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Exploring Volume by Janet Bowers (4)

Exploring Volume by Janet Bowers (4)
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2014-08-29 07:48
VectorSolid GeometryCalculusMatrixTransformation
This example demonstrates the generation of 3D volume by rotating a 2D shape. Key math concepts include volume calculation, transformation, vector operations, and matrix applications. GeoGebra commands and objects used in the construction are Circle, Vector, Translate, Intersect, Rotate, Polygon, Segment, OrthogonalLine, newbox, Point, and blackbox. Suitable for high school students and above to learn and understand the principles of solid geometry and volume integration in calculus, as well as for math competition training.

GeoGebra Commands & Parameters

  1. Circle(Origin, len - (2 * h)) → c
  2. Vector((width - (2 * h), 0)) → u
  3. Translate(A, u) → A'
  4. Intersect(c, xAxis) → B, C
  5. Rotate(C, α) → C'
  6. Vector(C') → v
  7. Translate(A, v) → A'_1
  8. Translate(A', v) → A''
  9. Polygon(A, A'_1, A'', A') → poly1, a, a'_1, a'', a'
  10. Vector((0, h)) → w
  11. Translate(A, w) → A'_2
  12. Translate(A', w) → A''_1
  13. Translate(A'_1, w) → D
  14. Translate(A'', w) → A'''
  15. Segment(A'_2, A) → b
  16. Segment(A''_1, A') → d
  17. Segment(D, A'_1) → e
  18. Segment(A''', A'') → f
  19. Polygon(A'_2, A''_1, A', A) → poly2, a'_2, a''_1, a'_3, a_1
  20. Polygon(D, A''', A'', A'_1) → poly3, d_1, a''', a''_2, a'_4
  21. Polygon(A''_1, A''', A'', A') → poly4, a''_3, a'''_1, a''_4, a'_5
  22. Polygon(D, A'_2, A, A'_1) → poly5, d_2, a'_6, a_2, a'_7
  23. Rotate(E, α) → E'
  24. Vector(E') → g
  25. Segment(W_1, A_2) → c_7
  26. Segment(A_2, Z_1) → d_7
  27. Segment(Z_1, V_1) → e_7
  28. Segment(V_1, W_1) → f_7
  29. Segment(Paper, U_1) → g_7
  30. Segment(U_1, S_1) → h_7
  31. Segment(S_1, T_1) → i_7
  32. Segment(T_1, Paper) → j_7
  33. OrthogonalLine(Z_1, g_7) → m_7
  34. OrthogonalLine(Z_1, h_7) → n_7
  35. OrthogonalLine(A_2, h_7) → a_8
  36. OrthogonalLine(W_1, i_7) → b_8
  37. Intersect(j_7, a_8) → B_2
  38. Intersect(i_7, b_8) → C_2
  39. Intersect(m_7, i_7) → D_2
  40. Intersect(h_7, a_8) → E_2
  41. Intersect(n_7, h_7) → F_2
  42. Intersect(m_7, g_7) → G_2
  43. Intersect(g_7, b_8) → H_2
  44. Intersect(n_7, j_7) → I_2
  45. Polygon(B_2, W_1, 4) → poly26, c_8, d_8, e_8, f_8, J_2, K_2
  46. Polygon(Paper, H_2, 4) → poly27, g_8, h_8, i_8, j_8, L_2, M_2
  47. Polygon(G_2, U_1, 4) → poly28, k_8, l_8, m_8, n_8, N_2, O_2
  48. Polygon(A_2, E_2, 4) → poly29, p_8, q_8, r_8, s_8, P_2, Q_2
  49. Polygon(V_1, Z_1, A_2, W_1) → poly31, v_1, z_1, a_9, w_1
  50. newbox(α, E, G) → n, p, c_1, e_1, f_1, g_1, h_1, i_1, j_1, k_1, J, q, H, poly6, poly7, r, s, t, b_1
  51. Intersect(r, i_1) → K
  52. Intersect(r, i_1) → L
  53. Intersect(r, e_1) → M
  54. Polygon(J, M, H, G) → poly8, j_2, m_1, h_2, g_2
  55. Polygon(B_2, W_1, V_1, I_2) → poly9, b_2, w_2, v_2, i_2
  56. Polygon(W_1, A_2, D_2, C_2) → poly10, w_3, a_3, d_3, c_2
  57. Point(w_3) → N
  58. Polygon(N, Z_1, F_2, E_2) → poly11, n_1, z_2, f_2, e_2
  59. Point(n_1) → O
  60. Point(w_2) → P
  61. Polygon(H_2, G_2, O, P) → poly12, h_3, g_3, o, p_1
  62. blackbox(α, E, Q) → poly13, poly14, r_1, s_1, t_1, k_2, l_2, m_2, n_2, p_2, q_2, r_2, poly15, s_2, t_2, i, j, k, l, m, l_1, q_1
  63. blackbox(α, E, R) → poly16, poly17, l_3, m_3, n_3, p_3, q_3, r_3, s_3, t_3, a_4, b_4, poly18, c_4, d_4, b_3, c_3, e_3, f_3, i_3, j_3, k_3
  64. blackbox(α, E, S) → poly19, poly20, l_4, m_4, n_4, p_4, q_4, r_4, s_4, t_4, a_5, b_5, poly21, c_5, d_5, e_4, f_4, g_4, h_4, i_4, j_4, k_4
  65. blackbox(α, E, T) → poly22, poly23, l_5, m_5, n_5, p_5, q_5, r_5, s_5, t_5, a_6, b_6, poly24, c_6, d_6, e_5, f_5, g_5, h_5, i_5, j_5, k_5
  66. blackbox(α, E, V) → poly33, poly34, c_9, d_9, e_9, f_9, g_9, h_9, i_9, j_9, k_9, l_9, poly35, m_9, n_9, p_7, q_7, r_7, s_7, t_7, t_8, b_9
  67. blackbox(α, E, W) → poly36, poly37, c_{10}, d_{10}, e_{10}, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}, poly38, m_{10}, n_{10}, p_9, q_9, r_9, s_9, t_9, a_{10}, b_{10}
  68. blackbox(α, E, A_1) → poly42, poly43, c_{12}, d_{12}, e_{12}, f_{12}, g_{12}, h_{12}, i_{12}, j_{12}, k_{12}, l_{12}, poly44, m_{12}, n_{12}, p_{11}, q_{11}, r_{11}, s_{11}, t_{11}, a_{12}, b_{12}
  69. blackbox(α, E, B_1) → poly45, poly46, c_{13}, d_{13}, e_{13}, f_{13}, g_{13}, h_{13}, i_{13}, j_{13}, k_{13}, l_{13}, poly47, m_{13}, n_{13}, p_{12}, q_{12}, r_{12}, s_{12}, t_{12}, a_{13}, b_{13}
  70. blackbox(α, E, C_1) → poly48, poly49, c_{14}, d_{14}, e_{14}, f_{14}, g_{14}, h_{14}, i_{14}, j_{14}, k_{14}, l_{14}, poly50, m_{14}, n_{14}, p_{13}, q_{13}, r_{13}, s_{13}, t_{13}, a_{14}, b_{14}
  71. blackbox(α, E, D_1) → poly51, poly52, c_{15}, d_{15}, e_{15}, f_{15}, g_{15}, h_{15}, i_{15}, j_{15}, k_{15}, l_{15}, poly53, m_{15}, n_{15}, p_{14}, q_{14}, r_{14}, s_{14}, t_{14}, a_{15}, b_{15}
  72. blackbox(α, E, E_1) → poly54, poly55, c_{16}, d_{16}, e_{16}, f_{16}, g_{16}, h_{16}, i_{16}, j_{16}, k_{16}, l_{16}, poly56, m_{16}, n_{16}, p_{15}, q_{15}, r_{15}, s_{15}, t_{15}, a_{16}, b_{16}
  73. blackbox(α, E, F_1) → poly57, poly58, c_{17}, d_{17}, e_{17}, f_{17}, g_{17}, h_{17}, i_{17}, j_{17}, k_{17}, l_{17}, poly59, m_{17}, n_{17}, p_{16}, q_{16}, r_{16}, s_{16}, t_{16}, a_{17}, b_{17}
  74. blackbox(α, E, G_1) → poly60, poly61, c_{18}, d_{18}, e_{18}, f_{18}, g_{18}, h_{18}, i_{18}, j_{18}, k_{18}, l_{18}, poly62, m_{18}, n_{18}, p_{17}, q_{17}, r_{17}, s_{17}, t_{17}, a_{18}, b_{18}

AI Drawing Prompt Reference

Create a 3D volume generation example showing how rotating a 2D shape forms a 3D volume. Start by drawing a circle, then use vectors and translations to create several points, determining key positions via rotations and intersections. Connect these points to form multiple polygons, creating the surface of a rotating solid. Finally, add some helper line segments and squares to make the shape clearer, fully demonstrating the volume generation process.
The prompt above was compiled from the original drawing commands and is for reference only.