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Dynamic Parallelogram and Sphere Demonstration

Dynamic Parallelogram and Sphere Demonstration
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2017-04-13 09:27
VectorFunctionSolid GeometryGeometric Construction
This example dynamically illustrates the combination of a parallelogram and a sphere, showcasing the application of the Pythagorean Theorem in three-dimensional space. Core mathematical concepts include properties of parallelograms, sphere surfaces, Pythagorean Theorem, function graphs, and interest calculations. Construction highlights the use of GeoGebra commands and objects such as Ray, Segment, Circle, Intersect, Point, Rotate, Dilate, Polygon, Midpoint, CircleArc, CircleSector, Vector, Translate, Line, Angle, Mirror, and OrthogonalLine. Suitable for high school and above students to learn spatial geometry and function applications.

GeoGebra Commands & Parameters

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Point(n_2) → K_1
  7. Segment(B_2, K_1) → r_2
  8. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  9. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  10. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  11. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  12. Midpoint(A, C) → V
  13. CircleArc(V, C, D) → g_2
  14. Point(g_2) → B
  15. Segment(C, B) → j_1
  16. Segment(A, B) → n
  17. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  18. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  19. Segment(R, A''') → f''
  20. Intersect(n, f'') → Z
  21. Segment(B, Z) → l_1
  22. Segment(A, A') → f
  23. Segment(A'_1, A'') → f'
  24. Segment(B_2, C) → h
  25. Segment(C, D) → i
  26. Circle(B_2, (0.2 * r_2)) → p
  27. Segment(D, A) → j
  28. Circle(C, Radius[p]) → k
  29. Circle(A, (0.1 * j)) → q
  30. Circle(D, (0.1 * j)) → r
  31. Segment(A, C) → l
  32. Segment(D, B_2) → m
  33. Intersect(r, j, 1) → F
  34. Intersect(r, m, 1) → G
  35. Intersect(r, i, 1) → H
  36. Intersect(k, i, 1) → I
  37. Intersect(k, l, 1) → J
  38. Intersect(k, h, 1) → K
  39. Intersect(p, h, 1) → L
  40. Intersect(p, m, 1) → M
  41. Intersect(p, g, 1) → N
  42. Intersect(q, g, 1) → O
  43. Intersect(q, l, 1) → P
  44. Intersect(q, j, 1) → Q
  45. CircleSector(B_2, L, N) → s
  46. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  47. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  48. Segment(R', S) → f'''
  49. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  50. Vector(B_2, B') → u
  51. Translate(s, u) → s'
  52. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  53. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  54. Vector(C, C') → v
  55. Translate(s'', v) → s'''
  56. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  57. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  58. Vector(D, D') → w
  59. Translate(t, w) → t'
  60. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  61. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  62. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  63. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  64. Vector(B_2, B'_1) → v_1
  65. Vector(B_2, B'_3) → w_1
  66. Vector(B_2, B'_2) → i_1
  67. Circle(A, n - j_1) → h_2
  68. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  69. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  70. Vector(B_2, B'_4) → u_1
  71. Translate(s, u_1) → s'_1
  72. Line(B, f') → a_1
  73. Point(a_1) → U
  74. Angle(U, B, C) → α
  75. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  76. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  77. Line(A, B_2) → h_1
  78. Angle(B, A, D) → β
  79. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  80. Mirror(C, D) → C'_1
  81. Rotate(C, ((-π)) / 3, B_2) → C'_2
  82. Segment(B_2, C'_2) → f_1
  83. Segment(C, C'_2) → g_1
  84. Intersect(k, g_1, 1) → E
  85. Intersect(k, h) → T,
  86. CircleSector(C, T, E) → d_1
  87. Rotate(D, 15°, C) → D'_2
  88. Angle(D, C, D'_2) → γ
  89. Ray(C, D'_2) → k_1
  90. Rotate(C, (-(15°)), D) → C'_3
  91. Angle(C'_3, D, C) → δ
  92. Ray(D, C'_3) → m_1
  93. Intersect(k_1, m_1) → W
  94. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  95. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  96. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  97. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  98. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  99. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  100. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  101. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  102. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  103. Rotate(T, 15°, C) → T'
  104. Angle(T, C, T') → ε
  105. Ray(C, T') → q_1
  106. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  107. Vector(C, C'_8) → u_2
  108. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  109. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  110. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  111. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  112. Segment(A, A'_2) → j_2
  113. Segment(B_2, G_1) → l_2
  114. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  115. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  116. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  117. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  118. OrthogonalLine(K_1, r_2) → s_2
  119. Circle(K_1, r_2) → t_2
  120. Intersect(t_2, s_2, 2) → L_1
  121. OrthogonalLine(L_1, h) → f_3
  122. Intersect(f_3, h) → M_1
  123. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  124. Segment(B_2, C_1) → n_1
  125. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  126. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  127. Segment(D_1, E_1) → p_1
  128. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  129. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  130. Segment(F_1, N_1) → r_1
  131. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  132. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  133. Segment(O_1, P_1) → s_1
  134. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  135. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  136. Vector(B_2, Q_1) → c_1
  137. Translate(s, c_1) → s'_2
  138. Segment(C, K_1) → e_1
  139. Line(A, M_1) → t_1
  140. Intersect(t_1, e_1) → R_1
  141. Segment(A, R_1) → f_2
  142. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  143. Segment(C, C'_6) → i_2
  144. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  145. Segment(A, A'_3) → p_2
  146. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  147. Segment(M_1, S_1) → e_2
  148. Dilate(B_2, If[18.5 < e ≤ 19.5, 19.5 - e, 0], R_1) → T_1
  149. Vector(B_2, T_1) → v_2
  150. Translate(s, v_2) → s'_3
  151. Line(R_1, p_1) → g_3
  152. Point(g_3) → U_1
  153. Angle(U_1, R_1, K_1) → ζ
  154. Rotate(s'_3, If[18.5 < e ≤ 19.5, ((e - 18.5) * (π / 2 + ζ)), π / 2 + ζ], T_1) → e_3
  155. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4