' some place: r ₁=| 1/k (i) |, " the radius at some place: r at o ₂=| 1/k (i+1) |, adjustment curve is long: l _s=l+l ₀, oo ' arc length: | r ₂L/ (r ₁-r ₂) |.Ask coordinate and the position angle of o point under x ' oy ' earlier: o ' some coordinate in the xoy coordinate system is: Rotation angle: rotateangle=|k (i) l ₀/ 2|, transition matrix: $A_2=\left[\begin{array}{cc}\cos \left(\mathrm{rotateangle}\right)& -\sin \left(\mathrm{rotateangle}\right)\\ \sin \left(\mathrm{rotateangle}\right)& \cos \left(\mathrm{rotateangle}\right)\end{array}\right],$ If the coordinate of o point in x ' oy ' coordinate system is (x ₀, y ₀), because: $\left[\begin{array}{c}0\\ 0\end{array}\right]=A_2\left[\begin{array}{c}{x}_0^{\text{'}}\\ y_0^{\text{'}}\end{array}\right]+\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right],$ Then have: $\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]-A_2\left[\begin{array}{c}{x}_0^{\text{'}}\\ y_0^{\text{'}}\end{array}\right]=-A_2\left[\begin{array}{c}{x}_0^{\text{'}}\\ y_0^{\text{'}}\end{array}\right],$

The coordinate of o point under global coordinate system is: $\left[\begin{array}{c}\mathrm{xz}\\ \mathrm{yz}\end{array}\right]=A\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]+\left[\begin{array}{c}{x}_i\\ y_i\end{array}\right],$ The rotation angle of xoy under global coordinate system is:

The rotateangle=pi/2-(azimuthangle (i)-| k _il ₀/ 2|), transition matrix: $A_3=\left[\begin{array}{cc}\cos \left(\mathrm{rotateangle}\right)& -\sin \left(\mathrm{rotateangle}\right)\\ \sin \left(\mathrm{rotateangle}\right)& \cos \left(\mathrm{rotateangle}\right)\end{array}\right],$ Terminal point o then " coordinate under the xoy coordinate system is:

X, the y coordinate under global coordinate system is: The non-complete adjustment curve terminal point k (i+1) that in like manner can try to achieve under other situation locates coordinate.

3) reverse adjustment curve

As k (i)＜0, k (i+1)＞0 o'clock, so long regard it by two sections complete adjustment curves as and constitute, be i.e. k (i)＜0 and k (0)=0 and k (i+1)＞0.At this moment, call complete adjustment curve subroutine respectively for twice and can try to achieve the coordinate that k (i+1) locates.In like manner can ask the situation of k (i)＞0 and k (i+1)＜0.

The computerized algorithm that known curvature figure uses Tinkertoy approach to carry out linear design is all introduced to this and is finished, and the author goes out Matlab operation program-Orthodoxline function according to above-mentioned algorithm development.

Instance analysis

Heavy mound, a certain mountain ridge Class III highway, the route starting point needs to cross a hilltop then for indicating leg-of-mutton place as shown in Figure 8, the place behind the arrival yamase.The round dot that marks among Fig. 9 is represented the reference mark of guiding route trend, these reference mark belong to the reference mark of virtual character, star point is that (some data point and level line are not lean on very near to the data point of obtaining by rectangle graticule mesh bicubic polynomial expression elevation interpolation (grade point), this is because due to the elevation system error that computing machine causes when forming level line), the design average gradient adopts 5.0%.Curve shown in Figure 8 be local coordinate system cubic spline curve match data point draw linear, the curvature chart of cubic spline curve is as shown in Figure 9.The minimum distance method of application mode identification is judged by calculating, the conversion template that obtains the sample curvature chart is the curvature chart of basic model linear combination, with transforming the linear curvature chart of conventional planar obtain behind the least square fitting as shown in figure 10, Figure 11 is that the utilization Tinkertoy approach conventional planar of laying out is linear.

From example as can be seen, when the utilization emulation technology is carried out the horizontal alignment design, do not need to be provided with leading line, saved great amount of manpower and material resources, linear design is accomplished quick, simple and direct, has realized the plane figure of highway in mountain area The Automation Design.

The mileage pile No. of route unique point, coordinate are as shown in table 3.

Practice shows, the analogue system that adopts this method to set up is used for the plane figure of highway in mountain area design fast, conveniently, and the designing quality height is effective, the automaticity height.Can satisfy technical standard, can reduce Route Design to environment damage well in conjunction with landform and environmental baseline again, be the automatic design and simulation technology of a kind of highway preferably, has application value widely.

List of references:

[1] publishing house of Highway Survey design University Of Chongqing, 1994 years are write by a team of four horses of grandson family

[2] Zhu write publishing house of road route CAD Tongji University 1998 years according to grand Chen Yu people etc.

[3] the male Wang Fujian of Wu state writes plane figure of highway shaped form method for designing People's Transportation Press

[4] Tu Shurong is applied to Horizontal Alignment Design Study of Some Problems Traffic Institutes Of Chongqing journal, 1994 about SPL

Table 1

Table 2

The unique point of route, mileage and coordinates table 3

Unique point	The mileage pile No.	Coordinate
				X	Y
		Starting point (ZH)	K17+540			?92.2296	?42.3198
HY	+583			?92.2099	?42.3117
		YH	+687.8			?92.1905	?42.3484
HZ	+730.8			?92.2083	?42.3601
		Terminal point	+734			?92.2097	?42.3608

Claims (4)

1, a kind of plane figure of highway in mountain area automatic simulation method for designing is characterized in that comprising the steps:

(1) set up digital terrain model, draw out hypsographic map on this basis, step is more specifically:

A) terrain data collection;

B) terrain data pre-service;

C) terrain data ordering and retrieval;

D) the elevation interpolation of digital terrain model adopts rectangle graticule mesh interpolation method;

E) fracture line are handled;

(2) adopt the method for rectangle graticule mesh cubic polynomial elevation interpolation to obtain grade point;

(3) looking these grade points is data point, utilizes the local coordinate cubic spline curve to carry out match, and the plane fitting that generates route is linear;

(4), generate the linear curvature chart of match to the linear curvature of calculating at a certain distance of the match of route;

(5) the linear curvature chart of match is discerned conversion module automatically, and transform the linear curvature chart of generation conventional planar with least square method;

(6) utilize Tinkertoy approach to lay horizontal alignment, calculate the line position of going forward side by side.

2, automatic simulation method for designing as claimed in claim 1 is characterized in that described terrain data ordered steps is:

(1) all known numbers is sorted to the x direction;

(2) according to the density of raw data, determine graticule mesh width dc automatically, obtain total columns of x direction, thereby can determine first and every row x direction true origin;

(3) from x _{0 (i)}＜x＜x _{0 (i)}In+dc the zone, determine the known topographic(al) point of every row;

(4) to the known terrain data of these row, the y direction is arranged;

(5) determine this row graticule mesh number according to this distribution range that is listed as known terrain data, and the local coordinate initial point y that should be listed as first lattice _{0 (i)}

(6) from y _{0 (i)}+ jdc＜x＜y _{0 (i)}+ (j+1) in the dc zone, this column data is write down the known point first point sequence number in each graticule mesh and always counts;

Repeat (3) ~ (6) of this claim, all known points in digital-to-analogue zone back that disposes is formed the digital-to-analogue data, on this basis, as long as provide the planimetric coordinates of any point to be located, just can retrieve this fast and put graticule mesh of living in, thus known topographic(al) point in this graticule mesh of rapid extraction.

3, automatic simulation method for designing as claimed in claim 1 is characterized in that described least square method transforms the linear curvature chart design procedure of generation conventional planar and is:

(1) determines that matched curve passes through the topography and geomorphology parameter γ in area;

(2) determine the parameters for shape characteristic λ of curvature chart according to the sample curvature chart, go to judge turning to of route according to continuous 0 or 1 number among the vectorial λ, thus the sign of the curvature chart mean curvature k that settles the standard;

(3) structure standard curvature chart: at first according to the combinations of values of parameter γ, as rough sledding appears, then increase and consider convex, C type, compound these three kinds special linear array modes, otherwise only consider these three kinds of linear combination modes commonly used of basic model, S type and ovum type; According to the mileage length of known road quality classification and sample curvature chart, determine l then _s, R _AveAnd l _sAnd l _yOptimum ratio, construct the standard curvature chart;

(4) to the vectorial value of the curvature K vector sum mileage L of standard curvature chart and sample curvature chart, and curvature K vector is carried out " normalization " handle, calculate the C value of each curvature chart;

(5) the C value is relatively reached a conclusion.

4, automatic simulation method for designing as claimed in claim 1, it is characterized in that institute's Tinkertoy approach is that the plane figure of highway that combination is complicated " breaks the whole up into parts ", resolve into several linear unit, if origin information such as coordinate, tangent line or normal direction and the radius-of-curvature of known route plane curve, then begin to be provided with arbitrary unit from starting point, extend along any direction, the information of this element terminal point such as coordinate, tangent line or normal line direction angle and radius-of-curvature can be calculated, and simultaneously its identical information as the next unit starting point are used;

The unit down calculates so one by one, and as playing with building blocks, each unit connects from beginning to end, constitutes the plane figure of highway of a continuous whole;

Wherein linear unit is that line element can be respectively straight line, adjustment curve and circular curve, also can be the linear unit that these three kinds of line elements combine.

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