平移镶嵌：用向量复制多边形构造平面密铺

几何画板及绘图内容加载中，请稍等……

2014-03-30 05:23

所属专题 Tesselation by Translation Tesselation by Translation

① 本范例演示如何通过平移变换，将一个基础多边形沿两个不同方向的向量反复复制，从而形成覆盖整个平面的镶嵌（tessellation）图案。② 核心数学概念是平移（translation）与向量（vector），以及它们如何作为几何变换生成周期性密铺结构；同时涉及多边形、旋转、步进函数等辅助概念。③ 作图要点：使用GeoGebra的Polygon命令创建基础多边形poly1，定义点A、B、C、D等，利用Vector命令构造平移向量u和v，再通过Translate命令将多边形沿u和v复制，得到stroke1、stroke2等副本，并借助AttachCopyToView保持视图同步，最终形成连续镶嵌。④ 适合中学几何或大学图形学课程，用于直观理解平移变换、向量运算和平面密铺的构造原理，也可作为艺术设计或数学实验的素材。

GeoGebra 命令及参数

Polygon(A, B, 4) → poly1, a, b, c, d, C, D
AttachCopyToView(PolyLine[(2.32, -3.66), (2.2600000000000002, -3.68), (2.2, -3.72), (2.16, -3.7800000000000002), (2.14, -3.84), (2.1, -3.92), (2.1, -4), (2.1, -4.08), (2.08, -4.14), (2.08, -4.22), (2, -4.24), (1.92, -4.24), (1.84, -4.24), (1.76, -4.24), (1.7, -4.28), (1.62, -4.3), (1.54, -4.32), (1.48, -4.34), (1.42, -4.36), (1.36, -4.42), (1.3, -4.5), (1.24, -4.5600000000000005), (1.2, -4.62), (1.18, -4.68), (1.16, -4.74), (1.08, -4.74), (1, -4.74), (0.9400000000000001, -4.78), (0.9, -4.84), (0.86, -4.9), (0.8, -4.92), (0.78, -4.98), (0.76, -5.04), (0.76, -4.96), (0.76, -4.88), (0.76, -4.8), (0.72, -4.86), (0.7000000000000001, -4.92), (0.68, -4.98), (0.64, -4.92), (0.64, -4.84), (0.64, -4.74), (0.68, -4.68), (0.74, -4.62), (0.8, -4.58), (0.86, -4.5600000000000005), (0.9400000000000001, -4.5600000000000005), (0.96, -4.5), (0.98, -4.42), (1, -4.36), (1.02, -4.3), (1.06, -4.24), (1.12, -4.22), (1.18, -4.16), (1.24, -4.14), (1.3, -4.1), (1.36, -4.0600000000000005), (1.42, -4.04), (1.48, -3.98), (1.54, -3.94), (1.6, -3.9), (1.54, -3.88), (1.46, -3.88), (1.3800000000000001, -3.88), (1.3, -3.88), (1.22, -3.88), (1.1400000000000001, -3.86), (1.08, -3.84), (1.08, -3.7600000000000002), (1.08, -3.68), (1.02, -3.62), (1.02, -3.54), (1.02, -3.46), (1.02, -3.38), (1.02, -3.2800000000000002), (1.04, -3.2), (0.98, -3.2600000000000002), (0.92, -3.3000000000000003), (0.84, -3.3000000000000003), (0.76, -3.3000000000000003), (0.72, -3.24), (0.72, -3.16), (0.72, -3.06), (0.68, -3), (0.68, -2.9), (0.62, -2.88), (0.56, -2.94), (0.5, -2.96), (0.42, -2.96), (0.36, -2.98), (0.28, -2.96), (0.26, -2.9), (0.22, -2.84), (0.22, -2.7600000000000002), (0.16, -2.7800000000000002), (0.1, -2.8000000000000003), (0.04, -2.82), (-0.04, -2.82), (-0.06, -2.7600000000000002), (-0.08, -2.7), (-0.1, -2.64), (-0.12, -2.58), (-0.16, -2.64), (-0.22, -2.68), (-0.28, -2.72), (-0.34, -2.74), (-0.42, -2.74), (-0.46, -2.68), (-0.48, -2.62), (-0.5, -2.56), (-0.52, -2.5), (-0.54, -2.56), (-0.58, -2.62), (-0.64, -2.68), (-0.7000000000000001, -2.74), (-0.78, -2.74), (-0.8200000000000001, -2.68), (-0.86, -2.62), (-0.88, -2.68), (-0.92, -2.74), (-0.9400000000000001, -2.8000000000000003), (-0.9400000000000001, -2.88), (-0.96, -2.94), (-0.98, -3), (-1.02, -3.06), (-1.08, -3.12), (-1.1, -3.18), (-1.12, -3.24), (-1.18, -3.2800000000000002), (-1.22, -3.34), (-1.28, -3.38), (-1.32, -3.44), (-1.36, -3.5), (-1.42, -3.54), (-1.48, -3.58), (-1.48, -3.66), (-1.4000000000000001, -3.66), (-1.32, -3.66), (-1.26, -3.68), 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[(-1.28, -3.6), (-1.3, -3.68), (-1.34, -3.74), (-1.4000000000000001, -3.7600000000000002), (-1.46, -3.7800000000000002), (-1.54, -3.7800000000000002), (-1.6, -3.72), (-1.62, -3.66), (-1.6400000000000001, -3.72), (-1.6400000000000001, -3.8000000000000003), (-1.7, -3.84), (-1.74, -3.7800000000000002), (-1.76, -3.72), (-1.76, -3.64), (-1.76, -3.56), (-1.76, -3.48), (-1.72, -3.4), (-1.7, -3.34), (-1.6400000000000001, -3.3200000000000003), (-1.56, -3.3200000000000003), (-1.54, -3.2600000000000002), (-1.52, -3.2), (-1.5, -3.14), (-1.48, -3.08), (-1.46, -3.02), (-1.44, -2.96), (-1.42, -2.9), (-1.3800000000000001, -2.84), (-1.36, -2.7800000000000002), (-1.34, -2.72), (-1.28, -2.66), (-1.22, -2.62), (-1.16, -2.56), (-1.12, -2.5), (-1.08, -2.44), (-1.02, -2.38), (-0.96, -2.32), (-0.92, -2.2600000000000002), (-0.88, -2.2), (-0.8200000000000001, -2.18), (-0.76, -2.14), (-0.7000000000000001, -2.1), (-0.68, -2.04), (-0.64, -1.98), (-0.62, -1.92), (-0.6, -1.86), (-0.58, -1.8), (-0.56, -1.74), (-0.56, -1.6600000000000001), (-0.62, -1.6), (-0.68, -1.56), (-0.74, -1.5), (-0.84, -1.5), (-0.92, -1.5), (-1, -1.5), (-1.08, -1.5), (-1.1400000000000001, -1.54), (-1.2, -1.6), (-1.26, -1.6600000000000001), (-1.28, -1.72), (-1.3, -1.78), (-1.36, -1.82), (-1.42, -1.86), (-1.46, -1.92), (-1.48, -1.98), (-1.48, -2.06), (-1.5, -2.12), (-1.48, -2.18), (-1.42, -2.2), (-1.36, -2.22), (-1.36, -2.14), (-1.36, -2.06), (-1.36, -1.98), (-1.3, -1.94), (-1.22, -1.94), (-1.22, -2.02), (-1.2, -2.08), (-1.1400000000000001, -2.04), (-1.12, -1.96), (-1.1, -1.9000000000000001), (-1.06, -1.96), (-1.06, -2.04), (-1.06, -2.12), (-1.08, -2.2), (-1.12, -2.2600000000000002), (-1.18, -2.2800000000000002), (-1.24, -2.32), (-1.3, -2.34), (-1.36, -2.36), (-1.42, -2.38), (-1.5, -2.38), (-1.56, -2.4), (-1.6400000000000001, -2.4), (-1.6600000000000001, -2.34), (-1.68, -2.2600000000000002), (-1.7, -2.2), (-1.72, -2.12), (-1.72, -2.04), (-1.72, -1.96), (-1.72, -1.8800000000000001), (-1.72, -1.8), (-1.6600000000000001, -1.76), (-1.62, -1.7), (-1.56, -1.6600000000000001), (-1.5, -1.6), (-1.46, -1.54), (-1.44, -1.48), (-1.42, -1.42), (-1.3800000000000001, -1.36), (-1.36, -1.3), (-1.3, -1.24), (-1.24, -1.2), (-1.18, -1.18), (-1.1, -1.18), (-1.04, -1.1400000000000001), (-0.98, -1.12), (-0.9, -1.12), (-0.8200000000000001, -1.12), (-0.76, -1.1), (-0.7000000000000001, -1.06), (-0.66, -1), (-0.62, -0.9400000000000001), (-0.56, -0.9), (-0.5, -0.88), (-0.42, -0.84), (-0.36, -0.78), (-0.32, -0.7000000000000001), (-0.26, -0.64), (-0.22, -0.58), (-0.18, -0.52), (-0.12, -0.48), (-0.06, -0.42), (0, -0.36), (0.06, -0.32), (0.1, -0.26), (0.12, -0.16), (0.14, -0.06), (0.18, 0), (0.24, 0.06), (0.18, 0.1), (0.12, 0.12), (0.06, 0.14), (0, 0.18), (-0.02, 0.24), (-0.08, 0.26), (-0.16, 0.26), (-0.24, 0.26), (-0.3, 0.28), (-0.32, 0.34), (-0.38, 0.38), (-0.42, 0.32), (-0.46, 0.26), (-0.52, 0.22), (-0.58, 0.18), (-0.64, 0.14), (-0.7000000000000001, 0.12), (-0.76, 0.08), (-0.8200000000000001, 0.02), (-0.9, -0.02), (-0.9400000000000001, -0.08), (-1.02, -0.08), (-1.08, -0.1), (-1.16, -0.1), 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'

AI 绘图提示语参考

画一个平面填充的镶嵌图案，通过沿着两个不同的向量方向重复平移一个基础多边形来创建。首先绘制一个四边形，然后定义两个点，并基于这两个点创建两个向量。使用第一个向量平移四边形得到第一个副本，再使用第二个向量平移第一个副本得到第二个副本。最后，将这两个副本排列在一起，形成一个周期性的镶嵌图案。

以上提示语为根据原始绘图命令整理而来，仅供参考。