平移镶嵌：用向量构造平行四边形网格

几何画板及绘图内容加载中，请稍等……

2014-03-30 05:26

所属专题 Tesselation by Translation Tesselation by Translation

该范例演示了如何通过平移操作生成平面镶嵌图案，即利用一个基本多边形（平行四边形）沿两个方向反复平移，形成覆盖整个平面的无缝网格。核心数学概念是平移变换、向量、平行四边形以及镶嵌（tessellation）的几何逻辑。作图要点：首先用Polygon命令创建基础多边形poly1，然后定义向量u和v（由点构造），使用Translate命令将poly1沿向量u和v分别平移，得到平移后的副本（如stroke1'、stroke2'等），再通过AttachCopyToView固定视图，并利用Point、Vector等对象控制平移方向和距离。该范例适合中学几何或数学拓展课程，帮助学生直观理解平移变换在图案设计中的应用，以及向量在几何构造中的核心作用。

GeoGebra 命令及参数

Polygon(A, B, 4) → poly1, a, b, c, d, C, D
AttachCopyToView(PolyLine[(3.7, -3.7600000000000002), (3.62, -3.7600000000000002), (3.54, -3.7800000000000002), (3.48, -3.8200000000000003), (3.42, -3.86), (3.36, -3.9), (3.3000000000000003, -3.94), (3.24, -3.98), (3.18, -4.04), (3.1, -4.1), (3.02, -4.16), (2.96, -4.18), (2.84, -4.24), (2.7800000000000002, -4.26), (2.7, -4.32), (2.64, -4.34), (2.56, -4.4), (2.48, -4.42), (2.4, -4.46), (2.32, -4.48), (2.2600000000000002, -4.5200000000000005), (2.2, -4.58), (2.14, -4.62), (2.08, -4.68), (2.02, -4.72), (2.02, -4.64), (2.02, -4.5600000000000005), (2.02, -4.48), (2.02, -4.38), (2.02, -4.3), (2.02, -4.22), (2.08, -4.18), (2.14, -4.16), (2.2, -4.14), (2.2600000000000002, -4.12), (2.32, -4.08), (2.38, -4.0600000000000005), (2.36, -4), (2.3000000000000003, -3.98), (2.24, -3.94), (2.16, -3.9), (2.1, -3.88), (2.04, -3.8200000000000003), (1.98, -3.7800000000000002), (1.92, -3.7600000000000002), (1.86, -3.74), (1.82, -3.68), (1.76, -3.64), (1.74, -3.58), (1.8, -3.54), (1.86, -3.48), (1.92, -3.46), (1.98, -3.42), (2.06, -3.38), (2.12, -3.3200000000000003), (2.18, -3.3000000000000003), (2.24, -3.2600000000000002), (2.32, -3.2600000000000002), (2.38, -3.22), (2.44, -3.18), (2.5, -3.16), (2.56, -3.1), (2.62, -3.08), (2.68, -3.06), (2.6, -3.06), (2.5, -3.06), (2.4, -3.06), (2.3000000000000003, -3.06), (2.22, -3.06), (2.14, -3.06), (2.06, -3.06), (1.98, -3.06), (1.9000000000000001, -3.06), (1.82, -3.02), (1.74, -3), (1.6400000000000001, -3), (1.58, -2.98), (1.5, -2.98), (1.42, -2.98), (1.34, -2.98), (1.26, -2.98), (1.18, -3), (1.1, -3), (1.02, -3), (0.9400000000000001, -3), (0.86, -3), (0.78, -3), (0.7000000000000001, -3), (0.64, -2.98), (0.56, -2.98), (0.48, -2.98), (0.42, -2.96), (0.36, -3), (0.3, -3.02), (0.22, -3.02), (0.16, -3.04), (0.1, -3.06), (0.02, -3.08), (-0.06, -3.08), (-0.14, -3.08), (-0.22, -3.08), (-0.22, -3.16), (-0.22, -3.24), (-0.24, -3.3000000000000003), (-0.26, -3.36), (-0.32, -3.4), (-0.38, -3.42), (-0.44, -3.46), (-0.5, -3.5), (-0.56, -3.52), (-0.62, -3.54), (-0.68, -3.56), (-0.74, -3.58), (-0.8, -3.6), (-0.8, -3.68), (-0.76, -3.74), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke1
Point(a) → E
Point(c) → F
Vector(E, F) → u
Translate(stroke1, u) → stroke1'
AttachCopyToView(PolyLine[(-0.78, -3.7600000000000002), (-0.86, -3.7600000000000002), (-0.92, -3.7800000000000002), (-0.98, -3.8000000000000003), (-1.06, -3.8200000000000003), (-1.12, -3.84), (-1.12, -3.74), (-1.12, -3.62), (-1.1, -3.54), (-1.06, -3.48), (-1, -3.42), (-0.9400000000000001, -3.38), (-0.88, -3.36), (-0.8200000000000001, -3.3200000000000003), (-0.76, -3.2800000000000002), (-0.74, -3.22), (-0.74, -3.14), (-0.74, -3.02), (-0.74, -2.94), (-0.74, -2.84), (-0.74, -2.7600000000000002), (-0.7000000000000001, -2.7), (-0.64, -2.68), (-0.56, -2.68), (-0.48, -2.64), (-0.4, -2.62), (-0.34, -2.6), (-0.28, -2.56), (-0.22, -2.52), (-0.16, -2.5), (-0.1, -2.46), (-0.04, -2.42), (0.02, -2.38), (0.06, -2.3000000000000003), (0.06, -2.22), (0.06, -2.14), (0.04, -2.08), (0.02, -2.02), (0.02, -1.94), (0.02, -1.86), (-0.06, -1.86), (-0.14, -1.86), (-0.2, -1.82), (-0.28, -1.82), (-0.36, -1.82), (-0.44, -1.82), (-0.5, -1.8), (-0.58, -1.78), (-0.64, -1.76), (-0.62, -1.82), (-0.58, -1.9000000000000001), (-0.52, -1.96), (-0.46, -2), (-0.42, -2.06), (-0.36, -2.1), (-0.34, -2.16), (-0.28, -2.18), (-0.36, -2.18), (-0.44, -2.18), (-0.52, -2.2), (-0.6, -2.22), (-0.68, -2.22), (-0.74, -2.2), (-0.8, -2.14), (-0.8200000000000001, -2.08), (-0.84, -2.02), (-0.88, -1.96), (-0.9, -1.9000000000000001), (-0.9400000000000001, -1.84), (-0.96, -1.76), (-0.98, -1.7), (-1, -1.6400000000000001), (-1.04, -1.58), (-1.06, -1.52), (-1.08, -1.46), (-1.02, -1.44), (-0.9400000000000001, -1.42), (-0.88, -1.4000000000000001), (-0.8200000000000001, -1.3800000000000001), (-0.74, -1.36), (-0.66, -1.34), (-0.6, -1.32), (-0.54, -1.3), (-0.48, -1.28), (-0.4, -1.28), (-0.32, -1.28), (-0.24, -1.28), (-0.16, -1.28), (-0.1, -1.26), (-0.04, -1.24), (0.04, -1.24), (0.1, -1.22), (0.12, -1.16), (0.14, -1.08), (0.2, -1), (0.24, -0.92), (0.28, -0.86), (0.3, -0.8), (0.34, -0.74), (0.38, -0.68), (0.42, -0.62), (0.48, -0.56), (0.54, -0.5), (0.6, -0.46), (0.66, -0.4), (0.72, -0.36), (0.78, -0.32), (0.8, -0.26), (0.8200000000000001, -0.2), (0.86, -0.14), (0.9, -0.08), (0.92, -0.02), (0.92, 0.06), (0.9, 0.14), (0.86, 0.2), (0.8200000000000001, 0.26), (0.8, 0.32), (0.84, 0.4), (0.88, 0.46), (0.9400000000000001, 0.54), (0.96, 0.6), (1.02, 0.66), (1.02, 0.74), (1.06, 0.8), (1, 0.84), (0.9, 0.8200000000000001), (0.8200000000000001, 0.8200000000000001), (0.74, 0.8200000000000001), (0.66, 0.8200000000000001), (0.58, 0.8200000000000001), (0.5, 0.8200000000000001), (0.42, 0.8200000000000001), (0.34, 0.8200000000000001), (0.26, 0.8200000000000001), (0.2, 0.8), (0.14, 0.74), (0.08, 0.7000000000000001), (0.04, 0.64), (-0.02, 0.62), (-0.08, 0.58), (-0.14, 0.54), (-0.2, 0.48), (-0.26, 0.44), (-0.34, 0.44), (-0.42, 0.44), (-0.48, 0.46), (-0.54, 0.5), (-0.62, 0.5), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke2
Point(d) → G
Point(b) → H
Vector(G, H) → v
Translate(stroke2, v) → stroke2'
Point(stroke1') → I
Point(stroke1') → J
Point(stroke1') → K
Point(stroke1') → L
Point(stroke1') → M
Point(stroke1') → N
Point(stroke1') → O
Point(stroke1') → P
Point(stroke1') → Q
Point(stroke1') → R
Point(stroke1') → S
Point(stroke1') → T
Point(stroke2') → U
Point(stroke2') → V
Point(stroke2') → W
Point(stroke2') → Z
Point(stroke2') → A_1
Point(stroke2') → B_1
Point(stroke2') → C_1
Point(stroke2') → D_1
Point(stroke2') → E_1
Point(stroke2') → F_1
Point(stroke2') → G_1
Point(stroke2') → I_1
Point(stroke2') → J_1
Point(stroke2') → K_1
Point(stroke2') → L_1
Point(stroke1) → M_1
Point(stroke1) → N_1
Point(stroke1) → O_1
Point(stroke1) → P_1
Point(stroke1) → Q_1
Point(stroke1) → R_1
Point(stroke1) → S_1
Point(stroke1) → T_1
Point(stroke1) → U_1
Point(stroke1) → V_1
Point(stroke2) → W_1
Point(stroke2) → Z_1
Point(stroke2) → A_2
Point(stroke2) → B_2
Point(stroke2) → C_2
Point(stroke2) → D_2
Point(stroke2) → E_2
Point(stroke2) → F_2
Point(stroke2) → G_2
Point(stroke2) → H_2
Point(stroke2) → I_2
Point(stroke2) → J_2
Point(stroke2) → K_2
Point(stroke2) → L_2
Point(stroke2) → M_2
Point(stroke2) → N_2
Polygon(D, I, R, S, T, C, U, V, W, Z, A_1, B_1, J, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, K_1, L_1, K, B, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1, U_1, L, V_1, A, W_1, Z_1, A_2, B_2, C_2, D_2, E_2, F_2, M, G_2, H_2, I_2, J_2, K_2, F, L_2, M_2, N_2, N, O, P, Q) → poly2, d_1, q, r, s, t, c_1, u_1, v_1, w, z, a_1, i, b_1, c_2, d_2, e_1, f_1, g_1, h_1, i_1, j_1, k_1, j, l_1, b_2, m_1, n_1, o_1, p_1, q_1, r_1, s_1, t_1, k, u_2, v_2, a_2, w_1, z_1, a_3, b_3, c_3, d_3, e_2, l, f_2, g_2, h_2, i_2, j_2, k_2, f, l_2, m_2, n_2, m, n, o, p
Vector(W_1, I) → e
Translate(stroke1, e) → stroke1'_1
Translate(stroke1', e) → stroke1''
Translate(stroke2, e) → stroke2'_1
Translate(stroke2', e) → stroke2''
Translate(D, e) → D'
Translate(I, e) → I'
Translate(J, e) → J'
Translate(K, e) → K'
Translate(L, e) → L'
Translate(M, e) → M'
Translate(N, e) → N'
Translate(O, e) → O'
Translate(P, e) → P'
Translate(Q, e) → Q'
Translate(R, e) → R'
Translate(S, e) → S'
Translate(T, e) → T'
Translate(C, e) → C'
Translate(U, e) → U'
Translate(V, e) → V'
Translate(W, e) → W'
Translate(Z, e) → Z'
Translate(A_1, e) → O_2
Translate(B_1, e) → P_2
Translate(C_1, e) → Q_2
Translate(D_1, e) → R_2
Translate(E_1, e) → S_2
Translate(F_1, e) → T_2
Translate(G_1, e) → U_2
Translate(H_1, e) → V_2
Translate(I_1, e) → W_2
Translate(J_1, e) → Z_2
Translate(K_1, e) → A_3
Translate(L_1, e) → B_3
Translate(B, e) → B'
Translate(M_1, e) → C_3
Translate(N_1, e) → D_3
Translate(O_1, e) → E_3
Translate(P_1, e) → F_3
Translate(Q_1, e) → G_3
Translate(R_1, e) → H_3
Translate(S_1, e) → I_3
Translate(T_1, e) → J_3
Translate(U_1, e) → K_3
Translate(V_1, e) → L_3
Translate(A, e) → A'
Translate(W_1, e) → M_3
Translate(Z_1, e) → N_3
Translate(A_2, e) → O_3
Translate(B_2, e) → P_3
Translate(C_2, e) → Q_3
Translate(D_2, e) → R_3
Translate(E_2, e) → S_3
Translate(F_2, e) → T_3
Translate(G_2, e) → U_3
Translate(H_2, e) → V_3
Translate(I_2, e) → W_3
Translate(J_2, e) → Z_3
Translate(K_2, e) → A_4
Translate(F, e) → F'
Translate(L_2, e) → B_4
Translate(M_2, e) → C_4
Translate(N_2, e) → D_4
Polygon(D', I', R', S', T', C', U', V', W', Z', O_2, P_2, J', Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, K', B', C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, K_3, L', L_3, A', M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, M', U_3, V_3, W_3, Z_3, A_4, F', B_4, C_4, D_4, N', O', P', Q') → poly2', d', q', r', s', t', c', u', v', w', z', o_2, i', p_2, q_2, r_2, s_2, t_2, u_3, v_3, w_2, z_2, a_4, j', b_4, b', c_4, d_4, e_3, f_3, g_3, h_3, i_3, j_3, k', k_3, l_3, a', m_3, n_3, o_3, p_3, q_3, r_3, s_3, l', t_3, u_4, v_4, w_3, z_3, a_5, f', b_5, c_5, d_5, m', n', o', p'
Translate(E, e) → E'
Translate(G, e) → G'
Translate(H, e) → H'
AttachCopyToView(PolyLine[(0, 5.5600000000000005), (0.08, 5.5600000000000005), (0.14, 5.6000000000000005), (0.2, 5.62), (0.26, 5.66), (0.32, 5.7), (0.38, 5.72), (0.46, 5.72), (NaN, NaN), (0.04, 5.58), (0.12, 5.6000000000000005), (0.18, 5.64), (0.22, 5.7), (0.28, 5.72), (0.34, 5.74), (0.42, 5.74), (0.5, 5.74), (0.58, 5.74), (NaN, NaN), (0.2, 5.5600000000000005), (0.26, 5.54), (0.32, 5.5600000000000005), (0.34, 5.62), (0.34, 5.7), (NaN, NaN), (-0.32, 5.3), (-0.24, 5.32), (-0.18, 5.34), (-0.16, 5.28), (-0.22, 5.24), (NaN, NaN), (-0.12, 5.1000000000000005), (-0.1, 5.16), (-0.08, 5.24), (-0.04, 5.3), (0.02, 5.32), (0.08, 5.36), (0.16, 5.36), (NaN, NaN), (2.4, 3.9), (2.42, 3.96), (2.46, 4.0200000000000005), (2.52, 4.0600000000000005), (2.58, 4.12), (2.66, 4.16), (2.72, 4.22), (2.7800000000000002, 4.26), (2.84, 4.28), (2.9, 4.32), (2.96, 4.34), (3.02, 4.36), (3.1, 4.38), (3.16, 4.42), (3.24, 4.46), (3.2600000000000002, 4.5200000000000005), (3.3000000000000003, 4.58), (3.36, 4.62), (3.44, 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3.7600000000000002), (4.2, 3.72), (4.26, 3.68), (4.3, 3.62), (4.34, 3.56), (4.4, 3.52), (4.44, 3.46), (4.5, 3.42), (4.5600000000000005, 3.4), (4.58, 3.34), (NaN, NaN), (3.52, 4.76), (3.5, 4.7), (3.5, 4.6000000000000005), (3.48, 4.5200000000000005), (3.46, 4.44), (3.46, 4.36), (3.46, 4.28), (3.46, 4.2), (3.46, 4.12), (3.46, 4.04), (3.46, 3.96), (3.46, 3.88), (3.46, 3.8000000000000003), (3.46, 3.72), (3.46, 3.64), (3.46, 3.56), (3.48, 3.5), (3.5, 3.44), (3.52, 3.38), (3.56, 3.3200000000000003), (3.58, 3.24), (3.58, 3.16), (NaN, NaN), (2.4, 3.96), (2.48, 3.96), (2.56, 3.96), (2.66, 3.96), (2.72, 3.92), (2.7800000000000002, 3.86), (2.84, 3.8200000000000003), (2.86, 3.7600000000000002), (2.92, 3.72), (2.98, 3.66), (3.02, 3.6), (3.04, 3.54), (3.1, 3.46), (3.12, 3.4), (3.16, 3.34), (3.22, 3.3000000000000003), (3.2800000000000002, 3.24), (3.34, 3.2), (3.4, 3.16), (3.46, 3.12), (3.52, 3.08), (3.56, 3.02), (3.62, 3), (NaN, NaN), (-0.1, 1.04), (-0.04, 1.06), (0.02, 1.08), (0.08, 1.1), (0.14, 1.1400000000000001), (0.2, 1.16), (0.28, 1.16), (NaN, NaN), (0.02, 1.02), (0.1, 1.04), (0.16, 1.08), (NaN, NaN), (-0.52, 0.62), (-0.46, 0.66), (NaN, NaN), (2.1, -0.66), (2.12, -0.56), (2.18, -0.48), (2.22, -0.4), (2.2800000000000002, -0.34), (2.36, -0.3), (2.4, -0.24), (2.48, -0.18), (2.56, -0.12), (2.62, -0.08), (2.68, -0.06), (2.7600000000000002, -0.06), (2.84, -0.06), (2.92, -0.06), (3, -0.06), (3.06, -0.04), (3.14, 0.02), (3.2, 0.04), (3.2600000000000002, 0.06), (3.3200000000000003, 0.1), (3.4, 0.14), (3.48, 0.14), (3.56, 0.14), (3.64, 0.16), (3.7, 0.22), (3.7800000000000002, 0.22), (3.84, 0.24), (3.92, 0.24), (4, 0.24), (4.0600000000000005, 0.26), (4.12, 0.3), (4.18, 0.32), (4.24, 0.34), (4.28, 0.4), (4.34, 0.46), (4.38, 0.52), (4.44, 0.58), (4.48, 0.64), (NaN, NaN), (3.7, 0.22), (3.7800000000000002, 0.22), (3.86, 0.22), (3.94, 0.22), (4.04, 0.22), (4.1, 0.2), (4.18, 0.2), (4.26, 0.18), (4.32, 0.16), (4.4, 0.14), (4.46, 0.12), (4.5200000000000005, 0.1), (4.58, 0.06), (4.64, 0), (4.7, -0.02), (4.76, -0.06), (4.82, -0.08), (4.9, -0.08), (4.96, -0.1), (5.0200000000000005, -0.12), (5.1000000000000005, -0.12), (5.18, -0.12), (NaN, NaN), (3.74, 0.22), (3.74, 0.14), (3.7800000000000002, 0.08), (3.8000000000000003, 0.02), (3.84, -0.04), (3.86, -0.1), (3.9, -0.16), (3.92, -0.22), (3.94, -0.28), (3.96, -0.34), (4, -0.4), (4.0600000000000005, -0.42), (4.08, -0.5), (4.12, -0.56), (4.14, -0.62), (4.16, -0.68), (4.18, -0.74), (4.22, -0.8), (4.24, -0.86), (4.26, -0.9400000000000001), (4.28, -1), (4.28, -1.08), (4.3, -1.1400000000000001), (4.32, -1.2), (4.34, -1.26), (NaN, NaN), (3.68, 0.24), (3.68, 0.12), (3.66, 0.02), (3.64, -0.06), (3.62, -0.16), (3.56, -0.28), (3.54, -0.34), (3.52, -0.4), (3.5, -0.46), (3.48, -0.56), (3.44, -0.64), (3.42, -0.7000000000000001), (3.42, -0.8), (3.4, -0.88), (3.4, -0.96), (3.4, -1.04), (3.38, -1.12), (3.36, -1.22), (3.36, -1.32), (3.36, -1.4000000000000001), (3.34, -1.48), (NaN, NaN), (2.12, -0.58), (2.22, -0.58), (2.2800000000000002, -0.6), (2.36, -0.62), (2.42, -0.66), (2.5, -0.7000000000000001), (2.56, -0.74), (2.62, -0.78), (2.68, -0.8200000000000001), (2.74, -0.86), (2.8000000000000003, -0.92), (2.84, -0.98), (2.9, -1.02), (2.96, -1.08), (3.04, -1.08), (3.1, -1.1400000000000001), (3.12, -1.2), (3.16, -1.26), (3.18, -1.32), (3.24, -1.36), (3.2600000000000002, -1.42), (3.3000000000000003, -1.48), true], 1, (-4.3, -7.96), (23.38, 6.3), (0, 713), (1384, 0)) → stroke3

AI 绘图提示语参考

绘制一个由平行四边形构成的平面镶嵌图案。先创建一个平行四边形作为基本单元，然后定义两个沿其边方向的平移向量。将平行四边形分别沿这两个向量平移，得到相邻的副本，并继续重复此过程，使整个平面被无缝覆盖。同时，在平移后的图形上添加一些标记点，以显示平移的对应关系。整体呈现为规则、连续的网格状图案，体现平移变换与向量在镶嵌中的应用。

以上提示语为根据原始绘图命令整理而来，仅供参考。