顶点在圆锥曲线上的正三角形中心的轨迹

数学星空

仿楼上wayne的相关参数：\(e =\frac{c}{a}, p = \frac{a^2}{c}\)

圆锥曲线：\((e^2p + x)^2 + y^2 - e^2(x + p)^2=0\)

圆锥曲线内接正三角形ABC三个顶点坐标\(A[x_1,y_1],B[x_2,y_2],C[x_3,y_3]\),三角形中心坐标\(G[x_0,y_0]\),正三角形外接圆外径\(r\)

则有G的轨迹曲线：\((-9e^6 + 33e^4 - 40e^2 + 16)x_0^2 + (e^4 - 8e^2 + 16)y_0^2 + e^8p^2 - p^2e^6=0\)

中心坐标\([x_0,y_0]\)与外接圆半径\(r\)的关系

\((e^4 - 8e^2 + 16)r^2 + (-16e^4 + 16e^2)x_0^2 + 16e^4p^2 - 16e^2p^2=0\)

\((-9e^4 + 24e^2 - 16)r^2 + 16p^2e^6 - 32e^4p^2 + 16e^2p^2 + 16e^2y_0^2=0\)

正三角形三个顶点\([x_1,y_1],[x_2,y_2],[x_3,y_3]\)与外接圆半径\(r\)的关系

下面简记\(x=x_i,y=y_i,i=1..3\)

\(-4096e^4(e - 1)^3(e + 1)^3x^6 + 768e^2(e - 1)^2(e + 1)^2(16e^4p^2 + 9e^4r^2 - 16e^2p^2 - 16e^2r^2 + 16r^2)x^4 - 48(e - 1)(e + 1)(256e^8p^4 + 544e^8p^2r^2 + 81e^8r^4 - 512e^6p^4 - 1824e^6p^2r^2 - 432e^6r^4 + 256e^4p^4 + 2304e^4p^2r^2 + 1200e^4r^4 - 1024e^2p^2r^2 - 1536e^2r^4 + 768r^4)x^2 + e^2(16e^4p^2 + e^4r^2 - 16e^2p^2 - 8e^2r^2 + 16r^2)(16e^2p^2 - 27e^2r^2 - 16p^2 + 36r^2)^2=0\)

\(e^2(4e^3p + 3e^2r - 4ep - 4r)(4e^3p - 3e^2r - 4ep + 4r)(16e^4p^2 - 32e^2p^2 - 9e^2r^2 + 16p^2 + 36r^2)^2 + 4096e^4y^6 + 768e^2(16e^6p^2 - 32e^4p^2 - 9e^4r^2 + 16e^2p^2 + 16e^2r^2 - 16r^2)y^4 + (12288e^{12}p^4 - 49152e^{10}p^4 - 13824e^{10}p^2r^2 + 73728e^8p^4 + 64512e^8p^2r^2 + 3888e^8r^4 - 49152e^6p^4 - 136704e^6p^2r^2 - 20736e^6r^4 + 12288e^4p^4 + 135168e^4p^2r^2 + 57600e^4r^4 - 49152e^2p^2r^2 - 73728e^2r^4 + 36864r^4)y^2=0\)

计算实例：

-(16*x^2)/9 + y^2 + 16=0

-(2704*x0^2)/81 + (121*y0^2)/81 + 10000/81=0

[{a = 3, b = 4, c = 5, e = 5/3, m = 0.4520285007, p = 9/5, r = 6.143867627, x = 1.982024171, x0 = 1.988380318, x1 = -4.035304953, x2 = 3.952967545, x3 = 6.047478364, y = 3.002689222, y0 = 2.389174275, y1 = 3.598440660, y2 = -3.432123386, y3 = 7.001205551},
{a = 3, b = 4, c = 5, e= 5/3, m = 0.5586335164, p = 9/5, r = 4.943409449, x = 1.572921850, x0 = 1.924062796, x1 = -3.017366852, x2 = 4.273626864, x3 = 4.515928375, y = 3.406116550, y0 = 0.2911322259, y1 = 0.431025070, y2 = -4.058217802, y3 = 4.500589410},
{a = 3, b = 4, c = 5, e= 5/3, m = 0.6758820332, p = 9/5, r = 5.482477646, x = 1.118569432, x0 = 1.951480403, x1 = -3.476192799, x2 = 5.334976301, x3 = 3.995657706, y = 3.711556109, y0 = -1.568221801, y1 = -2.341477750, y2 = -5.882096702, y3 = 3.518909049},
{a = 3, b = 4, c = 5, e= 5/3, m = 0.8074436091, p = 9/5, r = 9.614938681, x = 0.6320378091, x0 = 2.233344343, x1 = -6.919378329, x2 = 9.360398755, x3 = 4.259012603, y = 3.910220956, y0 = -5.368321605, y1 = -8.313608267, y2 = -11.82216861, y3 = 4.030812075},

数学星空

对于椭圆：\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)

内接等五边形重心轨迹：\(\frac{x^2}{m^2}+\frac{y^2}{n^2}=1\)

且\(m,n\)满足下列方程（对应最大等五边形与最小等五边形情形）：

-25*(a^2 + 15*b^2)*(a^4 + 10*a^2*b^2 + 5*b^4)*(b - a)^3*(a + b)^3*m^4 - 20*a^3*(5*a^6 + 89*a^4*b^2 + 351*a^2*b^4 + 195*b^6)*(b - a)^2*(a + b)^2*m^3 + 2*a^2*(a^2 + b^2)*(75*a^10 + 645*a^8*b^2 - 178*a^6*b^4 - 118*a^4*b^6 + 375*a^2*b^8 + 225*b^10)*m^2 - 4*a^3*(5*a^2 + 3*b^2)*(5*a^6 + 24*a^4*b^2 + 9*a^2*b^4 + 10*b^6)*(b - a)^2*(a + b)^2*m + a^4*(5*a^2 + 3*b^2)^2*(b - a)^4*(a + b)^4=0

-25*(15*a^2 + b^2)*(5*a^4 + 10*a^2*b^2 + b^4)*(a - b)^3*(a + b)^3*n^4 - 20*b^3*(195*a^6 + 351*a^4*b^2 + 89*a^2*b^4 + 5*b^6)*(a - b)^2*(a + b)^2*n^3 + 2*b^2*(a^2 + b^2)*(225*a^10 + 375*a^8*b^2 - 118*a^6*b^4 - 178*a^4*b^6 + 645*a^2*b^8 + 75*b^10)*n^2 - 4*b^3*(3*a^2 + 5*b^2)*(10*a^6 + 9*a^4*b^2 + 24*a^2*b^4 + 5*b^6)*(a - b)^2*(a + b)^2*n + b^4*(3*a^2 + 5*b^2)^2*(a - b)^4*(a + b)^4=0

对于最小等五边形边长L=4.614585235(注最大等五边形边长L=4.618361740)

(15*a^2 + b^2)^2*(5*a^4 + 10*a^2*b^2 + b^4)^2*(a - b)^4*(a + b)^4*L^8 - 16*a^4*b^2*(1125*a^12 - 10750*a^10*b^2 - 21605*a^8*b^4 - 4324*a^6*b^6 + 2315*a^4*b^8 + 450*a^2*b^10 + 21*b^12)*(a - b)^2*(a + b)^2*L^6 - 256*a^8*b^4*(625*a^12 + 1730*a^10*b^2 - 6985*a^8*b^4 - 1732*a^6*b^6 + 1999*a^4*b^8 + 354*a^2*b^10 - 87*b^12)*L^4 + 20480*a^12*b^6*(13*a^8 - 132*a^6*b^2 - 146*a^4*b^4 - 4*a^2*b^6 + 13*b^8)*L^2 + 1310720*a^16*b^8*(a^2 + b^2)^2=0

根据 https://bbs.emath.ac.cn/forum.ph ... 89&fromuid=1455 中的结论：

可以得到包络曲线方程：

L^8*a^10*b^2*y^4 + 2*L^8*a^8*b^4*x^2*y^2 - 2*L^8*a^8*b^4*y^4 + L^8*a^6*b^6*x^4 - 4*L^8*a^6*b^6*x^2*y^2 + L^8*a^6*b^6*y^4 - 2*L^8*a^4*b^8*x^4 + 2*L^8*a^4*b^8*x^2*y^2 + L^8*a^2*b^10*x^4 - 2*L^6*a^12*b^4*y^2 - 4*L^6*a^12*b^2*y^4 + 4*L^6*a^12*y^6 + 2*L^6*a^10*b^6*x^2 + 6*L^6*a^10*b^6*y^2 - 12*L^6*a^10*b^4*x^2*y^2 + 10*L^6*a^10*b^2*x^2*y^4 - 8*L^6*a^10*b^2*y^6 - 6*L^6*a^8*b^8*x^2 - 6*L^6*a^8*b^8*y^2 - 8*L^6*a^8*b^6*x^4 + 12*L^6*a^8*b^6*x^2*y^2 + 12*L^6*a^8*b^6*y^4 + 8*L^6*a^8*b^4*x^4*y^2 - 14*L^6*a^8*b^4*x^2*y^4 + 2*L^6*a^8*b^4*y^6 + 6*L^6*a^6*b^10*x^2 + 2*L^6*a^6*b^10*y^2 + 12*L^6*a^6*b^8*x^4 + 12*L^6*a^6*b^8*x^2*y^2 - 8*L^6*a^6*b^8*y^4 + 2*L^6*a^6*b^6*x^6 - 4*L^6*a^6*b^6*x^4*y^2 - 4*L^6*a^6*b^6*x^2*y^4 + 2*L^6*a^6*b^6*y^6 - 2*L^6*a^4*b^12*x^2 - 12*L^6*a^4*b^10*x^2*y^2 + 2*L^6*a^4*b^8*x^6 - 14*L^6*a^4*b^8*x^4*y^2 + 8*L^6*a^4*b^8*x^2*y^4 - 4*L^6*a^2*b^12*x^4 - 8*L^6*a^2*b^10*x^6 + 10*L^6*a^2*b^10*x^4*y^2 + 4*L^6*b^12*x^6 + L^4*a^14*b^6 + 8*L^4*a^14*b^4*y^2 - 8*L^4*a^14*b^2*y^4 - 4*L^4*a^12*b^8 - 12*L^4*a^12*b^6*x^2 - 12*L^4*a^12*b^6*y^2 + 22*L^4*a^12*b^4*x^2*y^2 + 56*L^4*a^12*b^4*y^4 - 20*L^4*a^12*b^2*x^2*y^4 - 32*L^4*a^12*b^2*y^6 + 6*L^4*a^10*b^10 + 28*L^4*a^10*b^8*x^2 - 12*L^4*a^10*b^8*y^2 + 22*L^4*a^10*b^6*x^4 + 4*L^4*a^10*b^6*x^2*y^2 - 66*L^4*a^10*b^6*y^4 - 32*L^4*a^10*b^4*x^4*y^2 - 44*L^4*a^10*b^4*x^2*y^4 + 72*L^4*a^10*b^4*y^6 + L^4*a^10*b^2*x^4*y^4 + 20*L^4*a^10*b^2*x^2*y^6 - 8*L^4*a^10*b^2*y^8 - 4*L^4*a^8*b^12 - 12*L^4*a^8*b^10*x^2 + 28*L^4*a^8*b^10*y^2 - 4*L^4*a^8*b^8*x^4 - 52*L^4*a^8*b^8*x^2*y^2 - 4*L^4*a^8*b^8*y^4 - 12*L^4*a^8*b^6*x^6 - 40*L^4*a^8*b^6*x^4*y^2 + 136*L^4*a^8*b^6*x^2*y^4 - 28*L^4*a^8*b^6*y^6 + 2*L^4*a^8*b^4*x^6*y^2 + 48*L^4*a^8*b^4*x^4*y^4 - 54*L^4*a^8*b^4*x^2*y^6 + 8*L^4*a^8*b^4*y^8 + L^4*a^6*b^14 - 12*L^4*a^6*b^12*x^2 - 12*L^4*a^6*b^12*y^2 - 66*L^4*a^6*b^10*x^4 + 4*L^4*a^6*b^10*x^2*y^2 + 22*L^4*a^6*b^10*y^4 - 28*L^4*a^6*b^8*x^6 + 136*L^4*a^6*b^8*x^4*y^2 - 40*L^4*a^6*b^8*x^2*y^4 - 12*L^4*a^6*b^8*y^6 + L^4*a^6*b^6*x^8 + 36*L^4*a^6*b^6*x^6*y^2 - 92*L^4*a^6*b^6*x^4*y^4 + 36*L^4*a^6*b^6*x^2*y^6 + L^4*a^6*b^6*y^8 + 8*L^4*a^4*b^14*x^2 + 56*L^4*a^4*b^12*x^4 + 22*L^4*a^4*b^12*x^2*y^2 + 72*L^4*a^4*b^10*x^6 - 44*L^4*a^4*b^10*x^4*y^2 - 32*L^4*a^4*b^10*x^2*y^4 + 8*L^4*a^4*b^8*x^8 - 54*L^4*a^4*b^8*x^6*y^2 + 48*L^4*a^4*b^8*x^4*y^4 + 2*L^4*a^4*b^8*x^2*y^6 - 8*L^4*a^2*b^14*x^4 - 32*L^4*a^2*b^12*x^6 - 20*L^4*a^2*b^12*x^4*y^2 - 8*L^4*a^2*b^10*x^8 + 20*L^4*a^2*b^10*x^6*y^2 + L^4*a^2*b^10*x^4*y^4 - 4*L^2*a^16*b^6 + 4*L^2*a^16*b^4*y^2 + 12*L^2*a^14*b^8 + 16*L^2*a^14*b^6*x^2 - 64*L^2*a^14*b^6*y^2 - 12*L^2*a^14*b^4*x^2*y^2 + 48*L^2*a^14*b^4*y^4 - 8*L^2*a^12*b^10 + 4*L^2*a^12*b^8*x^2 + 152*L^2*a^12*b^8*y^2 - 24*L^2*a^12*b^6*x^4 - 12*L^2*a^12*b^6*x^2*y^2 - 232*L^2*a^12*b^6*y^4 + 12*L^2*a^12*b^4*x^4*y^2 + 52*L^2*a^12*b^4*x^2*y^4 + 88*L^2*a^12*b^4*y^6 - 8*L^2*a^10*b^12 - 112*L^2*a^10*b^10*x^2 - 112*L^2*a^10*b^10*y^2 - 88*L^2*a^10*b^8*x^4 + 24*L^2*a^10*b^8*x^2*y^2 + 296*L^2*a^10*b^8*y^4 + 16*L^2*a^10*b^6*x^6 + 224*L^2*a^10*b^6*x^4*y^2 + 80*L^2*a^10*b^6*x^2*y^4 - 224*L^2*a^10*b^6*y^6 - 4*L^2*a^10*b^4*x^6*y^2 - 104*L^2*a^10*b^4*x^4*y^4 - 52*L^2*a^10*b^4*x^2*y^6 + 48*L^2*a^10*b^4*y^8 + 12*L^2*a^8*b^14 + 152*L^2*a^8*b^12*x^2 + 4*L^2*a^8*b^12*y^2 + 296*L^2*a^8*b^10*x^4 + 24*L^2*a^8*b^10*x^2*y^2 - 88*L^2*a^8*b^10*y^4 + 120*L^2*a^8*b^8*x^6 - 368*L^2*a^8*b^8*x^4*y^2 - 368*L^2*a^8*b^8*x^2*y^4 + 120*L^2*a^8*b^8*y^6 - 4*L^2*a^8*b^6*x^8 - 156*L^2*a^8*b^6*x^6*y^2 + 80*L^2*a^8*b^6*x^4*y^4 + 180*L^2*a^8*b^6*x^2*y^6 - 52*L^2*a^8*b^6*y^8 + 4*L^2*a^8*b^4*x^6*y^4 + 12*L^2*a^8*b^4*x^4*y^6 + 12*L^2*a^8*b^4*x^2*y^8 + 4*L^2*a^8*b^4*y^10 - 4*L^2*a^6*b^16 - 64*L^2*a^6*b^14*x^2 + 16*L^2*a^6*b^14*y^2 - 232*L^2*a^6*b^12*x^4 - 12*L^2*a^6*b^12*x^2*y^2 - 24*L^2*a^6*b^12*y^4 - 224*L^2*a^6*b^10*x^6 + 80*L^2*a^6*b^10*x^4*y^2 + 224*L^2*a^6*b^10*x^2*y^4 + 16*L^2*a^6*b^10*y^6 - 52*L^2*a^6*b^8*x^8 + 180*L^2*a^6*b^8*x^6*y^2 + 80*L^2*a^6*b^8*x^4*y^4 - 156*L^2*a^6*b^8*x^2*y^6 - 4*L^2*a^6*b^8*y^8 + 8*L^2*a^6*b^6*x^8*y^2 + 24*L^2*a^6*b^6*x^6*y^4 + 24*L^2*a^6*b^6*x^4*y^6 + 8*L^2*a^6*b^6*x^2*y^8 + 4*L^2*a^4*b^16*x^2 + 48*L^2*a^4*b^14*x^4 - 12*L^2*a^4*b^14*x^2*y^2 + 88*L^2*a^4*b^12*x^6 + 52*L^2*a^4*b^12*x^4*y^2 + 12*L^2*a^4*b^12*x^2*y^4 + 48*L^2*a^4*b^10*x^8 - 52*L^2*a^4*b^10*x^6*y^2 - 104*L^2*a^4*b^10*x^4*y^4 - 4*L^2*a^4*b^10*x^2*y^6 + 4*L^2*a^4*b^8*x^10 + 12*L^2*a^4*b^8*x^8*y^2 + 12*L^2*a^4*b^8*x^6*y^4 + 4*L^2*a^4*b^8*x^4*y^6 + 16*a^16*b^8 - 16*a^16*b^6*y^2 - 64*a^14*b^10 - 80*a^14*b^8*x^2 + 128*a^14*b^8*y^2 + 64*a^14*b^6*x^2*y^2 - 64*a^14*b^6*y^4 + 96*a^12*b^12 + 256*a^12*b^10*x^2 - 288*a^12*b^10*y^2 + 160*a^12*b^8*x^4 - 320*a^12*b^8*x^2*y^2 + 288*a^12*b^8*y^4 - 96*a^12*b^6*x^4*y^2 + 64*a^12*b^6*x^2*y^4 - 96*a^12*b^6*y^6 - 64*a^10*b^14 - 288*a^10*b^12*x^2 + 256*a^10*b^12*y^2 - 384*a^10*b^10*x^4 + 512*a^10*b^10*x^2*y^2 - 384*a^10*b^10*y^4 - 160*a^10*b^8*x^6 + 192*a^10*b^8*x^4*y^2 - 160*a^10*b^8*x^2*y^4 + 256*a^10*b^8*y^6 + 64*a^10*b^6*x^6*y^2 + 64*a^10*b^6*x^4*y^4 - 64*a^10*b^6*x^2*y^6 - 64*a^10*b^6*y^8 + 16*a^8*b^16 + 128*a^8*b^14*x^2 - 80*a^8*b^14*y^2 + 288*a^8*b^12*x^4 - 320*a^8*b^12*x^2*y^2 + 160*a^8*b^12*y^4 + 256*a^8*b^10*x^6 - 160*a^8*b^10*x^4*y^2 + 192*a^8*b^10*x^2*y^4 - 160*a^8*b^10*y^6 + 80*a^8*b^8*x^8 + 64*a^8*b^8*x^6*y^2 - 32*a^8*b^8*x^4*y^4 + 64*a^8*b^8*x^2*y^6 + 80*a^8*b^8*y^8 - 16*a^8*b^6*x^8*y^2 - 64*a^8*b^6*x^6*y^4 - 96*a^8*b^6*x^4*y^6 - 64*a^8*b^6*x^2*y^8 - 16*a^8*b^6*y^10 - 16*a^6*b^16*x^2 - 64*a^6*b^14*x^4 + 64*a^6*b^14*x^2*y^2 - 96*a^6*b^12*x^6 + 64*a^6*b^12*x^4*y^2 - 96*a^6*b^12*x^2*y^4 - 64*a^6*b^10*x^8 - 64*a^6*b^10*x^6*y^2 + 64*a^6*b^10*x^4*y^4 + 64*a^6*b^10*x^2*y^6 - 16*a^6*b^8*x^10 - 64*a^6*b^8*x^8*y^2 - 96*a^6*b^8*x^6*y^4 - 64*a^6*b^8*x^4*y^6 - 16*a^6*b^8*x^2*y^8=0

计算实例：

取\(a=5,b=3\) 代入上面方程可以得到

\(m=0.2796076822, n = 0.1249726941\)

\(L=4.614585235\)

包络曲线：

-1.290968655*x^10 - 8.128954048*x^8*y^2 - 18.50222811*x^6*y^4 - 19.64264659*x^4*y^6 - 9.839581760*x^2*y^8 - 1.861177894*y^10 + 60.61794488*x^8 + 26.94633460*x^6*y^2 - 397.8466489*x^4*y^4 - 148.6575094*x^2*y^6 + 34.98440350*y^8 - 1581.107003*x^6 + 3338.458060*x^4*y^2 + 6741.295221*x^2*y^4 - 1171.188994*y^6 + 23342.89706*x^4 - 15359.92906*x^2*y^2 + 12783.23746*y^4 - 208613.7652*x^2 - 161507.1205*y^2 + 864004.4450=0

绘图得到：

等五边形重心轨迹



包络曲线(神奇的是L=4.614585265与L=4.618361740对应的曲线完全重合在一起了?)：



重心轨迹和包络曲线：



可以猜测：内接椭圆的等边N边形的重心轨迹为椭圆

mathe

等五边形边长L是变量而不是常数，不同方向的等五边形边长L应该是不同的