2012年數(shù)學(xué)建模D題機(jī)器人避障問題論文

:第i段切線的長度。：第j段圓弧的長度。L：從原點(diǎn)到達(dá)最終目標(biāo)點(diǎn)的最短路徑的總長度。K:障礙物上的任一點(diǎn)與行走路徑之間的最短距離。模型的建立5.1模型猜想：猜想一：具有圓形限定區(qū)域的最短路徑是由兩部分組成的：一部分是平面上的直線段，另一部分是限定區(qū)域的部分邊界（即圓弧段），這兩部分是相切的，連續(xù)的。（即問題分析中的拉繩子拉到最緊時的狀況）證明：假設(shè)在平面中有A(-a,0)和B(a,0)兩點(diǎn)，中間有一個半圓形的障礙物，證明從A到B的最短路徑為AQUOTEB。平面上連接兩點(diǎn)最短的路徑是通過這兩點(diǎn)的直線段，但是連接兩點(diǎn)的線段于障礙物相交，所以設(shè)法嘗試折線路徑。在y軸上取一點(diǎn)C0,y，若y適當(dāng)大，則折線ACB與障礙物不相交，折線ACB的長度為：ACB=2a2+y2顯然ACB隨著y的減小而減小，減小y得y→y1，即C→C1,使得AC1與C1B都與障礙物相切，切點(diǎn)分別為E和F,顯然AC1B是這種折線路徑中最短的。由于滿足0<∠AC1O∠π2的角滿足,所以易知弧EF小于EC為了使結(jié)果更具有說服力，下面再考察一條不穿過障礙物的任何一條路徑，設(shè)其分別于OE和OF的延長線交與P、Q兩點(diǎn)，記A和P之間的路徑長度為APQUOTE，顯然QUOTEAP>|AP|,又因AEEO,所以|AP|>AE,從而APQUOTE>AE同理可得BQQUOTE再來比較PQ之間路徑長度QUOTEPQ和圓弧EF的長度的大小。若PQ之間的路徑可有極坐標(biāo)方程QUOTEρ=ρ(θ),則有QUOTEρ>1,可得：PQ即路徑APQB的長度超過路徑AEFB的長度。以上證明足以說明AEFB是滿足條件A到B的最短路徑。猜想二：假設(shè)一個圓環(huán)可以繞著環(huán)上一個定點(diǎn)轉(zhuǎn)動，那么過圓環(huán)外兩定點(diǎn)連接一根繩子，并以該圓環(huán)為支撐拉緊繩子，達(dá)到平衡時，圓心與該頂點(diǎn)以及兩條切線的延長線的交點(diǎn)共線圖2證明猜想：如圖2所示，E點(diǎn)就是圓環(huán)上的一個頂點(diǎn)，AQUOTEB就是拉緊的繩子，就是切線AC和BD的延長線的交點(diǎn)，證明、E、三點(diǎn)共線。我們可以用力學(xué)的知識進(jìn)行證明，因?yàn)槭抢o的繩子，所以兩邊的繩子拉力相等，設(shè)為，它們的合力設(shè)為，定點(diǎn)對圓環(huán)的作用力設(shè)為。那么由幾何學(xué)的知識我們可以知道一定與共線，而又由力的平衡條件可知：=-即與共線，、和三點(diǎn)一定共線。5.2模型一的準(zhǔn)備1）、有了5.1的定理，我們就可以認(rèn)為，無論起點(diǎn)到目標(biāo)點(diǎn)的途中有多少個障礙物，最短路徑都應(yīng)該是由若干個線圓結(jié)構(gòu)所組成的。根據(jù)本題中存在障礙物的情況，且障礙物在拐角處的危險區(qū)域是一個半徑為10的圓弧。所以結(jié)合定理5.1，求兩點(diǎn)之間的最短路徑時，我們應(yīng)該按照最小的轉(zhuǎn)彎半徑來計(jì)算才可達(dá)到最優(yōu)。線圓結(jié)構(gòu)5.21如線圓結(jié)構(gòu)5.21，設(shè)A點(diǎn)坐標(biāo)為(X1,Y1),B點(diǎn)坐標(biāo)為(X3,Y3)QUOTEQUOTE分別為機(jī)器人經(jīng)過拐點(diǎn)分別于隔離危險線拐角小圓弧的切點(diǎn)，圓的半徑為r，AB的長度為c，AO的長度為a，BO的長度為b,角度==QUOTE,=QUOTE，∠COD=QUOTE.求AQUOTEB的長度，設(shè)為解法如下：如上圖可得有以下關(guān)系：∵L=ACCDBDa=b=c=α=arccosβ=arccosγ=arccosθ=2π-α-β-γ∴L=2）、而對于下圖兩種情況我們不能直接采用線圓的結(jié)構(gòu)來解決，需要做簡單的變換。情況一如圖4所示：線圓結(jié)構(gòu)5.22我們假設(shè)A點(diǎn)坐標(biāo)為(X1,Y1)QUOTE，兩圓心坐標(biāo)分別為O(X2,Y2)和O(X3,Y3)這樣我們可以利用5.21中的方法，先求A到M，再求M到B，這樣把整個過程分成兩段進(jìn)行解即：L=+情形二：線圓結(jié)構(gòu)5.23這里我們依然設(shè)圓心坐標(biāo)分別為O1(x2,y2)QUOTEQUOTE和O2(x3,y3)QUOTE半徑為ACCDDE=EF=rBFbaOdbaOd這樣用D和E任意一點(diǎn)作為分割點(diǎn)都可以將上圖分割成兩個如圖2所示的線圓結(jié)構(gòu)，這樣就可以對其進(jìn)行求解。求解方法運(yùn)用MATLAB軟件進(jìn)行求解（程序見附錄）。同理有多個這樣的轉(zhuǎn)彎時，用同樣的方法都可以進(jìn)行分割計(jì)算。4）、模型一的建立假設(shè)機(jī)器人從原點(diǎn)O到達(dá)任意目標(biāo)點(diǎn)QUOTE,由5.2知機(jī)器人所走路徑一定是由直線段和圓弧組成。設(shè)有m條線段，n條圓弧，那么目標(biāo)函數(shù)可以表示為：Min=S.t=建立此模型運(yùn)用MATLAB軟件就可對原點(diǎn)到目標(biāo)點(diǎn)之間的最短路徑進(jìn)行優(yōu)化求解（見附表）。5)、模型二的建立圖6如圖6所示，建立直角坐標(biāo)系，C點(diǎn)坐標(biāo)為(0,300)，B點(diǎn)為(300,0)，坐標(biāo)系中陰影5為正方形，CB與OA相垂直，交點(diǎn)為D(x,y),陰影5的左上角的坐標(biāo)為(80,210)，半徑r在線段CB上，利用兩點(diǎn)之間的距離公式得出：（機(jī)器人行走線路與障礙物間的最近距離為10個單位），再利用求導(dǎo)公式求出半徑的最小值。a=b=c=cabdd=r六、模型的求解一、模型一的求解1、以下給出的是原點(diǎn)O到各個目標(biāo)點(diǎn)的可能的最短路徑：1）、如下圖7，解決的就是O到目標(biāo)點(diǎn)A的最短路徑問題。圖中給出了可能路徑的最短路徑（圖中標(biāo)注紅色箭頭的黑色線段），我們可以分別計(jì)算出兩條可能路徑的最短路徑的長度，然后進(jìn)行比較，求得的最小值就是O到A的最短路徑。圖72）、如下圖8，解決的是O到目標(biāo)B的最短路徑問題。圖中給出了兩條可能路徑的最短路徑（圖中標(biāo)注紅色箭頭的黑色線段），我們同樣可以分別計(jì)算出兩條可能的最短路徑，取最小值即為O到B得最短路徑。圖83）、如下圖9所示，解決的是O到C的最短路徑問題。圖中給出了兩條可能路徑的最短路徑（圖中的紅線所示），我們采用上述同樣的計(jì)算方法，分別計(jì)算出兩條可能的最短路徑，選擇最小值作為O到C得最短路徑。圖94）、如下圖10所示，解決的是原點(diǎn)的最短路徑問題。圖中給出了兩條可能路徑的最短路徑（圖中的紅線、黑線所示，有一部分路徑重疊），我們采用同樣的三中分法來計(jì)算，分別計(jì)算出兩條可能路徑的最短路徑，然后通過分析比較選出最小值作為的最短路徑。圖102、計(jì)算結(jié)果如下：1）、原點(diǎn)O到A點(diǎn)的可能的最短路徑有兩條，如圖7所示，運(yùn)用matlab編程求解O到A第一條路徑是繞過障礙物5，且走在其上方的總距離為471.0372；第二條路徑走在障礙物的下方的總距離為498.4259；比較得到O到目標(biāo)點(diǎn)A的最短路徑為：471.0372。（最短路徑詳細(xì)數(shù)據(jù)見下表）起點(diǎn)坐標(biāo)終點(diǎn)坐標(biāo)圓弧圓心坐標(biāo)圓弧半徑直線長或弧長機(jī)器人行走總時間OQUOTE線段一0,070.506,213.1406224.499444.89988圓弧一70.506,213.140676.6064,219.406680,210109.0513.6204線段二76.6064，219.4066300,300237.486847.49736機(jī)器人行走總距離471.037296.017642）、原點(diǎn)O到B點(diǎn)的可能的最短路徑有兩條，如圖8所示，一條是：機(jī)器人從障礙物6的左下頂點(diǎn)處經(jīng)過到達(dá)B點(diǎn)；另一條是走在了障礙物6的右上頂點(diǎn)到達(dá)B點(diǎn)。機(jī)器人繞過障礙物到達(dá)B點(diǎn)的這條路徑由六條線段和五段圓弧組成。直接用1）中的解發(fā)不能求解出來。于是我們采用三中分法把路徑分為如圖5.21、5.22、5.23的情況，分別求出每段線段（或圓弧）的長度然后總的相加，便可找出O到B的兩條路徑，即從障礙物6的左下頂點(diǎn)處經(jīng)過到達(dá)B點(diǎn)的路徑為877.384，從障礙物6的右上頂點(diǎn)到達(dá)B點(diǎn)的路徑為853.7001。最終求得最短路徑為853.7001。（最短路徑詳細(xì)數(shù)據(jù)見下表）起點(diǎn)坐標(biāo)終點(diǎn)坐標(biāo)圓弧圓心坐標(biāo)圓弧半徑直線長或弧長機(jī)器人行走總時間OQUOTE線段一0,050.1353,301.6396305.777761.15554圓弧一50.1353,301.6396,51.6795,305.547060,300104.23301.6932線段二51.6795,305.5470141.679,440.6795162.249832.44996圓弧二141.679,440.6795147.9621,444.7901150,435107.77563.11024線段三147.9621,444.7901222.0379,460.209975.663815.13276圓弧三222.0379,460.2099230,470220,4701013.65575.46228線段四230,470230,5306012圓弧四230,530225.4967,538.3538220,530109.88833.95532線段五225.4967,538.3538158.3538,591.646296.953619.39072圓弧五158.3538,591.6462140.6916,605.2542150,600106.14742.45896線段六140.6916,605.2542100,700111.355322.27106機(jī)器人行走總距離853.7001179.080043）、原點(diǎn)O到C點(diǎn)的可能的最短路徑同樣也是兩條，如圖9所示，我們同樣采用上述中三中分的方法把兩條路徑分割成如圖5.21、圖5.22、圖5.23的形式，再分別求每段路徑的長度然后總的相加，即走在障礙物5左上頂點(diǎn)處的距離為1098.9548，另一條走在了其下方距離為1090.8041，得到O到C的最短路徑為1090.8041。(最短路徑詳細(xì)數(shù)據(jù)見下表)起點(diǎn)坐標(biāo)終點(diǎn)坐標(biāo)圓弧圓心坐標(biāo)圓弧半徑直線長或弧長機(jī)器人行走總時間OQUOTE線段一0,0407.4009,90.2314421.900584.3801圓弧一407.4009，90.2314418.3448,107.7203410,100107.71663.08664線段二418.3448,107.7203491.6552,205.5103134.164126.83282圓弧二491.6552,205.5103508.5213,194.7666500,200103.31271.32508線段三508.5213,194.7666727.9377,525.2334388.201077.6402圓弧三727.9377,525.2334730,520720,520,106.53812.61524線段四730,520730,60079.372515.8745圓弧四730,600727.7178,606.3589720,600106.89162.75664線段五727.7178,606.3589700,64043.58908.7178機(jī)器人行走總距離1090.8041223.229024）、原點(diǎn)QUOTE的可能最短路徑有兩條，如圖10（圖中的紅線、黑線所示，有一部分路徑重疊），利用三中分法并結(jié)合線圓結(jié)構(gòu)（圖5.21、圖5.22、圖5.23），分別計(jì)算出兩條可能路徑的最短路徑，即黑色線為2716.0471，紅線距離為2772.0445，然后經(jīng)比較選出的最短路徑為2716.0471.（最短路徑詳細(xì)數(shù)據(jù)見下表)起點(diǎn)坐標(biāo)終點(diǎn)坐標(biāo)圓弧圓心坐標(biāo)圓弧半徑直線長或弧長機(jī)器人行走總時間OQUOTE線段一0,070.5060,213.1406224.499444.89988圓弧一70.5060,213.140676.6959,219.438480,210109.14593.65836線段二76.6959,219.4384286.5939,313.9504231.055746.21114圓弧二286.5939,313.9504300.7093,309.0475291.4707,305.22021015.36096.14436線段三300.7093,309.0475229.7525,532.2111235.019247.00384圓弧三229.7525,532.2111225.4967,538.3538220,530106.80982.72392線段四225.4967,538.3538144.5033,591.646296.953619.39072圓弧四144.5033,591.6462140.2475,597.7891150,600105.70622.28248線段五140.2475,597.789199.9635,688.4328101.113820.22276圓弧五99.9635,688.4328110.2329,703.9685108.14,694.19106.80982.72356線段六110.2329,703.9685269.6405,689.9935161.245232.24904圓弧六269.6405,689.6635272,689.7980270,680103.50391.40156線段七272,689.7980368，670.202097.979619.59592圓弧七368,670.2020370，670370,680102.01360.80544線段八370,670430,6706012圓弧八430,670435.5878,671.7068430,680105.92912.37164線段九435.5878,671.7068542.5741,738.2932119.163823.83276圓弧九542.5741,738.2932540,740540,730105.92912.37164線段十540,740670,74013026圓弧十670,740679.7788,732.0917670,7301011.48554.5942線段十一679.7788,732.0917700.1603,636.806297.440919.48818圓弧十一700.1603,636.8062702.7889,631.9068709.9391,638.8979106.74472.69788線段十二702.7889,631.9068727.1502,606.991034.84636.96926圓弧十二727.1502,606.9910730，600720,600107.32342.92936線段十三730，600730,5208016圓弧十三730,520711.4787，525.2334720,520106.53812.61524線段十四711.4787,525.2334492.0623,206.0822387.814477.56288圓弧十四492.0623,206.0822491.6552,205.5103500,200103.31271.32508線段十五491.6552,205.5103418.3448,94.4897133.041426.60828圓弧十五418.3448,94.4897412.1387,90.2314410,100107.71663.08664線段十六412.1387,90.23140，0421.544884.30896機(jī)器人行走總距離2716.0471564.075二、模型二的求解：圖11圖11A的坐標(biāo)為（），圓心O的坐標(biāo)（），B點(diǎn)坐標(biāo)為（），根據(jù)各線段之間的位置關(guān)系以及線段與角的位置關(guān)系，列出了以下的函數(shù)關(guān)系式：（1）（2）(3)(1)(2)取等號聯(lián)立利用導(dǎo)數(shù)求得r的極值利用MATLAB求解得=87.0711把（=87.0711）代入求得r=20;然后把x1=0,y1=0,x2=87.0711,y2=202.9289,x3=300,y3=300,r=20代入求得最短時間94.2697秒（Matlab程序見附錄模型二的求解）七、模型評價一、模型優(yōu)點(diǎn)1、本模型簡單易懂，便于實(shí)際的檢驗(yàn)與應(yīng)用，易于推廣。2、模型優(yōu)化后運(yùn)用解析幾何進(jìn)行求解，精確度比較高。3、運(yùn)用多個方案對路徑進(jìn)行優(yōu)化求解，對相對優(yōu)化的解進(jìn)行比較得到最優(yōu)解?？偟膩碚f，該模型科學(xué)合理，考慮仔細(xì)，精確度高。比如在求解過程中全面的考慮了機(jī)器人所要途經(jīng)的各種路線情況，然后依據(jù)題目要求，優(yōu)化選擇了與題意緊密相關(guān)的路線進(jìn)行求解分析，找出了符合題目要求的最佳路徑。二、模型改進(jìn)在障礙物較多時且形狀不規(guī)則時，模型還需進(jìn)一步改進(jìn)。在本題中有12個障礙物，從起點(diǎn)到終點(diǎn)的路徑是有限的，我們利用線圓結(jié)構(gòu)、枚舉法求解比較科學(xué)合理，求得結(jié)果較符合提意要求。當(dāng)障礙物增多時，我們可以考慮采用另為一種求解方式，如dijkstra算法等。八、模型推廣該模型科學(xué)合理，首先全面地考慮了機(jī)器人在到達(dá)各目標(biāo)點(diǎn)時所要行走的所有路線，然后根據(jù)題目要求，選擇與題目緊密關(guān)聯(lián)的路線進(jìn)行分析求解，最終求出符合題意要求的最短路徑。求解過程中簡單易懂，便于實(shí)際問題的求解與應(yīng)用，易于推廣。雖然本模型是為計(jì)算機(jī)器人行走的最短路問題而建立的，但根據(jù)本模型的特點(diǎn)，仍然可以運(yùn)用到現(xiàn)實(shí)生活中的最短路問題的求解。如：油氣管道的布置問題、城市交通的規(guī)劃問題、排水系統(tǒng)的規(guī)劃問題等。九、參考文獻(xiàn)[1]韓忠庚，數(shù)學(xué)建模使用教程，北京：高等教育出版社，2012.[2]尤承業(yè)，解析幾何，北京：北京大學(xué)出版社，2004.[3]黃玉清，梁靚，機(jī)器人導(dǎo)航系統(tǒng)中的路徑規(guī)劃算法[J],微計(jì)算機(jī)信息，2006.7：[4]王沫然，MATLAB與科學(xué)計(jì)算，北京：電子工業(yè)出版社，2004.[5]胡海星，RPG游戲中精靈的移動問題，雜志《程序員》，2011。[6]邦迪，圖論及其應(yīng)用，西安，西安科學(xué)出版社，1984十、附錄附表一計(jì)算線圓結(jié)構(gòu)模型5.21symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=300;y3=300;r=10;（輸入對應(yīng)已知數(shù)據(jù)）a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1（線圓結(jié)構(gòu)模型5.22分解為兩個模型5.21進(jìn)行計(jì)算）計(jì)算線圓結(jié)構(gòu)模型5.23Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=0;y1=0;x2=50;y2=40;x3=20;y3=23;x4=30;y4=20;r=10;（輸入對應(yīng)已知數(shù)據(jù)）a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)路徑O→A第一條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=300;y3=300;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=471.0372路徑O→A第二條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=230;y2=60;x3=300;y3=300;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=498.4259路徑O→B第一條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=157.5;y3=255;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=321.9278Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=157.5;y1=255;x2=235;y2=300;x3=220;y3=530;x4=185;y4=565;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=389.4767symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=185;y1=565;x2=150;y2=600;x3=100;y3=700;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=165.9795路徑O→B第二條：Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=0;y1=0;x2=60;y2=300;x3=150;y3=435;x4=185;y4=452.5;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=517.8680Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=185;y1=452.5;x2=220;y2=470;x3=220;y3=530;x4=185;y4=565;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=169.8526symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=185;y1=565;x2=150;y2=600;x3=100;y3=700;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=165.9795路徑O→C的第一條：Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=0;y1=0;x2=80;y2=210;x3=400;y3=330;x4=416.7;y4=343.3;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=602.0599Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=416.7;y1=343.3;x2=550;y2=450;x3=640;y3=520;x4=730;y4=560;r1=80;r2=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r1^2);b1=acos(r1/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r1*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r2/b);d2=2*pi-a2-b2-pi/2;n=r2*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r1^2)+r1*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r2*d2+sqrt(b^2-r2^2)L3=406.4143symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=730;y1=560;x2=720;y2=600;x3=700;y3=640;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=90.4806路徑O→C的第二條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=410;y2=100;x3=455;y3=150;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=496.1377>>symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=455;y1=150;x2=500;y2=200;x3=610;y3=360;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=263.7405>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=610;y1=360;x2=720;y2=520;x3=720;y3=600;x4=700;y4=640;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=330.9259路徑O→A→B→C→O的第一條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=185.7354;y3=257.6101;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=349.1733Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=185.7354;y1=257.6101;x2=291.4707;y2=305.2202;x3=220;y3=530;x4=185;y4=565;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=422.1021Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=185;y1=565;x2=150;y2=600;x3=108.14;y3=694.19;x4=189.07;y4=687.095;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=259.8770symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=189.07;y1=687.095;x2=270;y2=680;x3=320;y3=680;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=133.7345>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=320;y1=680;x2=370;y2=680;x3=430;y3=680;x4=485;y4=705;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=176.5144>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=485;y1=705;x2=540;y2=730;x3=670;y3=730;x4=689.9696;y4=684.4490;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=257.8321>>symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=689.9696;y1=684.4490;x2=709.9391;y2=638.8979;x3=714.9696;y3=619.4490;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=74.9790>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=714.9696;y1=619.4490;x2=720;y2=600;x3=720;y3=520;x4=610;y4=360;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=305.6100>>symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=610;y1=360;x2=500;y2=200;x3=455;y3=150;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=236.7405symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=455;y1=150;x2=410;y2=100;x3=0;y3=0;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=496.1377路徑O→A→B→C→O的第二條：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=190;y3=255;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=352.3992>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=190;y1=255;x2=300;y2=300;x3=220;y3=530;x4=185;y4=565;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=432.9754Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=185;y1=565;x2=150;y2=600;x3=100;y3=700;x4=167.5;y4=695;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=255.0408Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=167.5;y1=695;x2=235;y2=690;x3=500;y3=600;x4=570;y4=560;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=433.9867>>Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=570;y1=560;x2=640;y2=520;x3=720;y3=520;x4=730;y4=560;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=222.1430symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=730;y1=560;x2=720;y2=600;x3=700;y3=640;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=90.4806Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=700;y1=640;x2=720;y2=600;x3=720;y3=520;x4=610;y4=360;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;m=r*d1;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;n=r*d2;h=sqrt(b^2-r^2);L3=sqrt(a^2-r^2)+r*d1+sqrt((x3-x2)^2+(y3-y2)^2)+r*d2+sqrt(b^2-r^2)L3=330.9259>>symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=610;y1=360;x2=500;y2=200;x3=455;y3=150;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=263.7405>>symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=455;y1=150;x2=410;y2=100;x3=0;y3=0;r=10;a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=sqrt(b^2-r^2)+sqrt(a^2-r^2)+r*d1L1=496.1377計(jì)算線圓結(jié)構(gòu)模型5.21切點(diǎn)坐標(biāo)symsxx1x2yy1y2rx1=0;x2=80;y1=0;y2=210;r=10;（輸入已知對應(yīng)數(shù)據(jù)）(y-y2)*(y-y1)*(x-x2)^(-1)*(x-x1)^(-1)+1a='(x-x2)^2+(y-y2)^2-r^2=0';(x-x2)^2+(y-y2)^2-r^2b='ans=0';s=solve(a,b)double(s.x(1))double(s.x(2))double(s.y(1))double(s.y(2))計(jì)算線圓結(jié)構(gòu)模型5.23的切點(diǎn)symsxx1x2x3yy1y2y3rx1=0;y1=0;x2=80;y2=210;x3=300;y3=300;r=10;（輸入已知對應(yīng)數(shù)據(jù)）>>(y-y2)/(x-x2)*(y3-y2)/(x3-x2)+1ans=9/22*(y-210)/(x-80)+1>>a='9/22*(y-210)/(x-80)+1=0';>>(x-x2)^2+(y-y2)^2-r^2ans=(x-80)^2+(y-210)^2-100>>b='(x-80)^2+(y-210)^2-100=0';s=solve(a,b)s=x:[2x1sym]y:[2x1sym]>>double(s.x(1))ans=83.7863double(s.x(2))double(s.y(1))計(jì)算線圓結(jié)構(gòu)5.21各個對應(yīng)的弧長：symsabcx1x2x3y1y2y3a1b1c1d1rL1x1=0;y1=0;x2=80;y2=210;x3=300;y3=300;r=10;%輸入已知的數(shù)據(jù)a=sqrt((x2-x1)^2+(y2-y1)^2);b=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((b^2+a^2-c^2)/(2*b*a));b1=acos(r/b);c1=acos(r/a);d1=2*pi-a1-b1-c1;L1=r*d1%求對應(yīng)的弧長計(jì)算線圓結(jié)構(gòu)5.23各對應(yīng)弧長Symsx1x2x3x4y1y2y3y4abcdfghmna1a2b1b2d1d2L3x1=190;y1=255;x2=308.7976;y2=295.2457;x3=220;y3=530;x4=185;y4=565;r=10;%輸入已知數(shù)據(jù)a=sqrt((x2-x1)^2+(y2-y1)^2);g=sqrt(a^2-r^2);b1=acos(r/a);f=sqrt((x3-x2)^2+(y3-y2)^2);c=sqrt((x3-x1)^2+(y3-y1)^2);a1=acos((a^2+f^2-c^2)/(2*a*f));d1=2*pi-a1-b1-pi/2;b=sqrt((x4-x3)^2+(y4-y3)^2);d=sqrt((x4-x2)^2+(y4-y2)^2);a2=acos((f^2+b^2-d^2)/(2*f*b));b2=acos(r/b);d2=2*pi-a2-b2-pi/2;L3=r*d1或L3=r*d2%對應(yīng)弧1的長度和對應(yīng)弧2的長度計(jì)算路徑經(jīng)過A點(diǎn)時點(diǎn)A所在圓弧的圓心symsxyk1k2>>k1=(y-530)/(x-220);>>k2=(y-300)/(x-300);>>(k2-k1)/(1+k1*k2)-sqrt(3)/2ans=((y-300)/(x-300)-(y-530)/(x-220))/(1+(y-530)/(x-220)*(y-300)/(x-300))-