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三维立体图形综合演示：棱柱、棱锥、圆柱、圆锥与球体

三维立体图形综合演示：棱柱、棱锥、圆柱、圆锥与球体
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2017-08-10 09:30
多边形向量立体几何变换几何作图
所属专题 Elementary Geometry
该范例通过GeoGebra动态展示多种三维立体图形，包括棱柱、棱锥、圆柱、圆锥和球体，并涉及它们的组合与旋转。核心数学概念涵盖立体几何中的基本体（多面体与旋转体）、表面积与体积计算、三维向量以及二次方程（如球面方程）。作图要点包括使用Point、Polygon、Prism、Pyramid、Cylinder、Cone、Sphere等命令创建基本体，通过Rotate命令实现旋转，利用Line和PointIn辅助定位，并生成多个面（face1）以构成完整立体。适合中学高年级或大学低年级学生学习立体几何、空间想象和三维建模，也可用于教师课堂演示或学生自主探索。

GeoGebra 命令及参数

  1. Point(Ray[(0, 0, 0), (1, 0, 0)]) → A
  2. Polygon(A, Rotate[A, (360 * ° / N), Ray[(0, 0, 0), (0, 0, 1)]], N, xOyPlane) → poly1, e, l_1, m_1, n_1, E_2, C_2, D_2, E_3, F_2, G_2, H_2, l_2, I_1, J_1, K_1, m_2, f_1, g_1, h_1, i_1, j_1, k_1
  3. Point(Ray[A + (0, 0, 0.5), A + (0, 0, 1)]) → B
  4. Prism(poly1, B) → a, C_1, L, M, face1, face3, face2, face4, face5, face6, face7, face8, D, face9, face10, face11, face12, faceBCD, edge1, edge2, edge3, edge4, edge6, E, edge8, edge10, edge12, edge14, edge16, edge17, edge20, edgeBC, edgeCD, edgeBD, F, edge5, edge7, edge9, edge11, edge13, edge15, edge18, edge19, edge21, G, H, I, J, K
  5. Point(Ray[(0, 0, 0), (0, 1, 0)]) → C
  6. Polygon((x(A), y(C), 0), ((-x(A)), y(C), 0), ((-x(A)), (-y(C)), 0), (x(A), (-y(C)), 0)) → poly2, g, h, i, j
  7. Prism(poly2, (x(A), y(C), z(B))) → c, Q, edge23, edge24, edge25, edge26, edgeQR, edgeRS, edge27, R, S, face13, face15, face16, face14, face17, edge22
  8. Polygon((b, 0, 0), (0, b, 0), ((-b), 0, 0), (0, (-b), 0)) → poly3, k, l, m, n
  9. Prism(poly3, (b, 0, (sqrt(2) * b))) → d, O, edge29, edge30, edge31, edge32, edge33, edge34, edge35, T, U, face18, face19, face20, face21, face22, edge28
  10. PointIn(z = 0) → V
  11. Line(V, V + (0, 0, 1)) → f
  12. Cylinder(Circle[(0, 0, 0), A, xOyPlane], z(B)) → q, r, s
  13. Sphere((0, 0, b), b) → t
  14. Prism(poly2, B) → u, W, edge62, edge63, edge64, edgeBW, edgeWZ, edge65, edge61, Z, A_1, face47, face50, face51, face48, face49, edge60
  15. Prism(Polygon[Sequence[Rotate[A, ((i * 90))°], i, 0, 3]], B) → v, B_1, edge69, edge70, edge71, edge67, edge72, edge73, edge68, D_1, E_1, face52, face55, face56, face53, face54, edge66
  16. Pyramid(Polygon[Sequence[Rotate[A, ((i * 90))°], i, 0, 3]], z(B)) → w, F_1, face57, face59, face60, face58, edge74, edge75, edge76, edge77
  17. Pyramid(Polygon[A, Rotate[A, 120°, Ray[(0, 0, 0), (0, 0, 1)]], 3, xOyPlane], (sqrt(2) * b)) → a_1, G_1, face23, face24, face25, edge36, edge37, edge38
  18. Rotate(A, (360 * ° / N), Ray[(0, 0, 0), (0, 0, 1)]) → A'
  19. Pyramid(poly1, z(B)) → o, H_1, face34, face35, face36, face37, edge39, edge40, edge41, edge42, edge43, edge44, face26, edge45, edge46, edge47, edge48, edge49, edge50, face27, face28, face29, face30, face31, face32, face33
  20. Cone(Circle[(0, 0, 0), A, xOyPlane], z(B)) → o_1, c_2, u_1