果树种植最优解精美图形作法探讨

mathe

给上面一个图看起来对称一些的结果

mathe

@数学星空
你的变换出来结果
L(-5/8, -7/8), R(-35/41, -19/41), Q(-11/13, -7/13)
三点很接近，但是没有问题。
对应图片

局部放大后为:

数学星空

25棵32行结果：

初始数据：
[[B, F, N, T], [A, B, C, D], [A, F, J, P], [A, N, S, Y], [A, M, T, X], [B, E, M, W], [B, I, O, U], [B, J, K, X], [B, G, P, Y], [C, F, M, V], [C, I, N, W], [C, J, O, T], [C, H, P, X], [C, G, S, U], [D, G, I, M], [D, H, K, N], [D, P, U, W], [E, F, G, H], [E, I, K, P], [E, N, U, X], [E, O, Q, Y], [F, I, L, Y], [F, O, S, X], [G, J, N, V], [G, K, O, W], [H, L, O, V], [H, Q, T, W], [I, Q, R, X], [J, L, R, S], [M, N, O, P], [P, S, T, V], [V, W, X, Y]]
[[A, 1, 3/2, 1], [B, 0, 1, 0], [C, 1, 0, 1], [D, 1, 2/3, 1], [E, 3/2, 2, 1], [F, 1, 2, 0], [G, 3/4, 1/2, 1], [H, 5/6, 2/3, 1], [I, 0, 0, 1], [J, 1/2, 1/2, 1], [K, 1/2, 2/3, 1], [L, 5/12, 5/6, 1], [M, 3/2, 1, 1], [N, 1, 0, 0], [O, 0, 1, 1], [P, 3/4, 1, 1], [Q, 3/10, 6/5, 1], [R, 5/16, 5/4, 1], [S, 1/4, 3/2, 1], [T, 1, -1, 0], [U, 0, 2, 1], [V, 5/4, 1/2, 1], [W, 3/2, 0, 1], [X, 1/2, 2, 1], [Y, 3/4, 3/2, 1]]

变换基点
B[0, 1, 0]-->[0, sqrt(3)/3, 1]
E[3/2, 2, 1]--> [-1/2, -sqrt(3)/6, 1]
T[1, -1, 0]--> [1/2, -sqrt(3)/6, 1]
K[1/2, 2/3, 1]-->[0, 0, 1]

变换矩阵
[[1/6, 0, -1/12], [-1/126*sqrt(3), 1/21*sqrt(3), -1/36*sqrt(3)], [10/21, 1/7, -4/3]]

变换后坐标
[-7/54, -sqrt(3)/18, A], [0, sqrt(3)/3, B], [-7/72, sqrt(3)/24, C], [-7/64, sqrt(3)/192, D], [-1/2, -sqrt(3)/6, E], [7/32, (11*sqrt(3))/96, F],
[-7/152, (5*sqrt(3))/456, G], [-7/106, sqrt(3)/318, H], [1/16, sqrt(3)/48, I], [0, sqrt(3)/129, J], [0, 0, K], [7/512, -(13*sqrt(3))/1536, L],
[-7/20, -sqrt(3)/60, M], [7/20, -sqrt(3)/60, N], [7/100, -sqrt(3)/60, O], [-1/20, -sqrt(3)/60, P], [7/214, -(17*sqrt(3))/642, Q], [21/676, -(59*sqrt(3))/2028, R],
[1/24, -sqrt(3)/24, S], [1/2, -sqrt(3)/6, T], [7/88, -(17*sqrt(3))/264, U], [-3/16, sqrt(3)/48, V], [-7/26, (5*sqrt(3))/78, W], [0, -(4*sqrt(3))/51, X], [-7/128, -(19*sqrt(3))/384, Y]

画图得到

数学星空

22棵26行结果：

初始数据：
[[A, K, O, U], [A, B, C, D], [A, G, L, Q], [A, F, N, T], [B, F, J, S], [B, K, R, T], [B, G, P, U], [B, H, L, V], [B, E, I, M], [C, G, K, N], [C, H, O, T], [C, E, L, U], [C, I, Q, V], [D, E, G, J], [D, I, N, S], [D, L, O, R], [D, M, Q, U], [E, H, K, S], [F, G, H, I], [F, K, P, V], [I, L, P, T], [J, K, L, M], [J, N, R, U], [M, P, R, S], [N, O, P, Q], [S, T, U, V]]
[[A, 1, 1/3, 0], [B, 0, 0, 1], [C, 1, 1/3, 1], [D, 3/2, 1/2, 1], [E, 4/3, 1/3, 1], [F, 4, 2, 1], [G, 1, 0, 1], [H, 4/3, 2/9, 1], [I, 8/5, 2/5, 1], [J, 2, 1, 1], [K, 0, 1, 0], [L, 2, 1/3, 1], [M, 2, 1/2, 1], [N, 1, 1, 1], [O, 1, -1/3, 0], [P, 4, 0, 1], [Q, 5/2, 1/2, 1], [R, 0, 1, 1], [S, 4/3, 2/3, 1], [T, 0, 2/3, 1], [U, 1, 0, 0], [V, 4, 2/3, 1]]

变换基点：
O[1, -1/3, 0]-->[-1, 1, 1]
G[1, 0, 1]--> [-1, -1, 1]
P[4, 0, 1]-->[1, -1, 1]
H[4/3, 2/9, 1]--> [1, 1, 1]

变换矩阵
[[-1/2, 3, -1], [3/2, 0, -3], [-3/2, -9, 3]]

变换后坐标
[-1/9, -1/3, A], [-1/3, -1, B], [1/3, 1, C], [1/15, 1/5, D], [1/3, 1/2, E], [-1/7, -1/7, F], [-1, -1, G], [1, 1, H], [1/5, 1/5, I], [-1/9, 0, J],
[-1/3, 0, K], [1/3, 0, L], [1/9, 0, M], [-1/5, 1/5, N], [-1, 1, O], [1, -1, P], [1/7, -1/7, Q], [-1/3, 1/2, R], [-1/15, 1/5, S], [-1/3, 1, T], [1/3, -1, U], [1/9, -1/3, V]

画图得到：

数学星空

21棵24行结果：

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

变换基点
S[0, 1, 0]-->[0, sqrt(3)/3, 1]
Q[1, 3, 0]-->[-1/2, -sqrt(3)/6, 1]
K[-1, 2, 1]-->[1/2, -sqrt(3)/6, 1]
C[2/3, 2, 1]-->[0, 0, 1]

变换矩阵
[[1/10, 0, -1/15], [-1/30*sqrt(3), 1/45*sqrt(3), -1/45*sqrt(3)], [-2/5, 1/15, -13/15]]

变换后坐标
[1/68, (11*sqrt(3))/204, A], [5/11, -(4*sqrt(3))/33, B], [0, 0, C], [-1/42, sqrt(3)/14, D], [0, sqrt(3)/48, E], [7/34, -(5*sqrt(3))/102, F], [1/70, (13*sqrt(3))/210, G], [1/4, -sqrt(3)/24, H], [-1/34, sqrt(3)/102, I], [1/58, sqrt(3)/174, J], [1/2, -sqrt(3)/6, K], [0, sqrt(3)/18, L], [1/12, 0, M], [-1/38, (5*sqrt(3))/114, N], [-1/6, sqrt(3)/6, O], [1/14, sqrt(3)/21, P], [-1/2, -sqrt(3)/6, Q], [5/14, -sqrt(3)/42, R], [0, sqrt(3)/3, S], [1/13, sqrt(3)/39, T], [-1/4, sqrt(3)/12, U]

画图得到：

mathe

你少画直线了，如IQRS没做出来

数学星空

数学星空 发表于 2019-12-19 22:04
经过楼上mathe的指点终于得到了想要的图形：

（24棵30行）

现在给出代数表达式：

初始数据：
[[A, 1, v0, 0], [B, v2, v9, 1], [C, v2, v3, 1], [D, 1, 0, 1], [E, 0, 1, 1], [F, v5, v4, 1], [G, 0, v10, 1], [H, v6, v11, 1], [I, v12, v7, 1], [J, 1, v1, 0], [K, v8, 0, 1], [L, v13, v4, 1], [M, v14, 1, 1], [N, v5, v3, 1], [O, v2, v4, 1], [P, 1, 0, 0], [Q, v6, v15, 1], [R, v5, v16, 1], [S, v6, v3, 1], [T, 0, 0, 1], [U, v17, v7, 1], [V, v8, 1, 1], [W, 0, 1, 0], [X, v8, v7, 1]]

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

解坐标方程得到

[[A, 1, t, 0], [B, -t^2 + 6*t - 4, -2*t^2 + 11*t - 4, 1], [C, -t^2 + 6*t - 4, -2*t^2 + 11*t - 5, 1], [D, 1, 0, 1], [E, 0, 1, 1], [F, -4*t^2 + 23*t - 13, -t^2 + 5*t - 2, 1], [G, 0, t, 1], [H, -7*t^2 + 40*t - 23, -2*t^2 + 11*t - 6, 1], [I, 6*t^2 - 34*t + 19, t^2 - 6*t + 4, 1], [J, 1, -2*t^2 + 11*t - 3, 0], [K, 3*t^2 - 17*t + 10, 0, 1], [L, -2*t^2 + 12*t - 7, -t^2 + 5*t - 2, 1], [M, 2*t^2 - 11*t + 6, 1, 1], [N, -4*t^2 + 23*t - 13, -2*t^2 + 11*t - 5, 1], [O, -t^2 + 6*t - 4, -t^2 + 5*t - 2, 1], [P, 1, 0, 0], [Q, -7*t^2 + 40*t - 23, -3*t^2 + 17*t - 9, 1], [R, -4*t^2 + 23*t - 13, -t^2 + 6*t - 3, 1], [S, -7*t^2 + 40*t - 23, -2*t^2 + 11*t - 5, 1], [T, 0, 0, 1], [U, 9*t^2 - 51*t + 29, t^2 - 6*t + 4, 1], [V, 3*t^2 - 17*t + 10, 1, 1], [W, 0, 1, 0], [X, 3*t^2 - 17*t + 10, t^2 - 6*t + 4, 1]]

其中：t^3 - 6*t^2 + 5*t - 1=0

基点变换

Q[-7*t^2 + 40*t - 23, -3*t^2 + 17*t - 9, 1]-->[-1, 1, 1]
V[3*t^2 - 17*t + 10, 1, 1]--> [-1, -1, 1]
U[9*t^2 - 51*t + 29, t^2 - 6*t + 4, 1]--> [1, -1, 1]
N[-4*t^2 + 23*t - 13, -2*t^2 + 11*t - 5, 1]-->[1, 1, 1]

变换矩阵

[[(1656*t^2 - 9426*t + 5377)/(2065*t^2 - 11754*t + 6705), (-6705*t^2 + 38165*t - 21771)/(2065*t^2 - 11754*t + 6705), (-3721*t^2 + 21180*t - 12082)/(2065*t^2 - 11754*t + 6705)], [(-636*t^2 + 3620*t - 2065)/(2065*t^2 - 11754*t + 6705), (2575*t^2 - 14657*t + 8361)/(2065*t^2 - 11754*t + 6705), (-1429*t^2 + 8134*t - 4640)/(2065*t^2 - 11754*t + 6705)], [(182*t^2 - 1036*t + 591)/(2065*t^2 - 11754*t + 6705), (737*t^2 - 4195*t + 2393)/(2065*t^2 - 11754*t + 6705), (409*t^2 - 2328*t + 1328)/(2065*t^2 - 11754*t + 6705)]]

变换后坐标
1, A[(-409*t^2 + 2328*t - 1328)/(409*t^2 - 2328*t + 1328), (157*t^2 - 894*t + 510)/(409*t^2 - 2328*t + 1328), 1],
2, B[-(2065*t^2 - 11754*t + 6705)/(-591*t^2 + 3364*t - 1919), -(2065*t^2 - 11754*t + 6705)/(-591*t^2 + 3364*t - 1919), 1],
3, C[(4640*t^2 - 26411*t + 15066)/(-1328*t^2 + 7559*t - 4312), (-4640*t^2 + 26411*t - 15066)/(-1328*t^2 + 7559*t - 4312), 1],
4, D[-(2065*t^2 - 11754*t + 6705)/(591*t^2 - 3364*t + 1919), -(2065*t^2 - 11754*t + 6705)/(591*t^2 - 3364*t + 1919), 1],
5, E[(-10426*t^2 + 59345*t - 33853)/(1146*t^2 - 6523*t + 3721), 1, 1],
6, F[(-10426*t^2 + 59345*t - 33853)/(-1146*t^2 + 6523*t - 3721), (1146*t^2 - 6523*t + 3721)/(-1146*t^2 + 6523*t - 3721), 1],
7, G[(-5786*t^2 + 32934*t - 18787)/(636*t^2 - 3620*t + 2065), (-636*t^2 + 3620*t - 2065)/(636*t^2 - 3620*t + 2065), 1],
8, H[1, (-1429*t^2 + 8134*t - 4640)/(-3721*t^2 + 21180*t - 12082), 1],
9, I[-3721*t^2 + 21180*t - 12082)/(3721*t^2 - 21180*t + 12082), (-1429*t^2 + 8134*t - 4640)/(3721*t^2 - 21180*t + 12082), 1],
10, J[1, (-126*t^2 + 717*t - 409)/(328*t^2 - 1867*t + 1065), 1],
11, K[(4640*t^2 - 26411*t + 15066)/(1328*t^2 - 7559*t + 4312), (-4640*t^2 + 26411*t - 15066)/(1328*t^2 - 7559*t + 4312), 1],
12, L[(-5786*t^2 + 32934*t - 18787)/(-636*t^2 + 3620*t - 2065), 1, 1],
13, M[(-5786*t^2 + 32934*t - 18787)/(1656*t^2 - 9426*t + 5377), (-636*t^2 + 3620*t - 2065)/(1656*t^2 - 9426*t + 5377), 1],
14, N[1, 1, 1],
15, O[(-3721*t^2 + 21180*t - 12082)/(-409*t^2 + 2328*t - 1328), (-1429*t^2 + 8134*t - 4640)/(-409*t^2 + 2328*t - 1328), 1],
16, P[(1656*t^2 - 9426*t + 5377)/(182*t^2 - 1036*t + 591), (-636*t^2 + 3620*t - 2065)/(182*t^2 - 1036*t + 591), 1],
17, Q[(4640*t^2 - 26411*t + 15066)/(-4640*t^2 + 26411*t - 15066), 1, 1],
18, R[(-5786*t^2 + 32934*t - 18787)/(-1656*t^2 + 9426*t - 5377), (-636*t^2 + 3620*t - 2065)/(-1656*t^2 + 9426*t - 5377), 1],
19, S[(-10426*t^2 + 59345*t - 33853)/(-2984*t^2 + 16985*t - 9689), (1146*t^2 - 6523*t + 3721)/(-2984*t^2 + 16985*t - 9689), 1],
20, T[(-3721*t^2 + 21180*t - 12082)/(409*t^2 - 2328*t + 1328), (-1429*t^2 + 8134*t - 4640)/(409*t^2 - 2328*t + 1328), 1],
21, U[1, (-4640*t^2 + 26411*t - 15066)/(4640*t^2 - 26411*t + 15066), 1],
22, V[-1, -1, 1],
23, W[(-6705*t^2 + 38165*t - 21771)/(737*t^2 - 4195*t + 2393), (2575*t^2 - 14657*t + 8361)/(737*t^2 - 4195*t + 2393), 1],
24, X[ (-10426*t^2 + 59345*t - 33853)/(2984*t^2 - 16985*t + 9689), (1146*t^2 - 6523*t + 3721)/(2984*t^2 - 16985*t + 9689), 1]

取t=0.307978528369904,(注：另外两个根为0.643104132107791, 5.04891733952231）

[-1.000000000, 0.3840429432, A], [3.493959206, 3.493959206, B], [-3.493959207, 3.493959207, C], [-3.493959207, -3.493959207, D], [-9.097834679, 1., E], [9.097834679, -1.000000000, F], [-9.097834681, -1.000000000, G], [1., 0.3840429434, H], [-1.000000000, -0.3840429434, I], [1., -0.3840429434, J], [3.493959207, -3.493959207, K], [9.097834681, 1., L], [-3.493959207, -0.3840429431, M], [1., 1., N], [9.097834677, 3.493959208, O], [9.097834680, -3.493959206, P], [-1.000000000, 1., Q], [3.493959207, 0.3840429431, R], [3.493959207, -0.3840429432, S], [-9.097834677, -3.493959208, T], [1., -1.000000000, U], [-1., -1., V], [-9.097834675, 3.493959207, W], [-3.493959207, 0.3840429432, X]

画图得到

wayne

@数学星空， 可否提供一下楼上的代码。我学习学习。

dingjifen

20棵树种植问题的内容是啥？

mathe

dingjifen 发表于 2020-1-3 09:31
20棵树种植问题的内容是啥？

这个内容很复杂，你看一下综述吧:
https://emathgroup.github.io/blog/orchard-planting-problem