果树种植最优解精美图形作法探讨

dingjifen

mathe 发表于 2020-1-3 10:34
这个内容很复杂，你看一下综述吧:
https://emathgroup.github.io/blog/orchard-planting-problem

感谢数学大师的热心指导！能拜为好友否？

数学星空

20棵树计算结果与绘图(24棵30行）.rar (58.35 KB, 下载次数: 5)

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数学星空

数学星空 发表于 2019-12-25 18:57
对于92# 22棵28行计算结果：

第一种变换：

对于正七边形的图形计算我们得到代数表达式如下：

初始数据：

[[A, 1, 0, 0], [B, 1, t^2 - 2*t, 0], [C, 1, 1, 0], [D, t^2 - t - 1, -t^2 + 2*t + 2, 1], [E, 1, t^2 - t - 1, 1], [F, -t, -t^2 + 1, 1], [G, t^2, t^2 + t, 1], [H, t^2, t^2, 1], [I, -t, -t^2 + t + 1, 1], [J, t^2 - t, t, 1], [K, 0, 1, 1], [L, 1, 0, 1], [M, t^2 - t - 1, 1 + t, 1], [N, t^2 - t, 1 + t, 1], [O, 0, 0, 1], [P, t^2, 1 + t, 1], [Q, -t, 0, 1], [R, 1, 1, 1], [S, t^2 - t - 1, t, 1], [T, t^2 - t, 1, 1], [U, 0, t, 1], [V, 0, 1, 0]]

[[A, B, C, V], [K, O, U, V], [J, N, T, V], [D, M, S, V], [E, L, R, V], [G, H, P, V], [F, I, Q, V], [A, L, O, Q], [A, M, N, P], [A, J, S, U], [A, K, R, T], [J, K, P, Q], [E, H, K, N], [D, I, J, O], [F, H, J, L], [G, I, K, M], [D, E, T, U], [E, F, M, O], [D, G, L, N], [F, G, R, S], [C, H, O, R], [C, I, N, S], [B, H, M, U], [B, I, L, T], [B, E, Q, S], [B, D, P, R], [C, G, Q, U], [C, F, P, T]]

变换基点：

Q[-t, 0, 1]-->[-cos(1/14*Pi), -sin(1/14*Pi), 1]
C[1, 1, 0]-->[-cos(3/14*Pi), sin(3/14*Pi), 1]
P[t^2, 1 + t, 1]--> [0, 1, 1]
D[t^2 - t - 1, -t^2 + 2*t + 2, 1]--> [cos(1/14*Pi), -sin(1/14*Pi), 1]

变换矩阵：

[[1/2*((-408*t^2 + 1745*t + 816)*s^5 + (-404*t^2 + 1581*t + 808)*s^4 + (2236*t^2 - 9258*t - 4472)*s^3 + (-660*t^2 + 2758*t + 1320)*s^2 + (-380*t^2 + 1573*t + 760)*s + 48*t^2 - 199*t - 96)/((3*t^2 - 4*t - 5)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25)), -(929*s^5*t - 521*s^5 + 773*s^4*t - 369*s^4 - 4786*s^3*t + 2550*s^3 + 1438*s^2*t - 778*s^2 + 813*s*t - 433*s - 103*t + 55)*(1 + t)/((418*s^5 + 422*s^4 - 2324*s^3 + 692*s^2 + 394*s - 50)*(3*t^2 - 4*t - 5)), 1/2*((929*t^2 - 521*t - 521)*s^5 + (773*t^2 - 369*t - 369)*s^4 + (-4786*t^2 + 2550*t + 2550)*s^3 + (1438*t^2 - 778*t - 778)*s^2 + (813*t^2 - 433*t - 433)*s - 103*t^2 + 55*t + 55)/((3*t^2 - 4*t - 5)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25))], [1/2*((-2424*t^2 + 2731*t + 3894)*s^5 + (-2028*t^2 + 2281*t + 3274)*s^4 + (12572*t^2 - 14182*t - 20244)*s^3 + (-3788*t^2 + 4278*t + 6100)*s^2 + (-2140*t^2 + 2415*t + 3446)*s + 272*t^2 - 307*t - 438)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25)), 1/2*((1359*t^2 + 464*t - 699)*s^5 + (1077*t^2 + 572*t - 421)*s^4 + (-6958*t^2 - 2660*t + 3402)*s^3 + (2110*t^2 + 756*t - 1066)*s^2 + (1187*t^2 + 444*t - 587)*s - 151*t^2 - 56*t + 75)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25)), 1/2*((-2953*t^2 + 2097*t + 3469)*s^5 + (-2619*t^2 + 1879*t + 3083)*s^4 + (15610*t^2 - 11158*t - 18382)*s^3 + (-4682*t^2 + 3350*t + 5518)*s^2 + (-2653*t^2 + 1897*t + 3125)*s + 337*t^2 - 241*t - 397)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25))], [((-1085*t^2 + 399*t + 1046)*s^5 + (-847*t^2 + 245*t + 774)*s^4 + (5418*t^2 - 1806*t - 5096)*s^3 + (-1626*t^2 + 542*t + 1528)*s^2 + (-921*t^2 + 307*t + 866)*s + 117*t^2 - 39*t - 110)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25)), ((229*t^2 + 2169*t + 1522)*s^5 + (107*t^2 + 1975*t + 1446)*s^4 + (-966*t^2 - 11550*t - 8260)*s^3 + (294*t^2 + 3454*t + 2468)*s^2 + (165*t^2 + 1961*t + 1402)*s - 21*t^2 - 249*t - 178)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25)), ((137*t^2 - 1470*t - 1300)*s^5 + (115*t^2 - 1246*t - 1108)*s^4 + (-770*t^2 + 7672*t + 6832)*s^3 + (242*t^2 - 2312*t - 2064)*s^2 + (133*t^2 - 1306*t - 1164)*s - 17*t^2 + 166*t + 148)/((9*t^2 - 14*t - 17)*(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25))]]
  
注：t^3 - 2*t^2 - t + 1 = 0 ，t=0.554958132087371  (另外两根为-0.801937735804838, , 2.24697960371747）
为了便于软件做代数运算，我们做了三角代换s=tan(Pi/28)=0.112672939900111,并且满足方程   s^6 - 8*s^5 - 13*s^4 + 48*s^3 - 13*s^2 - 8*s + 1 = 0

变换后坐标：

[((14324*t^2 - 23344*t - 27945)*s^5 + (13596*t^2 - 21940*t - 26349)*s^4 + (-77280*t^2 + 125492*t + 150402)*s^3 + (22912*t^2 - 37244*t - 44622)*s^2 + (13132*t^2 - 21324*t - 25557)*s - 1660*t^2 + 2696*t + 3231)/((3532*t^2 - 7794*t - 8514)*s^5 + (2236*t^2 - 5478*t - 5822)*s^4 + (-16128*t^2 + 37156*t + 40124)*s^3 + (4832*t^2 - 11140*t - 12028)*s^2 + (2740*t^2 - 6314*t - 6818)*s - 348*t^2 + 802*t + 866), ((2424*t^2 - 2731*t - 3894)*s^5 + (2028*t^2 - 2281*t - 3274)*s^4 + (-12572*t^2 + 14182*t + 20244)*s^3 + (3788*t^2 - 4278*t - 6100)*s^2 + (2140*t^2 - 2415*t - 3446)*s - 272*t^2 + 307*t + 438)/((2170*t^2 - 798*t - 2092)*s^5 + (1694*t^2 - 490*t - 1548)*s^4 + (-10836*t^2 + 3612*t + 10192)*s^3 + (3252*t^2 - 1084*t - 3056)*s^2 + (1842*t^2 - 614*t - 1732)*s - 234*t^2 + 78*t + 220), A],

[-2*(39*t^2 - 56*t - 70)*(102*s^5 + 101*s^4 - 559*s^3 + 165*s^2 + 95*s - 12)/((3447*t^2 - 5236*t - 6398)*s^5 + (3749*t^2 - 5656*t - 6930)*s^4 + (-20006*t^2 + 30268*t + 37044)*s^3 + (5974*t^2 - 9036*t - 11060)*s^2 + (3395*t^2 - 5136*t - 6286)*s - 431*t^2 + 652*t + 798), ((-882*t^2 + 1617*t + 1715)*s^5 + (-686*t^2 + 1309*t + 1351)*s^4 + (4508*t^2 - 8344*t - 8792)*s^3 + (-1372*t^2 + 2528*t + 2672)*s^2 + (-770*t^2 + 1423*t + 1501)*s + 98*t^2 - 181*t - 191)/((666*t^2 - 705*t - 1123)*s^5 + (706*t^2 - 779*t - 1201)*s^4 + (-3808*t^2 + 4130*t + 6454)*s^3 + (1136*t^2 - 1234*t - 1926)*s^2 + (646*t^2 - 701*t - 1095)*s - 82*t^2 + 89*t + 139), B],

[(1337*s^5 + 1177*s^4 - 7022*s^3 + 2098*s^2 + 1193*s - 151)/(1712*s^5 + 1480*s^4 - 8904*s^3 + 2664*s^2 + 1512*s - 192), (1065*s^5 + 951*s^4 - 5614*s^3 + 1678*s^2 + 953*s - 121)/(1712*s^5 + 1480*s^4 - 8904*s^3 + 2664*s^2 + 1512*s - 192), C],

[(-204*s^5 - 202*s^4 + 1118*s^3 - 330*s^2 - 190*s + 24)/(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25), (-49*s^5 - 21*s^4 + 224*s^3 - 72*s^2 - 39*s + 5)/(209*s^5 + 211*s^4 - 1162*s^3 + 346*s^2 + 197*s - 25), D],

[((-12811*t^2 + 15058*t + 20318)*s^5 + (-14595*t^2 + 18142*t + 23938)*s^4 + (74178*t^2 - 89240*t - 119288)*s^3 + (-21566*t^2 + 25784*t + 34552)*s^2 + (-12611*t^2 + 15174*t + 20282)*s + 1589*t^2 - 1910*t - 2554)/((22612*t^2 - 37362*t - 44494)*s^5 + (21348*t^2 - 34878*t - 41690)*s^4 + (-123200*t^2 + 202412*t + 241500)*s^3 + (36960*t^2 - 60716*t - 72444)*s^2 + (20940*t^2 - 34402*t - 41046)*s - 2660*t^2 + 4370*t + 5214), ((-2894*t^2 + 5527*t + 6239)*s^5 + (-2342*t^2 + 4581*t + 5129)*s^4 + (15008*t^2 - 28742*t - 32410)*s^3 + (-4560*t^2 + 8694*t + 9818)*s^2 + (-2562*t^2 + 4899*t + 5527)*s + 326*t^2 - 623*t - 703)/((6402*t^2 - 5186*t - 8348)*s^5 + (5806*t^2 - 4894*t - 7724)*s^4 + (-34188*t^2 + 28252*t + 45024)*s^3 + (10252*t^2 - 8476*t - 13504)*s^2 + (5810*t^2 - 4802*t - 7652)*s - 738*t^2 + 610*t + 972), E],

[(18*t^2 - 13*t - 22)*(1337*s^5 + 1177*s^4 - 7022*s^3 + 2098*s^2 + 1193*s - 151)/((15528*t^2 - 2264*t - 11792)*s^5 + (12080*t^2 - 200*t - 7920)*s^4 + (-78008*t^2 + 8176*t + 56672)*s^3 + (23512*t^2 - 2672*t - 17248)*s^2 + (13280*t^2 - 1432*t - 9680)*s - 1688*t^2 + 184*t + 1232), ((6501*t^2 + 732*t - 3528)*s^5 + (5875*t^2 + 444*t - 3360)*s^4 + (-34398*t^2 - 3472*t + 18984)*s^3 + (10270*t^2 + 1072*t - 5640)*s^2 + (5837*t^2 + 596*t - 3216)*s - 741*t^2 - 76*t + 408)/((9736*t^2 + 3320*t - 3528)*s^5 + (8520*t^2 + 2560*t - 3360)*s^4 + (-50848*t^2 - 16632*t + 18984)*s^3 + (15200*t^2 + 5016*t - 5640)*s^2 + (8632*t^2 + 2832*t - 3216)*s - 1096*t^2 - 360*t + 408), F],

[((11989*t^2 + 3057*t - 5260)*s^5 + (11137*t^2 + 1705*t - 5796)*s^4 + (-64178*t^2 - 14006*t + 30048)*s^3 + (19070*t^2 + 4362*t - 8768)*s^2 + (10905*t^2 + 2377*t - 5108)*s - 1379*t^2 - 303*t + 644)/((18414*t^2 - 1268*t - 12860)*s^5 + (16474*t^2 - 2036*t - 12228)*s^4 + (-96572*t^2 + 7952*t + 68488)*s^3 + (28764*t^2 - 2160*t - 20232)*s^2 + (16374*t^2 - 1308*t - 11580)*s - 2078*t^2 + 164*t + 1468), ((4489*t^2 + 4887*t + 1045)*s^5 + (3911*t^2 + 4437*t + 1055)*s^4 + (-23422*t^2 - 25942*t - 5810)*s^3 + (7006*t^2 + 7750*t + 1730)*s^2 + (3977*t^2 + 4403*t + 985)*s - 505*t^2 - 559*t - 125)/((12376*t^2 + 3986*t - 4770)*s^5 + (10528*t^2 + 3574*t - 3910)*s^4 + (-64120*t^2 - 20916*t + 24500)*s^3 + (19224*t^2 + 6228*t - 7380)*s^2 + (10896*t^2 + 3546*t - 4170)*s - 1384*t^2 - 450*t + 530), G],

[((5260*t^2 + 1469*t - 2203)*s^5 + (5796*t^2 - 455*t - 4091)*s^4 + (-30048*t^2 - 4082*t + 16042)*s^3 + (8768*t^2 + 1534*t - 4406)*s^2 + (5108*t^2 + 689*t - 2731)*s - 644*t^2 - 91*t + 341)/((2276*t^2 + 9158*t + 5864)*s^5 + (2148*t^2 + 7698*t + 4776)*s^4 + (-11536*t^2 - 48188*t - 31136)*s^3 + (3312*t^2 + 14620*t + 9568)*s^2 + (1932*t^2 + 8222*t + 5336)*s - 244*t^2 - 1046*t - 680), ((1307*t^2 + 4227*t + 2404)*s^5 + (1185*t^2 + 3781*t + 2132)*s^4 + (-6846*t^2 - 22386*t - 12768)*s^3 + (2030*t^2 + 6706*t + 3840)*s^2 + (1159*t^2 + 3803*t + 2172)*s - 147*t^2 - 483*t - 276)/((7122*t^2 + 484*t - 4312)*s^5 + (6150*t^2 + 468*t - 3696)*s^4 + (-37156*t^2 - 2464*t + 22568)*s^3 + (11140*t^2 + 704*t - 6792)*s^2 + (6314*t^2 + 412*t - 3840)*s - 802*t^2 - 52*t + 488), H],

[((-17337*t^2 + 18969*t + 26357)*s^5 + (-15845*t^2 + 17461*t + 24189)*s^4 + (92266*t^2 - 101210*t - 140478)*s^3 + (-27462*t^2 + 30102*t + 41794)*s^2 + (-15677*t^2 + 17197*t + 23869)*s + 1983*t^2 - 2175*t - 3019)/((610*t^2 - 8162*t - 6932)*s^5 + (2246*t^2 - 9534*t - 9084)*s^4 + (-7028*t^2 + 47964*t + 42952)*s^3 + (1940*t^2 - 14108*t - 12552)*s^2 + (1162*t^2 - 8098*t - 7236)*s - 146*t^2 + 1026*t + 916), ((3319*t^2 + 72*t - 2169)*s^5 + (3149*t^2 - 212*t - 2283)*s^4 + (-17822*t^2 + 84*t + 12026)*s^3 + (5294*t^2 + 28*t - 3530)*s^2 + (3019*t^2 - 4*t - 2029)*s - 383*t^2 + 257)/((4482*t^2 - 182*t - 3070)*s^5 + (4142*t^2 - 546*t - 3146)*s^4 + (-23884*t^2 + 1820*t + 17052)*s^3 + (7116*t^2 - 508*t - 5052)*s^2 + (4050*t^2 - 302*t - 2886)*s - 514*t^2 + 38*t + 366), I],

[(9*t^2 - 14*t - 17)*(929*s^5 + 773*s^4 - 4786*s^3 + 1438*s^2 + 813*s - 103)/((2706*t^2 - 6300*t - 6810)*s^5 + (1438*t^2 - 4228*t - 4334)*s^4 + (-12460*t^2 + 30632*t + 32676)*s^3 + (3884*t^2 - 9384*t - 10052)*s^2 + (2146*t^2 - 5244*t - 5602)*s - 274*t^2 + 668*t + 714), ((-418*t^2 + 1594*t + 1803)*s^5 + (-422*t^2 + 1542*t + 1753)*s^4 + (2324*t^2 - 8652*t - 9814)*s^3 + (-692*t^2 + 2572*t + 2918)*s^2 + (-394*t^2 + 1466*t + 1663)*s + 50*t^2 - 186*t - 211)/((1908*t^2 - 732*t - 1686)*s^5 + (1564*t^2 - 444*t - 1226)*s^4 + (-9800*t^2 + 3472*t + 8372)*s^3 + (2952*t^2 - 1072*t - 2548)*s^2 + (1668*t^2 - 596*t - 1430)*s - 212*t^2 + 76*t + 182), J],

[-(4*t^2 - 8*t - 9)*(929*s^5 + 773*s^4 - 4786*s^3 + 1438*s^2 + 813*s - 103)/((5592*t^2 - 5304*t - 7878)*s^5 + (5832*t^2 - 6064*t - 8642)*s^4 + (-31024*t^2 + 30408*t + 44492)*s^3 + (9136*t^2 - 8872*t - 13036)*s^2 + (5240*t^2 - 5120*t - 7502)*s - 664*t^2 + 648*t + 950), ((-1594*t^2 + 2561*t + 2770)*s^5 + (-1542*t^2 + 2451*t + 2662)*s^4 + (8652*t^2 - 13818*t - 14980)*s^3 + (-2572*t^2 + 4106*t + 4452)*s^2 + (-1466*t^2 + 2341*t + 2538)*s + 186*t^2 - 297*t - 322)/((732*t^2 + 1398*t + 444)*s^5 + (444*t^2 + 1458*t + 676)*s^4 + (-3472*t^2 - 7756*t - 2856)*s^3 + (1072*t^2 + 2284*t + 808)*s^2 + (596*t^2 + 1310*t + 476)*s - 76*t^2 - 166*t - 60), K],

[((-7551*t^2 + 16527*t + 18115)*s^5 + (-8799*t^2 + 17687*t + 19847)*s^4 + (44130*t^2 - 93322*t - 103246)*s^3 + (-12798*t^2 + 27318*t + 30146)*s^2 + (-7503*t^2 + 15863*t + 17551)*s + 945*t^2 - 2001*t - 2213)/((9600*t^2 - 8212*t - 12758)*s^5 + (8936*t^2 - 8140*t - 12274)*s^4 + (-52472*t^2 + 46648*t + 71148)*s^3 + (15832*t^2 - 14136*t - 21516)*s^2 + (8936*t^2 - 7956*t - 12126)*s - 1136*t^2 + 1012*t + 1542), ((5377*t^2 - 4828*t - 7363)*s^5 + (4647*t^2 - 4160*t - 6357)*s^4 + (-28182*t^2 + 25340*t + 38626)*s^3 + (8470*t^2 - 7628*t - 11618)*s^2 + (4793*t^2 - 4312*t - 6571)*s - 609*t^2 + 548*t + 835)/((1896*t^2 + 2142*t + 508)*s^5 + (1464*t^2 + 2002*t + 668)*s^4 + (-9296*t^2 - 11732*t - 3472)*s^3 + (2768*t^2 + 3540*t + 1072)*s^2 + (1576*t^2 + 1998*t + 596)*s - 200*t^2 - 254*t - 76), L],

[((-19628*t^2 + 36965*t + 42269)*s^5 + (-18848*t^2 + 34693*t + 39945)*s^4 + (106348*t^2 - 198614*t - 227682)*s^3 + (-31492*t^2 + 58954*t + 67534)*s^2 + (-18072*t^2 + 33749*t + 38689)*s + 2284*t^2 - 4267*t - 4891)/((12486*t^2 - 7216*t - 13826)*s^5 + (13330*t^2 - 9976*t - 16582)*s^4 + (-71036*t^2 + 46424*t + 82964)*s^3 + (21084*t^2 - 13624*t - 24500)*s^2 + (12030*t^2 - 7832*t - 14026)*s - 1526*t^2 + 992*t + 1778), ((3365*t^2 - 673*t - 2790)*s^5 + (2683*t^2 - 167*t - 1942)*s^4 + (-17206*t^2 + 2870*t + 13832)*s^3 + (5206*t^2 - 950*t - 4248)*s^2 + (2933*t^2 - 505*t - 2370)*s - 373*t^2 + 65*t + 302)/((4536*t^2 + 2808*t - 734)*s^5 + (3472*t^2 + 3016*t + 118)*s^4 + (-22568*t^2 - 16016*t + 2044)*s^3 + (6792*t^2 + 4752*t - 668)*s^2 + (3840*t^2 + 2712*t - 358)*s - 488*t^2 - 344*t + 46), M],

[((-5304*t^2 + 13621*t + 14324)*s^5 + (-5252*t^2 + 12753*t + 13596)*s^4 + (29068*t^2 - 73122*t - 77280)*s^3 + (-8580*t^2 + 21710*t + 22912)*s^2 + (-4940*t^2 + 12425*t + 13132)*s + 624*t^2 - 1571*t - 1660)/((16018*t^2 - 15010*t - 22340)*s^5 + (15566*t^2 - 15454*t - 22404)*s^4 + (-87164*t^2 + 83580*t + 123088)*s^3 + (25916*t^2 - 24764*t - 36528)*s^2 + (14770*t^2 - 14146*t - 20844)*s - 1874*t^2 + 1794*t + 2644), ((941*t^2 + 2058*t + 1104)*s^5 + (655*t^2 + 2114*t + 1332)*s^4 + (-4634*t^2 - 11312*t - 6412)*s^3 + (1418*t^2 + 3328*t + 1852)*s^2 + (793*t^2 + 1910*t + 1076)*s - 101*t^2 - 242*t - 136)/((2366*t^2 + 3606*t + 1358)*s^5 + (1778*t^2 + 3506*t + 1666)*s^4 + (-11732*t^2 - 19628*t - 8148)*s^3 + (3540*t^2 + 5836*t + 2388)*s^2 + (1998*t^2 + 3326*t + 1374)*s - 254*t^2 - 422*t - 174), N],

[((6773*t^2 - 6817*t - 9830)*s^5 + (4797*t^2 - 4253*t - 6502)*s^4 + (-33150*t^2 + 32170*t + 47156)*s^3 + (10114*t^2 - 9926*t - 14476)*s^2 + (5629*t^2 - 5461*t - 8006)*s - 715*t^2 + 695*t + 1018)/((13132*t^2 - 16006*t - 21272)*s^5 + (11172*t^2 - 13618*t - 18096)*s^4 + (-68600*t^2 + 83804*t + 111272)*s^3 + (20664*t^2 - 25276*t - 33544)*s^2 + (11676*t^2 - 14270*t - 18944)*s - 1484*t^2 + 1814*t + 2408), ((-2953*t^2 + 2097*t + 3469)*s^5 + (-2619*t^2 + 1879*t + 3083)*s^4 + (15610*t^2 - 11158*t - 18382)*s^3 + (-4682*t^2 + 3350*t + 5518)*s^2 + (-2653*t^2 + 1897*t + 3125)*s + 337*t^2 - 241*t - 397)/((274*t^2 - 2940*t - 2600)*s^5 + (230*t^2 - 2492*t - 2216)*s^4 + (-1540*t^2 + 15344*t + 13664)*s^3 + (484*t^2 - 4624*t - 4128)*s^2 + (266*t^2 - 2612*t - 2328)*s - 34*t^2 + 332*t + 296), O],

[0, 1, P],

[(929*s^5 + 773*s^4 - 4786*s^3 + 1438*s^2 + 813*s - 103)/(954*s^5 + 782*s^4 - 4900*s^3 + 1476*s^2 + 834*s - 106), (-209*s^5 - 211*s^4 + 1162*s^3 - 346*s^2 - 197*s + 25)/(954*s^5 + 782*s^4 - 4900*s^3 + 1476*s^2 + 834*s - 106), Q],

[2*(26*t^2 - 39*t - 48)*(102*s^5 + 101*s^4 - 559*s^3 + 165*s^2 + 95*s - 12)/((4562*t^2 - 6549*t - 8196)*s^5 + (4034*t^2 - 5771*t - 7232)*s^4 + (-23576*t^2 + 33782*t + 42308)*s^3 + (6984*t^2 - 10006*t - 12532)*s^2 + (3990*t^2 - 5717*t - 7160)*s - 506*t^2 + 725*t + 908), ((2009*t^2 - 2646*t - 3332)*s^5 + (1785*t^2 - 2366*t - 2968)*s^4 + (-10612*t^2 + 14000*t + 17612)*s^3 + (3180*t^2 - 4192*t - 5276)*s^2 + (1803*t^2 - 2378*t - 2992)*s - 229*t^2 + 302*t + 380)/((719*t^2 - 1098*t - 1268)*s^5 + (625*t^2 - 974*t - 1112)*s^4 + (-3682*t^2 + 5684*t + 6524)*s^3 + (1090*t^2 - 1684*t - 1932)*s^2 + (623*t^2 - 962*t - 1104)*s - 79*t^2 + 122*t + 140), R],

[((22685*t^2 - 36350*t - 43738)*s^5 + (20553*t^2 - 32762*t - 39490)*s^4 + (-120354*t^2 + 192496*t + 231764)*s^3 + (35854*t^2 - 57376*t - 69068)*s^2 + (20449*t^2 - 32706*t - 39378)*s - 2587*t^2 + 4138*t + 4982)/((6238*t^2 - 14094*t - 15324)*s^5 + (3674*t^2 - 9706*t - 10156)*s^4 + (-28588*t^2 + 67788*t + 72800)*s^3 + (8716*t^2 - 20524*t - 22080)*s^2 + (4886*t^2 - 11558*t - 12420)*s - 622*t^2 + 1470*t + 1580), ((2006*t^2 - 1137*t - 2091)*s^5 + (1606*t^2 - 739*t - 1521)*s^4 + (-10248*t^2 + 5530*t + 10430)*s^3 + (3096*t^2 - 1706*t - 3182)*s^2 + (1746*t^2 - 949*t - 1783)*s - 222*t^2 + 121*t + 227)/((4078*t^2 - 1530*t - 3778)*s^5 + (3258*t^2 - 934*t - 2774)*s^4 + (-20636*t^2 + 7084*t + 18564)*s^3 + (6204*t^2 - 2156*t - 5604)*s^2 + (3510*t^2 - 1210*t - 3162)*s - 446*t^2 + 154*t + 402), S],

[(9*t^2 - 9*t - 13)*(1337*s^5 + 1177*s^4 - 7022*s^3 + 2098*s^2 + 1193*s - 151)/((120*t^2 + 4584*t + 3616)*s^5 + (-1240*t^2 + 5720*t + 5400)*s^4 + (2128*t^2 - 27440*t - 23464)*s^3 + (-464*t^2 + 7984*t + 6728)*s^2 + (-328*t^2 + 4616*t + 3928)*s + 40*t^2 - 584*t - 496), ((2241*t^2 - 1398*t - 2463)*s^5 + (2071*t^2 - 1458*t - 2409)*s^4 + (-11942*t^2 + 7756*t + 13370)*s^3 + (3558*t^2 - 2284*t - 3962)*s^2 + (2025*t^2 - 1310*t - 2263)*s - 257*t^2 + 166*t + 287)/((2888*t^2 - 104*t - 1816)*s^5 + (2600*t^2 - 400*t - 1880)*s^4 + (-15232*t^2 + 1176*t + 10080)*s^3 + (4544*t^2 - 312*t - 2976)*s^2 + (2584*t^2 - 192*t - 1704)*s - 328*t^2 + 24*t + 216), T],

[((-44*t^2 + 8405*t + 6773)*s^5 + (544*t^2 + 6413*t + 4797)*s^4 + (-980*t^2 - 42094*t - 33150)*s^3 + (188*t^2 + 12754*t + 10114)*s^2 + (168*t^2 + 7149*t + 5629)*s - 20*t^2 - 907*t - 715)/((3006*t^2 + 5580*t + 2548)*s^5 + (3154*t^2 + 3884*t + 1092)*s^4 + (-16436*t^2 - 27664*t - 11648)*s^3 + (4788*t^2 + 8496*t + 3744)*s^2 + (2766*t^2 + 4740*t + 2028)*s - 350*t^2 - 604*t - 260), ((229*t^2 + 2757*t + 2110)*s^5 + (107*t^2 + 2535*t + 2006)*s^4 + (-966*t^2 - 14714*t - 11424)*s^3 + (294*t^2 + 4394*t + 3408)*s^2 + (165*t^2 + 2497*t + 1938)*s - 21*t^2 - 317*t - 246)/((5528*t^2 + 562*t - 3058)*s^5 + (4608*t^2 + 614*t - 2430)*s^4 + (-28504*t^2 - 3108*t + 15596)*s^3 + (8568*t^2 + 900*t - 4716)*s^2 + (4848*t^2 + 522*t - 2658)*s - 616*t^2 - 66*t + 338), U],

[((3057*t^2 + 615*t - 1469)*s^5 + (1705*t^2 + 1931*t + 455)*s^4 + (-14006*t^2 - 6118*t + 4082)*s^3 + (4362*t^2 + 1578*t - 1534)*s^2 + (2377*t^2 + 1043*t - 689)*s - 303*t^2 - 129*t + 91)/((18724*t^2 - 21310*t - 29150)*s^5 + (17004*t^2 - 19682*t - 26738)*s^4 + (-99624*t^2 + 114212*t + 155764)*s^3 + (29800*t^2 - 34148*t - 46580)*s^2 + (16916*t^2 - 19390*t - 26446)*s - 2148*t^2 + 2462*t + 3358), ((1359*t^2 + 464*t - 699)*s^5 + (1077*t^2 + 572*t - 421)*s^4 + (-6958*t^2 - 2660*t + 3402)*s^3 + (2110*t^2 + 756*t - 1066)*s^2 + (1187*t^2 + 444*t - 587)*s - 151*t^2 - 56*t + 75)/((458*t^2 + 4338*t + 3044)*s^5 + (214*t^2 + 3950*t + 2892)*s^4 + (-1932*t^2 - 23100*t - 16520)*s^3 + (588*t^2 + 6908*t + 4936)*s^2 + (330*t^2 + 3922*t + 2804)*s - 42*t^2 - 498*t - 356), V]

代入{s = 0.112672939900111, t = 0.554958132087371}得到

[[-3.165571200, 2.524458851, A], [2.190644936, -1.746979509, B], [-0.7818317785, 0.6234900338, C], [0.9749277390, -0.2225208876, D], [0.4338839024, -0.9009685846, E], [-3.947405535, -0.9009686002, F], [0., 4.048914474, G], [-0., -2.801937733, H], [2.731687223, 0.6234899847, I], [-2.190644215, -1.746979102, J], [1.215715508, 2.524457369, K], [1.756758921, -3.647947731, L], [3.947401883, -0.9009688551, M], [0.7818315146, 0.6234900076, N], [-0.4338836247, -0.9009687151, O], [0, 1, P], [-0.9749283912, -0.2225210467, Q], [-1.215715565, 2.524458604, R], [-2.731687543, 0.6234898140, S], [3.165572936, 2.524457086, T], [-1.756759413, -3.647949374, U], [3.394638237*10^(-9), -1.068614566*10^(-8), V]]

画图得到：

数学星空

mathe 发表于 2019-12-31 09:23
转化自https://www2.stetson.edu/~efriedma/trees/ 的23棵树28行整数解

        A(+0,+0)

这个问题，还没有找到对称的构型

得到的计算结果如下：

初始数据：

[[A, 0, 0, 1], [B, 1/3, 1, 1], [C, 1/2, 3/2, 1], [D, 5/6, 5/2, 1], [E, -1/6, 1/2, 1], [F, 0, 1, 1], [G, 1/3, 2, 1], [H, 1/2, 5/2, 1], [I, -1/6, 3/2, 1], [J, 0, 2, 1], [K, -1/6, 5/2, 1], [L, -1, 4, 1], [M, 1, 1, 0], [N, 1, 0, 0], [O, 1, -1, 0], [P, 0, 1, 0], [Q, 1, 0, 1], [R, 1, 1, 1], [S, 5/6, 1/2, 1], [T, 1, 2, 1], [U, 5/6, 3/2, 1], [V, 1/2, 1/2, 1], [W, 1/3, 0, 1]]

[[M, N, O, P], [A, B, C, D], [E, F, G, H], [T, U, V, W], [A, F, J, P], [A, N, Q, W], [A, M, R, V], [B, E, M, U], [B, F, N, R], [B, I, O, S], [B, J, K, V], [B, G, P, W], [C, F, M, T], [C, I, N, U], [C, J, O, R], [C, G, Q, S], [C, H, P, V], [D, G, I, M], [D, H, K, N], [D, P, S, U], [E, I, K, P], [E, N, S, V], [F, O, Q, V], [F, I, L, W], [G, J, N, T], [G, K, O, U], [H, L, O, T], [P, Q, R, T]]

变换基点

P[0, 1, 0]-->[0, sqrt(3)/3, 1]
O[1, -1, 0]-->[-1/2, -sqrt(3)/6, 1]
T[1, 2, 1]--> [1/2, -sqrt(3)/6, 1]
B[1/3, 1, 1]-->[0, 0, 1]

变换矩阵

[[-1/4, 0, 1/12], [-1/60*sqrt(3), 1/15*sqrt(3), -11/180*sqrt(3)], [7/10, 1/5, -43/30]]

变换后坐标

[-5/86, (11*sqrt(3))/258, A], [0, 0, B], [5/94, -(11*sqrt(3))/282, C], [5/14, -(11*sqrt(3))/42, D], [-5/58, sqrt(3)/58, E], [-5/74, -sqrt(3)/222, F], [0, -sqrt(3)/12, G], [1/14, -sqrt(3)/6, H], [-1/10, -sqrt(3)/30, I], [-5/62, -(13*sqrt(3))/186, J], [-5/42, -(13*sqrt(3))/126, K], [-1/4, -sqrt(3)/6, L], [-5/18, sqrt(3)/18, M], [-5/14, -sqrt(3)/42, N], [-1/2, -sqrt(3)/6, O], [0, sqrt(3)/3, P], [5/22, (7*sqrt(3))/66, Q], [5/16, sqrt(3)/48, R], [1/6, sqrt(3)/18, S], [1/2, -sqrt(3)/6, T], [5/22, -sqrt(3)/22, U], [5/118, (13*sqrt(3))/354, V], [0, sqrt(3)/18, W]

画图得到：

mathe

这个23棵28行是无法对称的，但是去掉唯一的二度点L的图可以对称

数学星空

数学星空 发表于 2019-12-24 19:11
经过反复选点，我们终于得到了想要的图形

以 \(X[0.3819660120, 0.500000000000000, 1.]->[0,\frac{\sqr ...

24棵30行问题代数解

初始数据：
[[A, 0, 1, 1], [B, -t - 1, 1/2, 1], [C, 2*t + 1, (3*t)/2 + 1, 1], [D, 2*t - 1, t/2, 1], [E, 1 + t, -t/2, 1], [F, 1, 0, 1], [G, 1, 1 - t/2, 1], [H, -2*t - 1, -t/2, 1], [I, t, 1/2, 1], [J, 0, t/2 + 1/2, 1], [K, 1, -t/2, 0], [L, 1, t/2, 0], [M, -1, t/2, 1], [N, 2*t + 1, t/2 + 1, 1], [O, t, 1/2 - t/2, 1], [P, t, 0, 1], [Q, 1 + t, t/2 + 1/2, 1], [R, -t, t/2 + 1/2, 1], [S, 2*t + 1, -t/2, 1], [T, 0, 0, 1], [U, 0, 1, 0], [V, 1, 0, 0], [W, 1, t/2, 1], [X, 1 + t, 1/2, 1]]
[[K, L, U, V], [I, O, P, U], [L, N, O, X], [J, Q, R, V], [K, M, R, T], [H, N, Q, W], [G, M, P, S], [G, H, T, X], [I, J, S, W], [A, G, K, O], [B, H, L, R], [F, I, L, Q], [D, G, I, N], [C, H, J, M], [E, J, K, P], [A, J, T, U], [B, I, V, X], [F, G, U, W], [C, L, T, W], [D, K, S, X], [E, H, S, V], [A, R, W, X], [B, O, S, T], [E, Q, U, X], [D, M, V, W], [F, P, T, V], [C, N, S, U], [A, B, M, N], [E, F, O, R], [C, D, P, Q]]

变换基点
X[1 + t, 1/2, 1]-->[0, sqrt(3)/3, 1],
S[2*t + 1, -t/2, 1]-->[-1/2, -sqrt(3)/6, 1],
V[1, 0, 0]-->[1/2, -sqrt(3)/6, 1],
A[0, 1, 1]-->[0, 0, 1]

变换矩阵
[[1/12, (2 + 3*t)/(6 + 6*t), (-2 - 3*t)/(6 + 6*t)], [-1/36*sqrt(3), sqrt(3)*(1 + t)/(54 + 72*t), -sqrt(3)*(1 + t)/(54 + 72*t)], [1/6, (-7 - 7*t)/(9 + 12*t), (-5*t - 2)/(9 + 12*t)]]

变换后坐标
[0, 0, A], [(18 + 29*t)/(46 + 74*t), (-5*t - 3)*sqrt(3)/(54 + 84*t), B], [(-29 - 47*t)/(38 + 62*t), sqrt(3)*(2 + 3*t)/(42 + 72*t), C], [(-7*t - 4)/(20 + 34*t), sqrt(3)*(5*t + 3)/(18 + 42*t), D], [(-47 - 76*t)/(50 + 80*t), -sqrt(3)/6, E], [(29 + 47*t)/(14 + 26*t), sqrt(3)*(5 + 6*t)/(6 + 36*t), F], [(11 + 18*t)/(14 + 20*t), sqrt(3)*(2 + 3*t)/(24 + 18*t), G], [(47 + 76*t)/(22 + 36*t), -sqrt(3)/6, H], [(3 + 4*t)/(14 + 20*t), sqrt(3)*(5 + 8*t)/(42 + 60*t), I], [(-11 - 18*t)/(98 + 160*t), -sqrt(3)*t/(108 + 186*t), J], [(-11 - 18*t)/(28 + 46*t), (-2 - 3*t)*sqrt(3)/(54*t + 30), K], [(-18 - 29*t)/(14 + 24*t), sqrt(3)*(1 + t)/(30*t + 12), L], [(3 + 4*t)/(28*t + 14), -sqrt(3)/42, M], [(-29 - 47*t)/(34 + 54*t), sqrt(3)*(5 + 8*t)/(42 + 60*t), N], [(-7 - 11*t)/(4*t), -sqrt(3)*(5*t + 3)/(12*t), O], [(25 + 40*t)/(12*t + 6), sqrt(3)*(2 + 3*t)/(6*t), P], [(-29 - 47*t)/(62 + 102*t), sqrt(3)*(7 + 10*t)/(66 + 120*t), Q], [0, (-2*t - 1)*sqrt(3)/(33 + 57*t), R], [-1/2, -sqrt(3)/6, S], [(18 + 29*t)/(14 + 24*t), sqrt(3)*(1 + t)/(30*t + 12), T], [(-18 - 29*t)/(28 + 42*t), -sqrt(3)/42, U], [1/2, -sqrt(3)/6, V], [0, sqrt(3)*(1 + t)/(30*t + 12), W], [0, sqrt(3)/3, X]

t^2 - t - 1 = 0  得  t=1/2 + sqrt(5)/2, t=1/2 - sqrt(5)/2 取t=1/2 - sqrt(5)/2  得到

[[0., 0., A], [0.290089364147525, 0.0749007517504103, B], [-0.149627093977425, -0.101143945804483, C], [-0.322001788881577, 0.0196268107186746, D], [-0.0527864045000584, -0.288675134594814, E], [0.0230063541380380, -0.137696204920078, F], [-0.0760143110525830, 0.0196268107186740, G], [-0.118033988749936, -0.288675134594814, H], [0.322001788881574, 0.0196268107186730, I], [-0.140734607452550, -0.153928202283518, J], [-0.290089364147690, 0.0749007517504071, K], [0.0924746297157850, -0.101143945804478, L], [-0.159719141249983, -0.0412393049421162, M], [0.0760143110526519, 0.0196268107186730, N], [0.0815594803123328, -0.0210585672649966, O], [-0.196723314583159, -0.0681469551782790, P], [-0.0457902732839705, -0.173895061605221, Q], [0., -0.183524787833429, R], [-0.500000000000000, -0.288675134594814, S], [-0.0924746297157850, -0.101143945804478, T], [-0.0377045746428633, -0.0412393049421162, U], [0.500000000000000, -0.288675134594814, V], [0., -0.101143945804478, W], [0., 0.577350269189626, X]]

画图得到

数学星空

mathe 发表于 2019-12-31 14:38
对前面23棵28行添加一棵树两行得到24棵30行

有关139#得到的结果不太理想~~

初始数据
[[A, 0, 0, 1], [B, 1/3, 1, 1], [C, 1/2, 3/2, 1], [D, 5/6, 5/2, 1], [E, -1/6, 1/2, 1], [F, 0, 1, 1], [G, 1/3, 2, 1], [H, 1/2, 5/2, 1], [I, -1/6, 3/2, 1], [J, 0, 2, 1], [K, -1/6, 5/2, 1], [L, -1, 4, 1], [M, 1, 1, 0], [N, 1, 0, 0], [O, 1, -1, 0], [P, 0, 1, 0], [Q, 1, 0, 1], [R, 1, 1, 1], [S, 5/6, 1/2, 1], [T, 1, 2, 1], [U, 5/6, 3/2, 1], [V, 1/2, 1/2, 1], [W, 1/3, 0, 1], [X, 11/6, -3/2, 1]]

[[M, N, O, P], [A, B, C, D], [E, F, G, H], [T, U, V, W], [A, F, J, P], [A, N, Q, W], [A, M, R, V], [B, E, M, U], [B, F, N, R], [B, I, O, S], [B, J, K, V], [B, G, P, W], [C, F, M, T], [C, I, N, U], [C, J, O, R], [C, G, Q, S], [C, H, P, V], [D, G, I, M], [D, H, K, N], [D, P, S, U], [E, I, K, P], [E, N, S, V], [F, O, Q, V], [F, I, L, W], [G, J, N, T], [G, K, O, U], [H, L, O, T], [P, Q, R, T], [E, O, W, X], [H, R, U, X]]

变换基点
M[1, 1, 0]-->[0,(sqrt(3))/3,1],
A[0, 0, 1]-->[-1/2,-(sqrt(3))/6,1],
U[5/6, 3/2, 1]-->[1/2,-(sqrt(3))/6,1],
I[-1/6, 3/2, 1]-->[0,0,1]

变换矩阵
[[-1/15, 1/15, -1/9], [1/45*sqrt(3), 11/405*sqrt(3), -1/27*sqrt(3)], [13/15, -97/135, 2/9]]

变换后坐标
[-1/2, -sqrt(3)/6, A], [9/28, sqrt(3)/84, B], [2/19, -(2*sqrt(3))/57, C], [0, -(4*sqrt(3))/69, D], [9/38, (11*sqrt(3))/114, E], [6/67, (4*sqrt(3))/201, F], [0, -(2*sqrt(3))/75, G], [-3/154, -(17*sqrt(3))/462, H], [0, 0, I], [-3/164, -(7*sqrt(3))/492, J], [-9/232, -(11*sqrt(3))/696, K], [-6/95, -(4*sqrt(3))/285, L], [0, sqrt(3)/3, M], [-1/13, sqrt(3)/39, N], [-9/107, -sqrt(3)/321, O], [-9/97, -(11*sqrt(3))/291, P], [-8/49, -(2*sqrt(3))/147, Q], [-3/10, sqrt(3)/30, R], [-18/79, -(2*sqrt(3))/237, S], [6/47, -(16*sqrt(3))/141, T], [1/2, -sqrt(3)/6, U], [-3/8, -sqrt(3)/24, V], [-6/23, -(4*sqrt(3))/69, W], [-3/26, -sqrt(3)/78, X]

画图得到



原始数据图形



很显然原始数据图形要对称很多

@wayne 利用随机投影法
@mathe 利用群论计算方法

能否给出满意的构图（需将所有无穷远点化为普通点）？

mathe

中心对称映射为轴对称的统一规则为
i)将对称中心映射到无穷远点，比如上图中BC中点需要映射到无穷远点。
ii)无穷远直线映射为y轴作为对称轴
iii)将任意满足中心对称的两个点映射为关于对称轴左右对称关系，也就是两点纵坐标相等，横坐标互为相反数
比如图中A,D需要关于y轴对称；E,T需要关于y轴对称
iv)如果选择四点映射成矩形（比如ADTE），为了避免将已知点映射成无穷远点，那么需要保证这四点对边交点的连线上没有有用点。比如这个例子中由于AD和TE交点为对称中心，只要保证DT和AE的交点和对称中心连线上没有有用点就可以了，由于图像对称性，在原图，DT必然和AE平行，所以只要保证过中心做DT的平行线上没有有用点（包括无穷远点）即可。
所以如果我们换成选择HTEW就不行了，HT过中心的平行线经过HT上无穷远点，是有用的。事实上只要避开和原图上四个无穷远方向即可

wayne

数学星空 发表于 2020-1-9 19:45
有关139#得到的结果不太理想~~

初始数据

我这边的Mathematica代码的输入数据类似是这种格式

ACDFEFGKCIJKBEJLCEHMBFIMBCGNFHLNDGJOAHKOAGLPDINPBDHQAEIQAJMRBKPRCOQRDLMSENOSFPQS

然后你的格式不方便 大家交流, 我需要人眼+手工才能还原出出来,^_^

我通过编辑器字符串替换得到数据是

1. MNOPABCDEFGHTUVWAFJPANQWAMRVBEMUBFNRBIOSBJKVBGPWCFMTCINUCJORCGQSCHPVDGIMDHKNDPSUEIKPENSVFOQVFILWGJNTGKOUHLOTPQRTEOWXHRUX

复制代码

然后作图,找到一种最紧凑的图案,24-30:

数学星空

157#结果：