Mathematica程序计算两线段之间的最短距离

2022-08-02 14:21:11 文档大全网 [ 字体：小中大 ] [ 阅读： ]

【#文档大全网# 导语】以下是®文档大全网的小编为您整理的《Mathematica程序计算两线段之间的最短距离》，欢迎阅读！
短距离,线段,Mathematica,之间,计算
Mathematica程序计算两线段之间的最短距离

(*输入四个点坐标*)

(*将坐标输成7/2形式可以得到准确解,输成3.5只能得到有一定精确度的数值解.*)

(*如果两线段分别为AB,CD,按照以下规则输入坐标

:{xA,yA,xB,yB,xC,yC,xD,yD}

或

者

{{xA,yA},{xB,yB},{xC,yC},{xD,yD}}或者两种混和输入,必须保证按照顺序输入*)

zuobiao = {{0, 0}, {13, 9}, {6, 6}, {0, 8}}; zuobiao = Partition[Flatten[zuobiao], 2];

(*下面进行坐标旋转,目的是让两条直线均不与坐标轴平行,省去以后很多分情况的麻烦,一劳永逸*) xuanzj = {0, \[Pi]/6, \[Pi]/4}; brr = {}; err = {}; juli = {}; frr = {}; chuizuf =.;

For[i = 1, i <= Length[xuanzj], i++, For[j = 1, j <= Length[zuobiao], j++, brr = Append[

brr, {zuobiao[[j]][[1]]*Cos[xuanzj[[i]]] +

zuobiao[[j]][[2]]*Sin[xuanzj[[i]]], -zuobiao[[j]][[1]]* Sin[xuanzj[[i]]] + zuobiao[[j]][[2]]*Cos[xuanzj[[i]]]}]; ];

If[brr[[1]][[1]] != brr[[2]][[1]] &&

brr[[3]][[1]] != brr[[4]][[1]] && brr[[1]][[2]] != brr[[2]][[2]] && brr[[3]][[2]] != brr[[4]][[2]], \[Alpha] = -xuanzj[[i]]; Break[], brr = {} ]; ];

(*下面判断两直线是否平行*) pingxing =.;

If[(zuobiao[[1]][[1]] - zuobiao[[2]][[1]])*(zuobiao[[3]][[2]] - zuobiao[[4]][[2]]) == (zuobiao[[1]][[2]] -

zuobiao[[2]][[2]])*(zuobiao[[3]][[1]] - zuobiao[[4]][[1]]), If[(zuobiao[[1]][[1]] - zuobiao[[3]][[1]])*(zuobiao[[3]][[2]] - zuobiao[[4]][[2]]) == (zuobiao[[1]][[2]] -

zuobiao[[3]][[2]])*(zuobiao[[3]][[1]] - zuobiao[[4]][[1]]), pingxing = 2(*直线重合*), pingxing = 1(*直线平行*)

],

pingxing = 0(*直线相交*) ];

(*下面是两直线重合时,先判断两条线段是否有重合部分,再计算距离*)

If[pingxing == 2, drr = Transpose[brr];

If[Partition[Sort[drr[[1]], Less],

2] == {Sort[{drr[[1]][[1]], drr[[1]][[2]]}, Less], Sort[{drr[[1]][[3]], drr[[1]][[4]]}, Less]} || Partition[Sort[drr[[1]], Less],

2] == {Sort[{drr[[1]][[3]], drr[[1]][[4]]}, Less], Sort[{drr[[1]][[1]], drr[[1]][[2]]}, Less]}, juli = Simplify[

Norm[{Sort[drr[[1]], Less][[2]] - Sort[drr[[1]], Less][[3]], Sort[drr[[2]], Less][[2]] - Sort[drr[[2]], Less][[3]]}]]; frr = {{{Sort[drr[[1]], Less][[2]],

Sort[drr[[2]], Less][[2]]}, {Sort[drr[[1]], Less][[3]], Sort[drr[[2]], Less][[3]]}}}; Print["准确解"]; Print[juli]; Print["数值解"];

Print[juli // N],

Print["两线段有一部分重合,距离为0"] ] ]

(*下面是两直线相交时,先计算两直线交点坐标,再判断两条线段是否相交*)

(*下面定义一个函数jiaodian,用来计算两直线交点坐标*) jiaodian[zuobiao_] := Module[{arr},

arr = Flatten[zuobiao];

{-(-arr[[3]] arr[[5]] arr[[2]] + arr[[3]] arr[[7]] arr[[2]] + arr[[1]] arr[[5]] arr[[4]] - arr[[1]] arr[[7]] arr[[4]] + arr[[1]] arr[[7]] arr[[6]] - arr[[3]] arr[[7]] arr[[6]] - arr[[1]] arr[[5]] arr[[8]] +

arr[[3]] arr[[5]] arr[[8]])/(arr[[5]] arr[[2]] - arr[[7]] arr[[2]] - arr[[5]] arr[[4]] + arr[[7]] arr[[4]] - arr[[1]] arr[[6]] + arr[[3]] arr[[6]] + arr[[1]] arr[[8]] - arr[[3]] arr[[8]]), -(-arr[[3]] arr[[2]] arr[[6]] + arr[[7]] arr[[2]] arr[[6]] + arr[[1]] arr[[4]] arr[[6]] - arr[[7]] arr[[4]] arr[[6]] + arr[[3]] arr[[2]] arr[[8]] - arr[[5]] arr[[2]] arr[[8]] - arr[[1]] arr[[4]] arr[[8]] + arr[[5]] arr[[4]] arr[[8]])/(arr[[5]] arr[[2]] -

arr[[7]] arr[[2]] - arr[[5]] arr[[4]] + arr[[7]] arr[[4]] - arr[[1]] arr[[6]] + arr[[3]] arr[[6]] + arr[[1]] arr[[8]] - arr[[3]] arr[[8]])} ];

(*下面是两直线相交时,先计算两直线交点坐标,再判断两线段是否相交,如果相交,输出"两线段相交,最短距离为0",否则继续接着的步骤*) xiangjiao =.; If[pingxing == 0,

brr = Append[brr, jiaodian[brr]];(*计算两直线交点坐标*) If[(brr[[1]][[1]] - brr[[5]][[1]])*(brr[[5]][[1]] - brr[[2]][[1]]) >= 0 && (brr[[3]][[1]] - brr[[5]][[1]])*(brr[[5]][[1]] - brr[[4]][[1]]) >= 0, Print["两线段相交,最短距离为0"]; xiangjiao = 1, xiangjiao = 0 ]; ];

(*下面先定义一个函数chuizu,用来计算点到直线的垂足坐标*) chuizu[zuobiao_] := Module[{arr},

arr = Flatten[zuobiao];

{-(-arr[[1]] arr[[3]]^2 + 2 arr[[1]] arr[[3]] arr[[5]] - arr[[1]] arr[[5]]^2 - arr[[3]] arr[[2]] arr[[4]] + arr[[5]] arr[[2]] arr[[4]] - arr[[5]] arr[[4]]^2 + arr[[3]] arr[[2]] arr[[6]] - arr[[5]] arr[[2]] arr[[6]] + arr[[3]] arr[[4]] arr[[6]] + arr[[5]] arr[[4]] arr[[6]] - arr[[3]] arr[[6]]^2)/(arr[[3]]^2 - 2 arr[[3]] arr[[5]] + arr[[5]]^2 + arr[[4]]^2 - 2 arr[[4]] arr[[6]] + arr[[6]]^2), -(-arr[[1]] arr[[3]] arr[[4]] +

arr[[1]] arr[[5]] arr[[4]] + arr[[3]] arr[[5]] arr[[4]] - arr[[5]]^2 arr[[4]] - arr[[2]] arr[[4]]^2 + arr[[1]] arr[[3]] arr[[6]] - arr[[3]]^2 arr[[6]] - arr[[1]] arr[[5]] arr[[6]] + arr[[3]] arr[[5]] arr[[6]] + 2 arr[[2]] arr[[4]] arr[[6]] -

arr[[2]] arr[[6]]^2)/(arr[[3]]^2 - 2 arr[[3]] arr[[5]] + arr[[5]]^2 + arr[[4]]^2 - 2 arr[[4]] arr[[6]] + arr[[6]]^2)} ];

(*下面是两线段线不相交时,计算各个端点到所对线段的最短距离*) crr = {{brr[[1]], brr[[3]], brr[[4]]}, {brr[[2]], brr[[3]], brr[[4]]}, {brr[[3]], brr[[1]], brr[[2]]}, {brr[[4]], brr[[1]], brr[[2]]}}; If[xiangjiao == 0, For[i = 1, i <= 4, i++,

chuizuf = chuizu[crr[[i]]];

If[(crr[[i]][[2]][[1]] - chuizuf[[1]])/(crr[[i]][[2]][[1]] - crr[[i]][[3]][[1]]) >=

0 && (crr[[i]][[3]][[1]] - chuizuf[[1]])/(crr[[i]][[3]][[1]] - crr[[i]][[2]][[1]]) >= 0,

frr = Append[frr, {crr[[i]][[1]], chuizuf}]; juli =

Append[juli, Simplify[Norm[crr[[i]][[1]] - chuizuf]]](*A的垂足F在CD内,

取线段AF距离*) ];

If[(crr[[i]][[2]][[1]] - chuizuf[[1]])/(crr[[i]][[2]][[1]] - crr[[i]][[3]][[1]]) <

0 && (crr[[i]][[3]][[1]] - chuizuf[[1]])/(crr[[i]][[3]][[1]] - crr[[i]][[2]][[1]]) >= 0,

frr = Append[frr, {crr[[i]][[1]], crr[[i]][[2]]}]; juli = Append[juli,

Simplify[Norm[crr[[i]][[1]] - crr[[i]][[2]]]]](*A的垂足F在C的外侧,

取线段AC距离*) ];

If[(crr[[i]][[2]][[1]] - chuizuf[[1]])/(crr[[i]][[2]][[1]] - crr[[i]][[3]][[1]]) >=

0 && (crr[[i]][[3]][[1]] - chuizuf[[1]])/(crr[[i]][[3]][[1]] - crr[[i]][[2]][[1]]) < 0,

frr = Append[frr, {crr[[i]][[1]], crr[[i]][[3]]}]; juli = Append[juli,

Simplify[Norm[crr[[i]][[1]] - crr[[i]][[3]]]]](*A的垂足F在D的外侧,

取线段AD距离*) ]; ];

Print["准确解"]; Print[Sort[juli, Less][[1]]]; Print["数值解"]; Print[Sort[juli // N][[1]]]; ];

(*下面是作图*)

For[j = 1, j <= Length[brr], j++, err = Append[

err, {brr[[j]][[1]]*Cos[\[Alpha]] +

brr[[j]][[2]]*Sin[\[Alpha]], -brr[[j]][[1]]*Sin[\[Alpha]] +

brr[[j]][[2]]*Cos[\[Alpha]]}]; ];

err = Partition[err, 2]; If[frr != {}, grr = Table[

Norm[frr[[i]][[1]] - frr[[i]][[2]]] // N, {i, 1, Length[frr]}]; For[i = 1, i <= Length[grr], i++, If[grr[[i]] < grr[[1]], grr[[1]] = grr[[i]]; frr[[1]] = frr[[i]] ] ];

err = Append[err, frr[[1]]]; ];

Print["下图中蓝色为已知的两条直线,红色为连接两条已知线段的最短线段"];

ListLinePlot[err, AspectRatio -> Automatic,

PlotStyle -> {Blue, Blue, Red}, AxesOrigin -> {0, 0}, PlotRange -> All]