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Circle Reflection: Metric Space

Circle Reflection: Metric Space
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2013-11-27 23:05
CircleVectorTrigonometryCoordinate Geometry
In collections Pythagoras
This example demonstrates the geometric transformation of inversion around a circle. Key math concepts include circle, reflection, metric space, trigonometry, and vectors. Construction involves commands like Segment, Point, Vector, Circle, Midpoint, UnitOrthogonalVector, Line, UnitVector, DynamicCoordinates, Intersect, Sequence, Mirror, Zip, Translate, Ray, Angle, If, and CircumcircleArc. Suitable for high school and above students to explore geometric transformations and metric space properties.

GeoGebra Commands & Parameters

  1. Segment(A, B) → AB
  2. Point(AB) → C
  3. Segment(A, C) → AC
  4. Vector(A, B) → u
  5. Circle(A, A + v) → c
  6. Midpoint(B, C') → D
  7. Segment(B, D) → r
  8. Vector(B, D) → w
  9. UnitOrthogonalVector(u) → v'
  10. Line(C', v') → g
  11. Line(B, z) → l
  12. UnitVector(u) → u'
  13. Line(B + prm * v', u') → g_2
  14. Point(Segment[Intersect[g_2, l], Intersect[g_2, g]]) → V
  15. DynamicCoordinates(O, x(V), y(V)) → W
  16. Line(W, u') → k
  17. Intersect(k, l) → T
  18. Segment(T, T + 2 * w) → t
  19. UnitVector(t) → t'
  20. Vector(A, L1) → a
  21. UnitVector(a) → l'
  22. Vector(H - L1) → v_1
  23. Vector(L1, L2) → w_1
  24. UnitVector(w_1) → z_1
  25. Line(O, u') → e_1
  26. Intersect(e_1, l) → T_1
  27. Sequence(k / n4, k, -n4, n4) → list5
  28. Midpoint(A, B) → M2
  29. Circle(M2, A) → k_1
  30. Point(k_1) → R
  31. Intersect(Ray[A, R], l) → S
  32. Line(D, v') → d
  33. Sequence(m, m, -2 * n + 1, 2 * n - 1, 2) → n_j
  34. Sequence(k, k, n0, n - 1) → n_i
  35. Midpoint(A, C) → M1
  36. Circle(M1, A) → f
  37. Mirror(W, c) → L'_1
  38. Mirror(T_1, c) → P'
  39. Segment(A, C') → e_2
  40. Mirror(Z, c) → Z'
  41. Midpoint(A, Z') → G
  42. Mirror(H, c) → H'
  43. Sequence(L1 - k / n3 * v_1, k, 1, n3) → RingA2
  44. Mirror(RingA2, c) → RingA1'
  45. Sequence(L1 + k / n3 * v_1, k, 1, n3) → RingA1
  46. Mirror(RingA1, c) → RingA2'
  47. Zip(Circle[Midpoint[a, b], a], a, RingA1, b, RingA2) → plrArcs
  48. Zip(Circle[Midpoint[a, b], a], a, RingA1', b, RingA2') → plrArcs'
  49. Zip(S - z + (k + 1) * w, k, list5) → GridPtsw
  50. Zip(S + k * z, k, list5) → GridPtsz
  51. Zip(Midpoint[A, a], a, Mirror[Zip[S - u + k * z, k, list5], c]) → ringCen
  52. Mirror(Grid1, c) → Grid1'
  53. Translate(Dilate[Grid1, EH / r, S - z], S' - S + z) → list1
  54. Mirror(list1, c) → list1'
  55. Mirror(GridPtsz, c) → GridPtsw'
  56. Zip(CircleArc[a, If[sgn((b - B) * v) ≥ 0, A, b], If[sgn((b - B) * v) ≥ 0, b, A]], a, ringCen, b, GridPtsw') → ringPart
  57. Zip(Circle[a, A], a, ringCen) → ringFull
  58. Line(W, v') → h_1
  59. Mirror(h_1, c) → p_1
  60. Mirror(t, c) → t'_1
  61. Sequence(Rotate[H, k * 30 * ° / b_1, L1], k, 1, 12 * b_1) → plrPts
  62. Zip(Segment[L1, k], k, plrPts) → plrRad2
  63. Ray(L1, L2) → sdf
  64. Mirror(L1, c) → L1'
  65. Segment(A, L1) → l_1
  66. Segment(A, B_1) → m_1
  67. Angle(L1, B_1, A) → α
  68. Line(A, v') → lineV
  69. Line(A, B) → lineU
  70. Mirror(B_1, c) → F
  71. If(v' ⊗ (L1 - B) ≥ 0, Intersect[l, sdf], L1) → L
  72. Intersect(g, sdf) → P
  73. Segment(L, P) → q_1
  74. Mirror(q_1, c) → s_1
  75. Midpoint(A, F) → Q
  76. Mirror(L2, c) → L2'
  77. Segment(A, L2) → t_1
  78. CircumcircleArc(L1', L2', A) → sdf'
  79. Circle(Q, L1') → c_2
  80. Segment(L1, B_1 - 0.25 * z_1) → r_1
  81. Segment(A, F) → a_2
  82. Ray(A, l') → b_2
  83. Angle(A, Z, W) → γ
  84. Segment(A, Z) → f_2
  85. Mirror(Z, c) → Z'_1
  86. Intersect(Line[V, v'], lineU) → C_1
  87. Segment(A, T) → d_2
  88. Segment(A, T + 2 * w) → f_1
  89. Circle(Midpoint[A, Z'_1], A) → d_1
  90. Sequence(Rotate[H, (k * 30)°, L1], k, 1, 12) → plrPts2
  91. Zip(Segment[L1, k], k, plrPts2) → plrRad1
  92. Mirror(plrRad2, c) → plrRad2'
  93. Mirror(plrRad1, c) → plrRad1'
  94. Ray(L1, Vector[-z_1]) → i_2
  95. Zip(Ray[a + 2 * w, u'], a, GridPtsz) → list2
  96. Zip(Line[a + 2 * w, u'], a, GridPtsz) → list4
  97. Intersect(l, b_2) → U
  98. Intersect(b_2, g) → A_1
  99. Segment(U, A_1) → j_1
  100. Point(Ray[L1, l']) → D_1
  101. Mirror(k_2, c) → list6
  102. Segment(Mirror[E_1, c], Mirror[D_1, c]) → j_2
  103. Segment(Mirror[H, c], Mirror[2 * L1 - H, c]) → h_2
  104. Midpoint(t) → N
  105. Zip(A + (1 - ε) / (1 + ((m + k2)^(2) - 1) * ε^(2)) * (u + (m + k2) * ε * v), m, n_j + 1) → CyclCen
  106. Zip(A + (1 - ε) / (1 + (m + k2)^(2) * ε^(2)) * (u + (m + k2) * ε * v), m, n_j) → CyclPts
  107. Zip(Circle[a, b], a, CyclCen, b, CyclPts) → list3
  108. Segment(A, T_1) → c_1
  109. Point(CyclCen) → E
  110. Segment(E, Intersect[Ray[E, u'], d]) → e
  111. Segment(D - r / 2 * v', D + r / 2 * v') → p

AI Drawing Prompt Reference

Draw a circle and demonstrate geometric transformations around it. Start by creating a line segment and marking a point on it. Connect the start and this point with another line. Then, draw a circle centered at the start point with the end point as the radius. Find the midpoint and perpendicular line of the segment, and draw unit orthogonal vectors. Use these vectors to create lines and rays, identifying their intersections. Finally, apply mirror and reflection operations to showcase the circle's symmetry and geometric transformations, connecting relevant points with arcs to form a complete visual representation.
The prompt above was compiled from the original drawing commands and is for reference only.