梯子上的猫：椭圆与双曲线的几何演示

几何画板及绘图内容加载中，请稍等……

2015-07-04 06:09

变换几何作图轨迹

该范例通过一个梯子（线段）上放置的“猫”（点）的镜像与中点构造，生动演示了椭圆与双曲线的生成过程。核心数学概念包括椭圆、双曲线、相似三角形、几何分布以及整数方程。作图要点：使用GeoGebra的Intersect（交点）、Midpoint（中点）、Mirror（镜像）、Segment（线段）、Ellipse（椭圆）、Polygon（多边形）等命令，创建了A、B、C、D等多个点对象以及A1至A7、B1至B7等系列点，并通过按钮（button2）控制动态变化。该范例适合高中或大学低年级学生，用于直观理解圆锥曲线的定义、几何变换与轨迹关系，也可作为数学实验课或探究性学习的素材。

GeoGebra 命令及参数

Intersect(xAxis, yAxis) → A
Midpoint(C, D) → A1
Midpoint(A1, C) → A2
Midpoint(A2, C) → A3
Midpoint(A2, B) → A4
Midpoint(B, D) → A5
Midpoint(A5, D) → A6
Midpoint(A5, B) → A7
Mirror(A1, yAxis) → C1
Mirror(A2, yAxis) → C2
Mirror(A3, yAxis) → C3
Mirror(A4, yAxis) → C4
Mirror(A5, yAxis) → C5
Mirror(A6, yAxis) → C6
Mirror(A7, yAxis) → C7
Mirror(B1, yAxis) → D1
Mirror(B2, yAxis) → D2
Mirror(B3, yAxis) → D3
Mirror(B4, yAxis) → D4
Mirror(B5, yAxis) → D5
Mirror(B6, yAxis) → D6
Mirror(B7, yAxis) → D7
Mirror(D, yAxis) → D'
Segment(C, D) → e
Ellipse(E1, F1, G1) → H1
Polygon(F1, G1, 3) → I1, f, h, i, F
Polygon(F2, G2, 3) → I2, m, n, p, H
Polygon(F3, G3, 3) → I3, q, s, a_1, I
Polygon(F4, G4, 3) → I4, b_1, c_1, d_1, J
Polygon(F5, G5, 3) → I5, e_1, f_1, g_1, K
Polygon(F6, G6, 3) → I6, h_1, i_1, j_1, L
Polygon(F7, G7, 3) → I7, k_1, l_1, m_1, M
Polygon(F8, G8, 3) → I8, n_1, p_1, q_1, N
Polygon(F9, G9, 3) → I9, r_1, s_1, t_1, O
Polygon(G1, E1, 3) → N1, j, k, l, G
Polygon(G2, E2, 3) → N2, a_2, b_2, c_2, P
Polygon(G3, E3, 3) → N3, d_2, e_2, f_2, Q
Polygon(G4, E4, 3) → N4, g_2, h_2, i_2, R
Polygon(G5, E5, 3) → N5, j_2, k_2, l_2, S
Polygon(G6, E6, 3) → N6, m_2, n_2, p_2, T
Polygon(G7, E7, 3) → N7, q_2, r_2, s_2, U
Polygon(G8, E8, 3) → N8, t_2, a_3, b_3, V
Polygon(G9, E9, 3) → N9, c_3, d_3, e_3, W
Ellipse(E2, F2, G2) → H2
Ellipse(E3, F3, G3) → H3
Ellipse(E4, F4, G4) → H4
Ellipse(E5, F5, G5) → H5
Ellipse(E6, F6, G6) → H6
Ellipse(E7, F7, G7) → H7
Ellipse(E8, F8, G8) → H8
Ellipse(E9, F9, G9) → H9