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Spider and Fly Mathematical Model

Spider and Fly Mathematical Model
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2023-03-14 01:01
VectorCoordinate GeometrySolid GeometryCalculusGeometric Construction
In collections geometriadinamica.es
This example dynamically demonstrates the spider catching the fly, showcasing spatial geometry and vector operations. It involves core math concepts like vectors, coordinate geometry, rotation, and integration. GeoGebra commands used include Ellipse, Point, Vector, Rotate, and Integral. Suitable for high school and above to understand spatial geometry and calculus applications.

GeoGebra Commands & Parameters

  1. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  2. Point(mira1) → A
  3. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  4. Point(mira2) → B
  5. punto3d(A, B, M, O_0, O_r, -4, 0.12121212121212122, 1, U, L) → E
  6. punto3d(A, B, M, O_0, O_r, -4, 1, 1, U, L) → V5
  7. punto3d(A, B, M, O_0, O_r, 1, -1, 1, U, L) → V3
  8. punto3d(A, B, M, O_0, O_r, 1, 1, 1, U, L) → V2
  9. Segment(V2, V5) → cubA8
  10. Segment(V2, V3) → cubA2
  11. punto3d(A, B, M, O_0, O_r, -4, 3, -1, U, L) → P
  12. punto3d(A, B, M, O_0, O_r, 1, 3, -1, U, L) → W
  13. Line(W, P) → l1
  14. punto3d(A, B, M, O_0, O_r, -4.166666666666667, 4, -1, U, L) → C
  15. punto3d(A, B, M, O_0, O_r, 1, 0.12121212121212122, -1, U, L) → D
  16. Line(D, C) → f1
  17. Intersect(f1, l1) → S
  18. Line(S, cubA2) → i1
  19. punto3d(A, B, M, O_0, O_r, 1, 1, -1, U, L) → V1
  20. Segment(V1, V2) → cubA1
  21. punto3d(A, B, M, O_0, O_r, -4, 1, -1, U, L) → V6
  22. Segment(V1, V6) → cubA5
  23. Intersect(cubA5, i1) → T
  24. Line(T, cubA1) → j1
  25. Intersect(cubA8, j1) → V
  26. Intersect(f1, cubA5) → K
  27. punto3d(A, B, M, O_0, O_r, ((-sqrt(4 - (t - 2)^(2)))) / 12 - 4, (2 * t), (t - 14) / 12 + sqrt(4 - t^(2)) + sqrt((8 * t) - (4 * t^(2))) / 2, U, L) → J
  28. punto3d(A, B, M, O_0, O_r, 1 + t / 12, 0, sqrt(4 - t^(2)) / 12 - 1, U, L) → R1a'
  29. punto3d(A, B, M, O_0, O_r, -4 - (t * 11 / 12), 0, (sqrt(4 - t^(2)) * 11 / 12) - 1, U, L) → R1c'
  30. punto3d(A, B, M, O_0, O_r, -4, 0, -1, U, L) → R1b
  31. punto3d(A, B, M, O_0, O_r, 1, 0, -1, U, L) → R1a
  32. punto3d(A, B, M, O_0, O_r, 1, 0, -0.8333333333333334, U, L) → Araña
  33. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  34. Angle(M, O_r, B) → θ
  35. Angle(U, O_r, A) → φ
  36. Vector(O_r, O_0) → u
  37. Translate(Xr, u) → X0
  38. Translate(Zr, u) → Z0
  39. Translate(Yr, u) → Y0
  40. Vector(O_0, Z0) → v
  41. Vector(O_0, Y0) → w
  42. Vector(O_0, X0) → z
  43. Circle(O_0, L) → c
  44. Line(O_0, Y0) → a
  45. Line(O_0, yAxis) → b
  46. Line(O_0, X0) → g
  47. Line(Z0, O_0) → j
  48. Translate(P1r, u) → P1
  49. Rotate(A, botonpaso, O_r) → botonA
  50. Rotate(B, botonpaso, O_r) → botonB
  51. Segment(Mira3, Mira2) → k
  52. Segment((-1, 0), U) → l
  53. Segment((0, -1), M) → m
  54. punto3d(A, B, M, O_0, O_r, 1, -1, -1, U, L) → V4
  55. punto3d(A, B, M, O_0, O_r, -4, -1, 1, U, L) → V7
  56. punto3d(A, B, M, O_0, O_r, -4, -1, -1, U, L) → V8
  57. Midpoint(V1, V2) → cub1M
  58. Midpoint(V3, V2) → cub2M
  59. Midpoint(V1, V6) → cub4M
  60. Midpoint(V7, V8) → cub5M
  61. Angle(V3, cub1M, V4) → cub1Ang
  62. If(cub1Ang < 180°, V2, V3) → cubV2b
  63. If(cub1Ang < 180°, V1, V4) → cubV1b
  64. Polygon(V3, V4, cubV1b, cubV2b) → cub1, cub1v3, cub1v4, cub1v1, cub1v2
  65. Angle(V6, cub1M, V5) → cub3Ang
  66. If(cub3Ang < 180°, V2, V5) → cubV2c
  67. If(cub3Ang < 180°, V1, V6) → cubV1c
  68. Polygon(V6, V5, cubV2c, cubV1c) → cub3, cub3v6, cub3v5, cub3v2, cub3v1
  69. Angle(V5, cub5M, V6) → cub5Ang
  70. If(cub5Ang < 180°, V5, V7) → cubV5b
  71. If(cub5Ang < 180°, V6, V8) → cubV6b
  72. Polygon(V8, V7, cubV5b, cubV6b) → cub5, cub5v8, cub5v7, cub5v5, cub5v6
  73. Angle(V4, cub5M, V3) → cub6Ang
  74. If(cub6Ang < 180°, V7, V3) → cubV7b
  75. If(cub6Ang < 180°, V8, V4) → cubV8b
  76. Polygon(V4, V3, cubV7b, cubV8b) → cub6, cub6v4, cub6v3, cub6v7, cub6v8
  77. Angle(V5, cub2M, V7) → cub2Ang
  78. If(cub2Ang < 180°, V3, V7) → cubV3b
  79. If(cub2Ang < 180°, V2, V5) → cubV2d
  80. Polygon(V7, V5, cubV2d, cubV3b) → cub2, cub2v7, cub2v5, cub2v2, cub2v3
  81. Angle(V4, cub4M, V8) → cub4Ang
  82. Segment(V2, V1) → cubA1'
  83. Segment(V2, V3) → cubA2'
  84. Segment(V3, V4) → cubA3'
  85. Segment(V4, V1) → cubA4'
  86. Segment(V1, V6) → cubA5'
  87. Segment(V8, V6) → cubA6'
  88. Segment(V4, V8) → cubA7'
  89. Segment(V2, V5) → cubA8'
  90. Segment(V5, V7) → cubA9'
  91. Segment(V3, V7) → cubA10'
  92. Segment(V5, V6) → cubA11'
  93. Segment(V7, V8) → cubA12'
  94. Segment(V3, V4) → cubA3
  95. Segment(V4, V1) → cubA4
  96. Segment(V6, V8) → cubA6
  97. Segment(V8, V4) → cubA7
  98. Segment(V5, V7) → cubA9
  99. Segment(V7, V3) → cubA10
  100. Segment(V6, V5) → cubA11
  101. Segment(V7, V8) → cubA12
  102. If(cub4Ang < 180°, V1, V4) → cubV1d
  103. If(cub4Ang < 180°, V6, V8) → cubV6c
  104. Polygon(V8, V4, cubV1d, cubV6c) → cub4, cub4v8, cub4v4, cub4v1, cub4v6
  105. punto3d(A, B, M, O_0, O_r, 1, 1 + t, sqrt(4 - t^(2)) - 1, U, L) → V2b
  106. punto3d(A, B, M, O_0, O_r, 1, -1 - t, sqrt(4 - t^(2)) - 1, U, L) → V3c
  107. punto3d(A, B, M, O_0, O_r, 1 + t, -1, sqrt(4 - t^(2)) - 1, U, L) → V3d
  108. punto3d(A, B, M, O_0, O_r, 1 + t, 1, sqrt(4 - t^(2)) - 1, U, L) → V2d
  109. Polygon(V4, V3d, V2d, V1) → poly2, v4, v3d_1, v2d_1, v1
  110. Polygon(V1, V4, V8, V6) → poly6, v1_2, v4_2, v8_2, v6_2
  111. punto3d(A, B, M, O_0, O_r, 1, (3 * t) - 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V3f
  112. punto3d(A, B, M, O_0, O_r, 1, (-3 * t) + 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V2e
  113. punto3d(A, B, M, O_0, O_r, -4, -1 - t, sqrt(4 - t^(2)) - 1, U, L) → V7c
  114. punto3d(A, B, M, O_0, O_r, -4, (-3 * t) + 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V5e
  115. punto3d(A, B, M, O_0, O_r, -4, 1 + t, sqrt(4 - t^(2)) - 1, U, L) → V5b
  116. punto3d(A, B, M, O_0, O_r, -4 - t, -1, sqrt(4 - t^(2)) - 1, U, L) → V7a
  117. punto3d(A, B, M, O_0, O_r, -4 - t, 1, sqrt(4 - t^(2)) - 1, U, L) → V5a
  118. punto3d(A, B, M, O_0, O_r, -4, (3 * t) - 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V7f
  119. punto3d(A, B, M, O_0, O_r, (-sqrt(4 - (t - 2)^(2))) - 4, 1 + t, t + sqrt(4 - t^(2)) - 3, U, L) → v6g
  120. punto3d(A, B, M, O_0, O_r, (-sqrt(4 - (t - 2)^(2))) - 4, (3 * t) - 1, t + sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 3, U, L) → V8g
  121. Polygon(V4, V3c, V7c, V8) → poly1, v4_1, v3c, v7c, v8
  122. Polygon(V3c, V2e, V5e, V7c) → poly3, v3c_1, v2e, v5e, v7c_1
  123. Polygon(V8, V7a, V5a, V6) → poly4, v8_1, v7a, v5a, v6
  124. Polygon(V6, V5b, V2b, V1) → poly5, v6_1, v5b, v2b, v1_1
  125. Polygon(V5b, V7f, V3f, V2b) → poly7, v5b_1, v7f, v3f, v2b_1
  126. Polygon(V5b, v6g, V8g, V7f) → poly8, v5b_2, v6g_1, v8g, v7f_1
  127. punto3d(A, B, M, O_0, O_r, -4, 0, 0.8333333333333334, U, L) → Mosca
  128. Segment(R1b, Mosca) → r1c
  129. Segment(Araña, R1a) → r1a
  130. Segment(R1a, R1b) → r1b
  131. Segment(R1a', R1a) → r1a'
  132. Segment(R1b, R1c') → r1c'
  133. Segment(V2, V6) → d
  134. Segment(V1, V5) → e
  135. Midpoint(d) → F
  136. Segment(V2b, V6) → f
  137. Segment(V1, V5b) → h
  138. Midpoint(f) → I
  139. Segment(R1a', D) → i
  140. Segment(Araña, D) → s
  141. Segment(R1a, R1b) → e1
  142. Segment(K, V) → k1
  143. Segment(E, Mosca) → d1
  144. Line(T, v6_1) → m1
  145. Intersect(v5b, v2b_1) → Z
  146. Intersect(v5b, v2b_1) → A1
  147. Intersect(v5b, v2b_1) → B1
  148. Intersect(m1, v5b) → C1
  149. Segment(D, K) → n
  150. Segment(K, C1) → q
  151. Segment(V, E) → b1
  152. punto3d(A, B, M, O_0, O_r, -4, 1 + t + (29 * (t - 1) / 33), sqrt(4 - t^(2)) - 1 + (29 * sqrt((2 * t) - t^(2)) / 33), U, L) → G
  153. Segment(G, J) → p
  154. Segment(C1, G) → r
  155. Segment(D, K) → a1

AI Drawing Prompt Reference

In a 3D coordinate space, start by drawing two vertical ellipses with the same center O_r, using axis-offset endpoints to define two control points. Use these points together with the provided parameters to construct several 3D points and connect them with line segments, forming interconnected spatial parts that represent the spider and the fly model. Next, create a couple of guiding lines from selected points, find their intersection points, and use those intersections as endpoints to assemble additional segments that link the components. Add two angle measurements with O_r as the vertex: one between M and B, and one between U and A. Finally, compute direction vectors from translated reference points and apply corresponding translations to place other key points in the correct spatial relation, illustrating the catching/approach geometry.
The prompt above was compiled from the original drawing commands and is for reference only.