四边形的定值轨迹问题

数学星空

已知在一个平面上的四点$A,B,C,D $有两条线段$AB=2,CD=1$
$AB$线段固定,$CD$为动线段且$C,D$两点位于$AB$的同一侧
$CD$的中点为$P$,求同时满足下面两个条件的$P$点的轨迹?
(1)$AB+BC+CD+DA=L$(周长定值)
(2)$S(ABCD)=S$(面积为定值)

数学星空

设$A(-1,0), B(1,0),C(r1*cos(t1),r1*sin(t2)),D(r2*cos(t2),r2*sin(t2)),P(x,y),$则

$sqrt((r1*cos(t1)-1)^2+(r1*sin(t1))^2)+sqrt((r2*cos(t2)+1)^2+(r2*sin(t2))^2)=L-3$

$r1*sin(t1)+r2*sin(t2)+r1*r2*sin(t2-t1)=2*S$

$r1^2+r2^2-2*r1*r2*cos(t2-t1)=1$

$x=1/2*(r1*cos(t1)+r2*cos(t2))$

$y=1/2*(r1*sin(t1)+r2*sin(t2))$

可以消元$r1,r2,t1,t2$得到关于$x,y$的函数关系.

数学星空

有谁有兴趣取$S=1.4,L=5$,利用数值计算绘出$P(x,y)$的轨迹?

数学星空

我们对3#方程去根号展开有:
$r1^2+r2^2-2*r1*r2*cos(t2)*cos(t1)-2*r1*r2*sin(t2)*sin(t1)-1=0$

$r2*sin(t2)+r1*sin(t1)+r1*r2*sin(t2)*cos(t1)-r1*r2*cos(t2)*sin(t1)-2*s=0$

$-4*r1*cos(t1)*L^2+4*r2*cos(t2)*L^2-4*r1^2*cos(t1)^2-r1^4-4*r1*cos(t1)*r2^2+2*r1^2*r2^2-r2^4+4*r1^3*cos(t1)-$
$8*r1*r2*cos(t2)*cos(t1)+4*L^2+4*r1^2*r2*cos(t2)+2*r1^2*L^2+2*r2^2*L^2-4*r2^3*cos(t2)-4*r2^2*cos(t2)^2-L^4=0$

将$s=1.4,L=5-3=2$代入,并利用数值计算
并得到如下图形
线段$CD$ 的运动轨迹及中点$P$的轨迹


$C,D,P$点的运动轨迹


中点$P$的运动轨迹

数学星空

奇怪的是:将5#的计算结果代入3#的三个方程中
得到的第二个方程
sqrt((r1*cos(t1)-1)^2+(r1*sin(t1))^2)+sqrt((r2*cos(t2)+1)^2+(r2*sin(t2))^2)-L(共取了53个样本点)
{1.549508772}, {1.422145287}, {1.305716755}, {1.199647877}, {1.103555632}, {1.017243431}, {.940695637}, {.874065438},
{.817644658}, {.771799798}, {.736859593}, {.712954817}, {.699846292}, {.696815312}, {.702688135}, {.715999496},
{.735218604}, {.758940594}, {.785992716}, {.815460500}, {.846664919}, {.879119988}, {.912488716}, {.946545304},
{.981145591}, {1.016205130}, {1.051683401}, {1.087572728}, {1.123890636}, {1.160674728}, {1.197979430}, {1.235874122},
{1.274442389}, {1.313782178}, {1.354006795}, {1.395246758}, {1.437652564}, {1.481398584}, {1.526688364}, {1.573761834},
{1.622905150}, {1.674464328}, {1.728864527}, {1.786637972}, {1.848465754}, {1.915242709}, {1.988183014}, {2.069002397},
{2.160257257}, {2.266043677}, {2.393665568}, {2.558662429}, {2.808970412}
所有的结果并不为0,计算误差很大.
而代入5#的第三个方程
-4*r1*cos(t1)*L^2+4*r2*cos(t2)*L^2-4*r1^2*cos(t1)^2-r1^4-4*r1*cos(t1)*r2^2+2*r1^2*r2^2-r2^4+4*r1^3*cos(t1)-8*r1*r2*cos(t2)*cos(t1)+
4*L^2+4*r1^2*r2*cos(t2)+2*r1^2*L^2+2*r2^2*L^2-4*r2^3*cos(t2)-4*r2^2*cos(t2)^2-L^4 =0
注:代入其它方程均近似于0,只有这个含有根号的方程例外....

我利用5#方程算出的数值解为(共53组解)

{L = 2, m = 1/100, n = .2937046429, r1 = 1.774296268, r2 = 1.880375024, s = 1.4, t1 = 0.1999933337e-1, t2 = .5713425552},
{L = 2, m = 1/50, n = .3142603636, r1 = 1.709147912, r2 = 1.829803199, s = 1.4, t1 = 0.3999466795e-1, t2 = .6089756226},
{L = 2, m = 3/100, n = .3346241373, r1 = 1.648302020, r2 = 1.785080345, s = 1.4, t1 = 0.5998200972e-1, t2 = .6458236557},
{L = 2, m = 1/25, n = .3547234060, r1 = 1.591287398, r2 = 1.745829143, s = 1.4, t1 = 0.7995737424e-1, t2 = .6817530809},
{L = 2, m = 1/20, n = .3744670602, r1 = 1.537698908, r2 = 1.711756053, s = 1.4, t1 = 0.9991679144e-1, t2 = .7166067071},
{L = 2, m = 3/50, n = .3937409840, r1 = 1.487193274, r2 = 1.682644582, s = 1.4, t1 = .1198563102, t2 = .7501981117},
{L = 2, m = 7/100, n = .4124032970, r1 = 1.439489088, r2 = 1.658348216, s = 1.4, t1 = .1397720033, t2 = .7823058708},
{L = 2, m = 2/25, n = .4302803500, r1 = 1.394371264, r2 = 1.638780154, s = 1.4, t1 = .1596599714, t2 = .8126692726},
{L = 2, m = 9/100, n = .4471656900, r1 = 1.351699479, r2 = 1.623895750, s = 1.4, t1 = .1795163484, t2 = .8409888269},
{L = 2, m = 1/10, n = .4628257426, r1 = 1.311418324, r2 = 1.613662693, s = 1.4, t1 = .1993373050, t2 = .8669369553},
{L = 2, m = 11/100, n = .4770169864, r1 = 1.273563476, r2 = 1.608015674, s = 1.4, t1 = .2191190535, t2 = .8901854592},
{L = 2, m = 3/25, n = .4895177195, r1 = 1.238254416, r2 = 1.606799237, s = 1.4, t1 = .2388578520, t2 = .9104533487},
{L = 2, m = 13/100, n = .5001706708, r1 = 1.205664311, r2 = 1.609715396, s = 1.4, t1 = .2585500081, t2 = .9275682727},
{L = 2, m = 7/50, n = .5089221372, r1 = 1.175966995, r2 = 1.616303011, s = 1.4, t1 = .2781918830, t2 = .9415196317},
{L = 2, m = 3/20, n = .5158381354, r1 = 1.149277452, r2 = 1.625968995, s = 1.4, t1 = .2977798952, t2 = .9524753729},
{L = 2, m = 4/25, n = .5210876056, r1 = 1.125611565, r2 = 1.638063490, s = 1.4, t1 = .3173105244, t2 = .9607500467},
{L = 2, m = 17/100, n = .5249016822, r1 = 1.104880633, r2 = 1.651965046, s = 1.4, t1 = .3367803143, t2 = .9667398485},
{L = 2, m = 9/50, n = .5275290714, r1 = 1.086915239, r2 = 1.667141961, s = 1.4, t1 = .3561858765, t2 = .9708551022},
{L = 2, m = 19/100, n = .5292030779, r1 = 1.071500824, r2 = 1.683177517, s = 1.4, t1 = .3755238931, t2 = .9734724323},
{L = 2, m = 1/5, n = .5301245517, r1 = 1.058409709, r2 = 1.699766682, s = 1.4, t1 = .3947911197, t2 = .9749116248},
{L = 2, m = 21/100, n = .5304572937, r1 = 1.047423045, r2 = 1.716698256, s = 1.4, t1 = .4139843884, t2 = .9754310437},
{L = 2, m = 11/50, n = .5303305230, r1 = 1.038342817, r2 = 1.733833437, s = 1.4, t1 = .4331006099, t2 = .9752331675},
{L = 2, m = 23/100, n = .5298442086, r1 = 1.030996693, r2 = 1.751086505, s = 1.4, t1 = .4521367760, t2 = .9744738909},
{L = 2, m = 6/25, n = .5290748674, r1 = 1.025238613, r2 = 1.768409560, s = 1.4, t1 = .4710899614, t2 = .9732721026},
{L = 2, m = 1/4, n = .5280807514, r1 = 1.020947244, r2 = 1.785781288, s = 1.4, t1 = .4899573263, t2 = .9717180608},
{L = 2, m = 13/50, n = .5269060948, r1 = 1.018023587, r2 = 1.803199044, s = 1.4, t1 = .5087361171, t2 = .9698801456},
{L = 2, m = 27/100, n = .5255844316, r1 = 1.016388458, r2 = 1.820673430, s = 1.4, t1 = .5274236690, t2 = .9678100888},
{L = 2, m = 7/25, n = .5241411235, r1 = 1.015980184, r2 = 1.838224688, s = 1.4, t1 = .5460174061, t2 = .9655469303},
{L = 2, m = 29/100, n = .5225952587, r1 = 1.016752669, r2 = 1.855880366, s = 1.4, t1 = .5645148440, t2 = .9631199780},
{L = 2, m = 3/10, n = .5209610681, r1 = 1.018673862, r2 = 1.873673894, s = 1.4, t1 = .5829135890, t2 = .9605510059},
{L = 2, m = 31/100, n = .5192489757, r1 = 1.021724648, r2 = 1.891643814, s = 1.4, t1 = .6012113401, t2 = .9578558773},
{L = 2, m = 8/25, n = .5174663751, r1 = 1.025898119, r2 = 1.909833496, s = 1.4, t1 = .6194058890, t2 = .9550457448},
{L = 2, m = 33/100, n = .5156181974, r1 = 1.031199249, r2 = 1.928291251, s = 1.4, t1 = .6374951209, t2 = .9521279125},
{L = 2, m = 17/50, n = .5137073208, r1 = 1.037644944, r2 = 1.947070770, s = 1.4, t1 = .6554770135, t2 = .9491064727},
{L = 2, m = 7/20, n = .5117348557, r1 = 1.045264500, r2 = 1.966231887, s = 1.4, t1 = .6733496388, t2 = .9459827231},
{L = 2, m = 9/25, n = .5097003288, r1 = 1.054100519, r2 = 1.985841688, s = 1.4, t1 = .6911111612, t2 = .9427554475},
{L = 2, m = 37/100, n = .5076017825, r1 = 1.064210349, r2 = 2.005976011, s = 1.4, t1 = .7087598382, t2 = .9394210495},
{L = 2, m = 19/50, n = .5054357939, r1 = 1.075668176, r2 = 2.026721451, s = 1.4, t1 = .7262940199, t2 = .9359735637},
{L = 2, m = 39/100, n = .5031974162, r1 = 1.088567963, r2 = 2.048178006, s = 1.4, t1 = .7437121476, t2 = .9324045376},
{L = 2, m = 2/5, n = .5008800325, r1 = 1.103027520, r2 = 2.070462614, s = 1.4, t1 = .7610127542, t2 = .9287027743},
{L = 2, m = 41/100, n = .4984751045, r1 = 1.119194134, r2 = 2.093713940, s = 1.4, t1 = .7781944621, t2 = .9248538969},
{L = 2, m = 21/50, n = .4959717830, r1 = 1.137252456, r2 = 2.118098960, s = 1.4, t1 = .7952559831, t2 = .9208396910},
{L = 2, m = 43/100, n = .4933563252, r1 = 1.157435716, r2 = 2.143822238, s = 1.4, t1 = .8121961166, t2 = .9166371149},
{L = 2, m = 11/25, n = .4906112250, r1 = 1.180042031, r2 = 2.171139346, s = 1.4, t1 = .8290137491, t2 = .9122168358},
{L = 2, m = 9/20, n = .4877138949, r1 = 1.205458850, r2 = 2.200376918, s = 1.4, t1 = .8457078522, t2 = .9075410083},
{L = 2, m = 23/50, n = .4846346118, r1 = 1.234200944, r2 = 2.231963770, s = 1.4, t1 = .8622774814, t2 = .9025598228},
{L = 2, m = 47/100, n = .4813331777, r1 = 1.266972207, r2 = 2.266481547, s = 1.4, t1 = .8787217746, t2 = .8972058869},
{L = 2, m = 12/25, n = .4777531869, r1 = 1.304772214, r2 = 2.304752124, s = 1.4, t1 = .8950399503, t2 = .8913845830},
{L = 2, m = 49/100, n = .4738114264, r1 = 1.349094291, r2 = 2.348000308, s = 1.4, t1 = .9112313064, t2 = .8849562406},
{L = 2, m = 1/2, n = .4693762054, r1 = 1.402333152, r2 = 2.398189323, s = 1.4, t1 = .9272952180, t2 = .8776996692},
{L = 2, m = 51/100, n = .4642161116, r1 = 1.468755976, r2 = 2.458821709, s = 1.4, t1 = .9432311357, t2 = .8692259080},
{L = 2, m = 13/25, n = .4578471358, r1 = 1.557423390, r2 = 2.537354230, s = 1.4, t1 = .9590385840, t2 = .8587208238},
{L = 2, m = 53/100, n = .4488069039, r1 = 1.696193334, r2 = 2.656808538, s = 1.4, t1 = .9747171589, t2 = .8437226072}
注:sin(t1)=2*m/(1+m^2),sin(t2)=2*n/(1+n^2)

hujunhua

设A(-1,0), B(1,0),C(r1*cos(t1),r1*sin(t2)),D(r2*cos(t2),r2*sin(t2)),P(x,y),则 ...
数学星空 发表于 2012-3-30 21:38

我将图形放大了一倍，AB=4，CD=2
设A(-2,0), B(2,0), P(x,y), CD的倾角为t, 则C(x+cost, y+sint), D(x-cost, y-sint)

数学星空

根据楼上的巧妙设参数简化了很多运算
我们设$A(-2,0), B(2,0), P(x,y), CD$的倾角为$t$, 则$C(x+cost, y+sint), D(x-cost, y-sint)$
$2*y-x*sin(t)+cos(t)*y=s$
$sqrt((x+cos(t)-2)^2+(y+sin(t))^2)+sqrt((x-cos(t)+2)^2+(y-sin(t))^2)=L$很容易消元得到
$4992*y^4*L^2+6912*x^4*y^2+6912*x^2*y^4-1152*x^4*L^2+320*y^4*L^4+384*y^6*L^2+16*y^6*L^4-8*y^4*L^6+$
$16*x^6*L^4-8*x^4*L^6-12544*x^2*L^2*y^2-896*y^2*L^2*x^4+640*y^2*L^4*x^2+128*y^4*L^2*x^2-640*x^6*L^2+$
$320*x^4*L^4+48*y^2*L^4*x^4-16*y^2*L^6*x^2+48*y^4*L^4*x^2+2304*y^6+2304*x^6+2704*y^2*L^4-104*y^2*L^6+$
$y^2*L^8+144*x^2*L^4-40*x^2*L^6+x^2*L^8-8192*y^3*L^2*s-6144*x^4*y*s-512*y^5*L^2*s+3712*y^2*L^2*s^2-1408*x^2*s^2*L^2+$
$128*x^4*s^2*L^2+128*y^4*L^2*s^2-1664*y*L^4*s+1536*s^2*x^4+256*s^2*L^4-6144*y^5*s+5632*y^4*s^2-2048*y^3*s^3+$
$256*s^4*y^2+256*x^2*s^4+8192*x^2*L^2*y*s-1024*x^2*y^3*s*L^2+256*x^2*y^2*s^2*L^2-512*y*x^4*L^2*s+32*L^6*y*s-$
$2048*x^2*y*s^3-12288*x^2*y^3*s+7168*x^2*y^2*s^2-512*s^3*L^2*y-32*s^2*L^4*y^2-32*s^2*L^4*x^2=0$

取$s=3*sqrt(3),L=4$代入得到

$1769472-86016*x^2*y^3*sqrt(3)-559104*y^3*sqrt(3)-1548288*y*sqrt(3)-43008*y^5*sqrt(3)-43008*x^4*y*sqrt(3)+$
$227328*x^2*y*sqrt(3)+336384*y^4-704256*x^2+1900800*y^2+201728*x^2*y^2+127488*x^4+4864*x^4*y^2+$
$21248*x^2*y^4+12544*y^6-3840*x^6=0$

画图:

取局部坐标放大$-5<x<0,0<y<1.7$,画图



通过数值计算得到(检验一下上面的符号计算结果的正确性)
四边形$ABCD$的运行轨迹

四边形$ABCD$及$P$的运行轨迹

$P$的运行轨迹

zeroieme

线太多有点昏