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Implicit Differential Equations Example: Geometric and Trigonometric Application

Implicit Differential Equations Example: Geometric and Trigonometric Application
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2015-08-17 05:40
TriangleCirclePolygonAngleVector
In collections Complex Math
This example demonstrates the application of implicit differential equations in geometric figures and trigonometric functions. Key mathematical concepts include implicit differential equations, trigonometric functions, and geometric constructions. The construction involves using GeoGebra commands such as Vector, Point, Polygon, Rotate, Angle, and Circle to build geometric shapes and perform transformations. It is suitable for high school students or college undergraduates to learn the integration of calculus and geometry, and to master GeoGebra drawing techniques.

GeoGebra Commands & Parameters

  1. Vector((a, 0)) → u
  2. Vector((a, 0), (a, b)) → v
  3. Vector((0, 0), (a, b)) → w
  4. Vector((0, 0), (a, c)) → d
  5. Vector((a, 0), (a, c)) → e
  6. Vector((o, 0)) → s
  7. Vector((0, o)) → u_1
  8. Vector((o, 0), (0, o)) → v_1
  9. Point(u_1) → A
  10. Point(s) → B
  11. Polygon(C, B, A) → poly1, a_2, c_1, b_2
  12. Rotate(h_4, -80*°, C) → e_1
  13. Point(e_1) → D
  14. Point(e_1) → E
  15. Polygon(C, E, D) → poly2, d_2, c_2, e_2
  16. Segment(C, D) → g_1
  17. Segment(C, E) → i_1
  18. Angle(E, C, D) → α_i
  19. Point(d) → F
  20. Point(u) → G
  21. Circle(C, F) → p_1
  22. Circle(C, A) → q_1
  23. Point(u) → H
  24. Circle(C, H) → r_1
  25. Circle(C, z_1) → s_1
  26. Circle(C, z_2) → t_1
  27. Circle(C, z_3) → f_2
  28. Circle(C, z_4) → g_2
  29. Circle(C, z_5) → h_2
  30. Distance(C, z_1) → distanceCz1
  31. Distance(C, z_2) → distanceCz2
  32. Distance(C, z_3) → distanceCz3
  33. Distance(C, z_4) → distanceCz4
  34. Distance(C, z_5) → distanceCz5
  35. Line(I, v_1) → a_3
  36. Mirror(C, a_3) → C'
  37. Mirror(B, a_3) → B'
  38. Mirror(A, a_3) → A'
  39. Mirror(C, a_3) → C'_1
  40. Mirror(B, a_3) → B'_1
  41. Mirror(A, a_3) → A'_1
  42. Polygon(C'_1, B'_1, A'_1) → poly1', a'_1, c'_1, b'_1
  43. Point(c_1) → J
  44. Point(c'_1) → K
  45. Point(a_2) → L
  46. Point(a'_1) → M
  47. Vector(K, J) → w_2
  48. Vector(M, L) → u_3
  49. Mirror(d, a_2) → d'
  50. Mirror(w, a_2) → w'
  51. Mirror(v, a_2) → v'
  52. Mirror(e, a_2) → e'
  53. Distance(K, J) → distanceKJ
  54. Distance(K, M) → distanceKM
  55. Mirror(v_1, xAxis) → v_3
  56. Mirror(u_1, xAxis) → w_3
  57. Point(s) → N
  58. Point(u_1) → O
  59. Circle(N, O) → d_3
  60. Circle(A, x_7) → k_3
  61. Circle(A, x_6) → p_3
  62. Segment(P, Q) → j_3

AI Drawing Prompt Reference

This is a geometric figure consisting of circles, polygons, angles, and line segments. The figure includes multiple circles such as p_1, q_1, r_1, and triangles and polygons formed by points A, B, C, D, E, etc., including poly1 and poly2. There are also angles α_i and line segments g_1, i_1. Additionally, mirrored points C', B', A' are obtained, and the final polygon poly1' is constructed. The construction order roughly involves defining vectors and points, building polygons, rotating points, drawing circles, calculating distances, drawing line segments, performing mirror operations, and constructing the final polygon.
The prompt above was compiled from the original drawing commands and is for reference only.