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Four Isometries

Four Isometries
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2023-03-22 20:40
VectorCoordinate GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric effects and properties of four isometries (reflection, rotation, translation, scaling) in a 2D plane. It involves core mathematical concepts such as transformation, symmetry, vectors, coordinate geometry, and geometric construction. Key GeoGebra commands and objects used include Ray, OrthogonalLine, Point, Distance, Min, Circle, Intersect, Line, Vector, Length, Translate, Rotate, CircleSector, Angle, LineBisector, CircleArc, and Mirror. Suitable for high school and above students to learn and understand geometric transformations and their applications.

GeoGebra Commands & Parameters

  1. Ray(B, A) → i
  2. OrthogonalLine(A, i) → j
  3. Point(j) → A1
  4. Ray(A, A1) → m
  5. Distance(A, A1) → l3
  6. Min(l3, ((a - 9.5) * l3)) → d3
  7. Circle(A, d3) → s
  8. Intersect(s, m) → W_1, K_2
  9. OrthogonalLine(K_2, m) → b_1
  10. Line(F_2, b_1) → c_1
  11. Vector(F_2, O_2) → v_1
  12. Vector(B_{18}, C_{18}) → u
  13. Length(u) → l1
  14. Min(l1, ((a - 0.5) * l1)) → d1
  15. Circle(D_1, d1) → h
  16. Line(F, u) → b
  17. Circle(F, d1) → d
  18. Intersect(d, b, 2) → E_1
  19. Vector(F, E_1) → w
  20. Translate(pic2, w) → pic2'
  21. Min(180, ((a - 3.5) * 180)) → r1
  22. Rotate(pic3, ((-r1))°, F_1) → pic3'
  23. Ray(A, B) → e
  24. OrthogonalLine(G_1, e) → f
  25. Point(f) → H_1
  26. Rotate(H_1, r1°, G_1) → U_1
  27. Rotate(H_1, ((-r1))°, G_1) → V_1
  28. CircleSector(G_1, V_1, H_1) → g
  29. Angle(V_1, G_1, H_1) → α
  30. OrthogonalLine(A1, j) → k
  31. Min(c, ((a - 7) * c)) → d2
  32. Circle(A, d2) → p
  33. Circle(B, d2) → q
  34. Circle(A1, d2) → r
  35. Intersect(r, k, 2) → Z_1
  36. Intersect(p, e) → A_2, I_2
  37. Intersect(q, i) → B_2, J_2
  38. Circle(A1, d3) → t
  39. Ray(A1, A) → l
  40. Intersect(t, l) → C_2, L_2
  41. OrthogonalLine(W_1, m) → n
  42. Line(Z_1, m) → a_1
  43. Intersect(n, a_1) → D_2
  44. LineBisector(A, B) → d_1
  45. LineBisector(A1, A) → e_1
  46. Intersect(a_1, b_1) → E_2
  47. OrthogonalLine(F_2, c_1) → f_1
  48. Point(f_1) → H_2
  49. Length(v_1) → l4
  50. Min(l4, ((a - 10.5) * l4)) → d4
  51. Circle(H_2, d4) → k_1
  52. Line(N_2, j) → i_1
  53. Line(P_2, c_1) → j_1
  54. Circle(P_2, d4) → p_1
  55. Intersect(p_1, j_1, 1) → G_2
  56. Vector(P_2, G_2) → z
  57. Translate(pic7, z) → pic7'
  58. Intersect(p, e, 2) → M_2
  59. Point(d_1) → Q_2
  60. CircleArc(Q_2, A, B) → g_1
  61. OrthogonalLine(M_2, e) → h_1
  62. Intersect(g_1, h_1, 1) → S
  63. Intersect(e, d_1) → S_2
  64. Mirror(S, S_2) → S'
  65. Point(e_1) → R_2
  66. CircleArc(R_2, A1, A) → q_1
  67. OrthogonalLine(L_2, j) → l_1
  68. Intersect(q_1, l_1) → TT,
  69. Intersect(j, e_1) → U_2
  70. Mirror(TT, U_2) → TT'

AI Drawing Prompt Reference

Draw a geometric figure with reflections, rotations, translations, and scalings. Start by drawing points A and B, then draw a ray from A, followed by its perpendicular line and taking the foot of the perpendicular as A1. Draw another ray through A1. Draw the distance between A and A1, take the minimum value as the radius of a circle, and draw the circle, finding its intersection with the ray at points W1 and K2. Draw a perpendicular line through W1 and a parallel line through K2, finding their intersection at F2. Draw vectors v1 and u, taking the length of u as the radius of a circle, drawing the circle and finding its intersection with the vector at point E1. Translate and rotate the figure, draw an arc and an angle, and finally draw perpendicular lines and angle bisectors, finding their intersections to complete the figure.
The prompt above was compiled from the original drawing commands and is for reference only.