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Scutoides 几何构造

Scutoides 几何构造
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2018-07-31 12:07
解析几何立体几何微积分几何作图
所属专题 From POST-IT to 3D PRINTING: playing with squares and cubes
①本范例演示一种新型几何形状 Scutoides 的构造与性质；②涉及核心数学概念包括面积、不等式、微积分、圆锥截面；③构造过程中运用了正交线、交点、线段、点、圆、旋转、多边形、点在对象内、正交平面、中点、直线、射线、样条、曲面、笛卡尔曲线等 GeoGebra 命令与对象；④适合高中及以上数学学习者探索立体几何与微积分应用。

GeoGebra 命令及参数

  1. OrthogonalLine(A, xOyPlane) → l
  2. OrthogonalLine(P, xOyPlane) → f
  3. Intersect(f, a) → Q
  4. Segment(P, Q) → g
  5. Point(g) → R
  6. Circle(zAxis, A) → d
  7. Rotate(A, (2 * π / n), zAxis) → A'
  8. Polygon(A, A', n, xOyPlane) → pol1, i, C, D, B_1, D_1, F_1, j, k, m, p, t_1, b_1, d_1, B
  9. Intersect(zAxis, a) → E
  10. PointIn(a) → F
  11. Circle(zAxis, F) → e
  12. Rotate(F, (2 * π / (n + 1)), zAxis) → F'
  13. Polygon(F, F', n + 1, a) → pol2, q, G, H, I, J, C_1, E_1, G_1, r, s, t, b, f_1, a_1, c_1, e_1
  14. OrthogonalPlane(R, g) → o
  15. Midpoint(q) → K
  16. Line(K, D) → g_1
  17. Intersect(g_1, o) → L
  18. Intersect(zAxis, o) → M
  19. Ray(M, L) → h_1
  20. Point(h_1) → N
  21. Polygon(F, N, F') → t1, f', f_2, n_1
  22. Segment(N, D) → i_1
  23. Line(J, C) → j_1
  24. Line(I, B) → k_1
  25. Line(H, A') → l_1
  26. Line(G, A) → m_1
  27. Intersect(k_1, o) → O
  28. Intersect(j_1, o) → S
  29. Intersect(l_1, o) → T
  30. Intersect(m_1, o) → U
  31. Ray(M, T) → p_1
  32. Ray(M, O) → q_1
  33. Ray(M, S) → r_1
  34. Ray(M, U) → s_1
  35. Point(s_1) → V
  36. Point(p_1) → W
  37. Point(q_1) → Z
  38. Point(r_1) → A_1
  39. Spline({H, W, A'}, 3) → c2
  40. Spline({I, Z, B}, 3) → c3
  41. Spline({J, A_1, C}, 3) → c4
  42. Spline({G, V, A}, 3) → c1
  43. Surface((k * c1(t)) + ((1 - k) * c2(t)), k, 0, 1, t, 0, 1) → s1
  44. Surface((k * c3(t)) + ((1 - k) * c2(t)), k, 0, 1, t, 0, 1) → s3
  45. Surface((k * c3(t)) + ((1 - k) * c4(t)), k, 0, 1, t, 0, 1) → s2
  46. CurveCartesian(F + (t * (N - F)), t, 0, 1) → c5
  47. CurveCartesian(F' + (t * (N - F')), t, 0, 1) → c6
  48. CurveCartesian(N + (t * (D - N)), t, 0, 1) → c7
  49. Surface((k * c4((u * t))) + ((1 - k) * c5(t)), k, 0, 1, t, 0, 1) → s4
  50. Surface((k * c4(((1 - u) * t) + u)) + ((1 - k) * c7(t)), k, 0, 1, t, 0, 1) → s5
  51. Surface((k * c1(((1 - u) * t) + u)) + ((1 - k) * c7(t)), k, 0, 1, t, 0, 1) → s7
  52. Surface((k * c1((u * t))) + ((1 - k) * c6(t)), k, 0, 1, t, 0, 1) → s6