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Geometry-based construction for optimization and fitting: cone-related configura

Geometry-based construction for optimization and fitting: cone-related configura
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2024-07-15 22:29
CircleAngleTransformationOptimizationGeometric Construction
In collections Simplifying Radicals: Geometric Thin Slice
This GeoGebra example uses a sequence of linked geometric constructions and parameter-driven transformations to illustrate how circle/line interactions and polygonal scaffolds can be assembled into a configuration suited for optimization and intuitive fitting (including a linear-regression style modeling mindset). It centers on core ideas such as: constructing intersections between circles and lines, measuring angles, using perpendicular constraints, representing relationships with vectors, and deriving meaningful target quantities from the geometry. Key construction workflow: Polygon, Segment, Line, and Point build the base geometric framework; OrthogonalLine introduces right-angle constraints; Circle together with Intersect computes intersection points; Angle captures critical angular features; Midpoint provides averaged geometric quantities; Rotate, Translate, Dilate, and Vector implement rotation, translation, scaling, and directional relationships. The model also includes parameter-like objects (e.g., A, B, C, t1, c, d, E, I, etc.) to maintain dependencies among quantities so that the configuration evolves consistently, supporting a geometry-to-objective approach. Suitable for high school through early university levels, especially for lessons on geometric construction, solid/solid-to-planar reasoning, and introductory optimization/fitting concepts.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. OrthogonalLine(C, b) → f
  3. Circle(C, 1) → d
  4. Intersect(d, f, 1) → D
  5. Segment(A, D) → g
  6. OrthogonalLine(D, g) → h
  7. Circle(D, 1) → e
  8. Intersect(e, h, 2) → E
  9. Segment(A, E) → i
  10. OrthogonalLine(E, i) → j
  11. Circle(E, 1) → k
  12. Intersect(k, j, 2) → F
  13. Segment(A, F) → l
  14. OrthogonalLine(F, l) → m
  15. Circle(F, 1) → p
  16. Intersect(p, m, 2) → G
  17. Segment(A, G) → n
  18. OrthogonalLine(G, n) → q
  19. Circle(G, 1) → r
  20. Intersect(r, q, 2) → H
  21. Segment(A, H) → s
  22. OrthogonalLine(H, s) → t
  23. Circle(H, 1) → c_1
  24. Intersect(c_1, t, 2) → I
  25. Segment(A, I) → f_1
  26. OrthogonalLine(I, f_1) → g_1
  27. Circle(I, 1) → d_1
  28. Intersect(d_1, g_1, 2) → J
  29. Segment(A, J) → h_1
  30. OrthogonalLine(J, h_1) → i_1
  31. Circle(J, 1) → e_1
  32. Intersect(e_1, i_1, 2) → K
  33. Segment(A, K) → j_1
  34. OrthogonalLine(K, j_1) → k_1
  35. Circle(K, 1) → p_1
  36. Intersect(p_1, k_1, 2) → L
  37. Segment(A, L) → l_1
  38. OrthogonalLine(L, l_1) → m_1
  39. Circle(L, 1) → q_1
  40. Intersect(q_1, m_1, 2) → M
  41. Segment(A, M) → n_1
  42. OrthogonalLine(M, n_1) → r_1
  43. Circle(M, 1) → s_1
  44. Intersect(s_1, r_1, 2) → N
  45. Segment(A, N) → t_1
  46. OrthogonalLine(N, t_1) → a_1
  47. Circle(N, 1) → c_2
  48. Intersect(c_2, a_1, 2) → O
  49. Segment(A, O) → b_1
  50. OrthogonalLine(O, b_1) → f_2
  51. Circle(O, 1) → d_2
  52. Intersect(d_2, f_2, 2) → P
  53. Segment(A, P) → g_2
  54. OrthogonalLine(P, g_2) → h_2
  55. Circle(P, 1) → e_2
  56. Intersect(e_2, h_2, 2) → Q
  57. Segment(A, Q) → i_2
  58. OrthogonalLine(Q, i_2) → j_2
  59. Circle(Q, 1) → k_2
  60. Intersect(k_2, j_2, 2) → R
  61. Segment(A, R) → l_2
  62. Segment(C, D) → m_2
  63. Segment(D, E) → n_2
  64. Segment(E, F) → p_2
  65. Segment(F, G) → q_2
  66. Segment(G, H) → r_2
  67. Segment(H, I) → s_2
  68. Segment(I, J) → t_2
  69. Segment(J, K) → a_2
  70. Segment(K, L) → b_2
  71. Segment(L, M) → f_3
  72. Segment(M, N) → g_3
  73. Segment(N, O) → h_3
  74. Segment(O, P) → i_3
  75. Segment(P, Q) → j_3
  76. Segment(Q, R) → k_3
  77. Angle(C, B, A) → α
  78. Angle(D, C, A) → β
  79. Angle(E, D, A) → γ
  80. Angle(F, E, A) → δ
  81. Angle(G, F, A) → ε
  82. Angle(H, G, A) → ζ
  83. Angle(I, H, A) → η
  84. Angle(J, I, A) → θ
  85. Angle(K, J, A) → ι
  86. Angle(L, K, A) → κ
  87. Angle(M, L, A) → λ
  88. Angle(N, M, A) → μ
  89. Angle(O, N, A) → ν
  90. Angle(P, O, A) → ξ
  91. Angle(Q, P, A) → ο
  92. Angle(R, Q, A) → ρ
  93. Segment(A, B) → l_3
  94. Segment(C, B) → m_3
  95. Segment(C, A) → n_3
  96. RotateText(text1_{1}, atand(Slope[n_3])) → text2
  97. RotateText(text3, atand(Slope[g])) → text4
  98. Midpoint(f_1) → S
  99. Midpoint(n_1) → T
  100. RotateText(text5, pi / 4) → text6
  101. RotateText(text7, atan(Slope[g])) → text8
  102. RotateText(text9, atand(Slope[i])) → text10
  103. Rotate(D, If[0 < o ≤ 1, ((-o) * pi / 3), ((-pi)) / 3], E) → D'
  104. Rotate(E, If[0 < o ≤ 1, ((-o) * pi / 3), ((-pi)) / 3], E) → E'
  105. Segment(D', E') → p_3
  106. Rotate(D', If[1 < o ≤ 2, ((-(o - 1)) * pi), (-pi)], D') → D''
  107. Rotate(E', If[1 < o ≤ 2, ((o - 1) * pi), pi], D') → E''
  108. Segment(D'', E'') → q_3
  109. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], I) → A'
  110. Vector(A, A') → u
  111. Translate(C, u) → C'
  112. Translate(A, u) → A'_{1}
  113. Segment(C', A'_{1}) → r_3
  114. Translate(C, u) → C'_{1}
  115. Angle(A, I, C'_{1}) → σ
  116. Rotate(C', If[4 < o ≤ 5, ((-(o - 4)) * σ), (-σ)], I) → C''
  117. Rotate(A'_{1}, If[4 < o ≤ 5, ((-(o - 4)) * σ), (-σ)], I) → U
  118. Segment(C'', U) → s_3
  119. Segment(A, S) → t_3
  120. Segment(S, I) → a_3
  121. Rotate(C'', If[5 < o ≤ 6, ((o - 5) * pi), pi], S) → C'''
  122. Rotate(U, If[5 < o ≤ 6, ((o - 5) * pi), pi], S) → U'
  123. Segment(C''', U') → b_3
  124. Angle(I, A, M) → τ
  125. Rotate(U', If[7 < o ≤ 8, ((o - 7) * τ), τ], A) → U''
  126. Dilate(A, If[7 < o ≤ 8, 8 - o, 0], M) → A'_{2}
  127. Vector(A, A'_{2}) → v
  128. Translate(A, v) → A'_{3}
  129. Translate(D, v) → D'_{1}
  130. Segment(A'_{3}, D'_{1}) → g'
  131. Segment(M, T) → c_3
  132. Segment(T, A) → d_3
  133. Line(M, A) → e_3
  134. Point(e_3) → V
  135. Angle(D, A, V) → χ
  136. Rotate(A'_{3}, If[8 < o ≤ 9, ((o - 8) * χ), χ], M) → W
  137. Rotate(D'_{1}, If[8 < o ≤ 9, ((o - 8) * χ), χ], M) → Z
  138. Segment(W, Z) → g''
  139. Rotate(W, If[9 < o ≤ 10, ((o - 9) * pi), pi], Z) → W'
  140. Rotate(Z, If[9 < o ≤ 10, ((o - 9) * pi), pi], Z) → Z'
  141. Segment(W', Z') → g'''
  142. Polygon(A, B, C) → t2, c_4, a_4, b_4
  143. Polygon(A, C, D) → t3, d_4, a_5, c_5
  144. Polygon(A, D, E) → t4, e_4, a_6, d_5
  145. OrthogonalLine(R, l_2) → f_4
  146. Circle(R, 1) → g_4
  147. Intersect(g_4, f_4, 2) → A_1
  148. Segment(A, A_1) → h_4
  149. Segment(A_1, R) → i_4
  150. Angle(A_1, R, A) → υ
  151. Circle(A, Distance[A, C]) → k_4
  152. Intersect(k_4, h_4, 1) → B_1
  153. Circle(B_1, Distance[A, C]) → p_4
  154. Intersect(p_4, h_4, 2) → C_1
  155. Segment(A, B_1) → j_4
  156. Segment(B_1, C_1) → l_4
  157. Segment(C_1, A_1) → m_4
  158. Polygon(A, H, I) → t5, i_5, a_7, h_5
  159. Polygon(A, L, M) → t6, m_5, a_8, l_5
  160. Line(A_1, A) → n_4
  161. Point(n_4) → D_1
  162. Angle(C, A, D_1) → ϕ
  163. Dilate(A, If[11 < o ≤ 12, 12 - o, 0], C_1) → A'_{4}
  164. Vector(A, A'_{4}) → w
  165. Translate(C, w) → C'_{2}
  166. Translate(A, w) → A'_{5}
  167. Segment(C'_{2}, A'_{5}) → q_4
  168. Angle(A_1, A'_{4}, C'_{2}) → ψ
  169. Rotate(C'_{2}, If[12 < o ≤ 13, ((o - 12) * ϕ), ϕ], C_1) → E_1
  170. Rotate(A'_{5}, If[12 < o ≤ 13, ((o - 12) * ϕ), ϕ], C_1) → F_1
  171. Segment(E_1, F_1) → r_4
  172. Rotate(E_1, If[13 < o ≤ 14, ((-(o - 13)) * pi), (-pi)], A'_{5}) → G_1
  173. Rotate(F_1, If[13 < o ≤ 14, ((-(o - 13)) * pi), (-pi)], A'_{5}) → H_1
  174. Segment(G_1, H_1) → s_4
  175. Rotate(E_1, If[14 < o ≤ 15, ((o - 14) * pi), pi], E_1) → I_1
  176. Rotate(F_1, If[14 < o ≤ 15, ((o - 14) * pi), pi], E_1) → J_1
  177. Segment(I_1, J_1) → t_4
  178. RotateText(text12, atand(Slope[l_4])) → text13

AI Drawing Prompt Reference

Start from a base triangle defined by three points, forming a polygon. From one vertex, draw a line perpendicular to the associated side, then construct a circle centered at that vertex with radius 1 and intersect it with the perpendicular line to obtain a new point. Connect this point back to the corresponding original vertex with a segment, erect a perpendicular line to that segment, and again take intersections between the perpendicular line and a circle centered at the new circle point with radius 1 (using the given intersection multiplicity order). Repeat this chained workflow—circle/line intersections, segment connection, then a perpendicular constraint—creating a sequence of points connected by segments emanating from the original reference vertex, capturing the key angle-related geometric configuration for optimization/fitting.
The prompt above was compiled from the original drawing commands and is for reference only.