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Geometric Construction of the Presbyterian Church (USA) Logo

Geometric Construction of the Presbyterian Church (USA) Logo
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2020-01-10 01:01
PolygonVectorTransformationParametric EquationGeometric Construction
① This GeoGebra example precisely reconstructs the official logo of the Presbyterian Church (USA), demonstrating how a complex emblem can be decomposed into computable mathematical elements using geometric construction tools. ② Core mathematical concepts include vector operations, orthogonal vectors, angle bisectors, perpendicular lines, polygons, midpoints, mirror symmetry, parametric curves, circular arcs, semicircles, centroids, and spline curves, spanning plane geometry, analytic geometry, and transformation geometry. ③ Construction highlights: from the summary, commands such as Vector, OrthogonalVector, LineBisector, Line, OrthogonalLine, Polygon, Midpoint, Intersect, Mirror, CurveCartesian, CircleArc, Semicircle, Translate, Centroid, Spline, and Segment are used, with objects like A, B, u, v, F, m, s, b, poly1, etc., progressively building points, lines, polygons, and curves to form the complete logo. ④ Suitable for high school or university geometry teaching, mathematical modeling, or computer-aided design courses, as well as interdisciplinary projects merging art and mathematics, helping students understand practical applications of geometric construction and parametric equations.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. OrthogonalVector(u) → v
  3. LineBisector(A, B) → m
  4. Line(A, B) → s
  5. OrthogonalLine(A, s) → b
  6. Polygon(A, B, 4) → poly1, f, h, j, k, D, E
  7. Midpoint(E, A) → J
  8. OrthogonalLine(J, m) → i
  9. OrthogonalLine(L_2, b) → h_3
  10. Intersect(m, h_3) → M_2
  11. OrthogonalLine(V, b) → g_3
  12. OrthogonalLine(K, i) → l
  13. OrthogonalLine(U, l) → c
  14. OrthogonalLine(F, l) → n
  15. Intersect(m, n) → H
  16. Mirror(n, i) → n'
  17. Intersect(m, n') → W
  18. Line(A_1, I) → f_1
  19. Intersect(g_3, f_1) → L_3
  20. Intersect(f_1, h_3) → N_2
  21. Intersect(f_1, g_3) → K_2
  22. Line(Z, P) → d
  23. Intersect(d, g_3) → J_2
  24. Polygon(Z, J_2, K_2, I_2, M_2, N_2) → poly4, z_2, i_3, j_3, k_3, l_3, m_3
  25. CurveCartesian(((1 - t)^(4) * Z) + (((4 * (1 - t)^(3)) * t) * O_2) + (((6 * (1 - t)^(2)) * t^(2)) * P_2) + (((4 * (1 - t)) * t^(3)) * M_2) + (t^(4) * N_2), t, 0, 1) → w
  26. Intersect(m_3, w) → Q_2, R_2, S_2, T_2
  27. Mirror(T_2, m) → J_3
  28. LineBisector(G_3, D_3) → h_4
  29. Intersect(m, h_4) → V_2
  30. OrthogonalLine(B, u, xOyPlane) → g
  31. Mirror(J, m) → J'
  32. Intersect(m, i) → C
  33. Mirror(l, m) → l'
  34. OrthogonalLine(G, l) → p
  35. Intersect(l, n) → Q
  36. LineBisector(R, P) → q
  37. OrthogonalLine(L, l) → r
  38. OrthogonalLine(M, l) → t
  39. OrthogonalLine(S, l) → a
  40. Line(O, d) → e
  41. Intersect(d, r) → B_1
  42. Intersect(e, i) → C_1
  43. Intersect(e, p) → D_1
  44. Polygon(L, B_1, P, O, C_1, K) → poly2, l_1, b_1, p_1, o_1, c_1, k_1
  45. Intersect(f_1, i) → E_1
  46. Intersect(e, f_1) → F_1
  47. Intersect(f_1, p) → G_1
  48. Polygon(E_1, C, H, O, D_1, G_1) → poly3, e_1, c_2, h_1, o_2, d_1, g_1
  49. Mirror(L, m) → L'
  50. Mirror(B_1, m) → H_1
  51. Mirror(P, m) → P'
  52. Mirror(O, m) → O'
  53. Mirror(C_1, m) → I_1
  54. Mirror(K, m) → K'
  55. Polygon(L', H_1, P', O', I_1, K') → poly2', i_1, j_1, p', o', m_1, k'
  56. Mirror(E_1, m) → J_1
  57. Mirror(C, m) → C'
  58. Mirror(H, m) → H'
  59. Mirror(O, m) → O'_1
  60. Mirror(D_1, m) → K_1
  61. Mirror(G_1, m) → L_1
  62. Polygon(J_1, C', H', O'_1, K_1, L_1) → poly3', n_1, c', h', o'_1, q_1, r_1
  63. Intersect(e, t) → M_1
  64. Mirror(M_1, m) → N_1
  65. CircleArc(P_1, M_1, N_1) → s_1
  66. Mirror(O_1, m) → Q_1
  67. Semicircle(Q_1, O_1) → t_1
  68. Polygon(M, M_1, C, O_1, S) → q1, a_1, f_2, i_2, j_2, k_2
  69. Mirror(M, m) → M'
  70. Mirror(M_1, m) → R_1
  71. Mirror(O_1, m) → S_1
  72. Mirror(S, m) → S'
  73. Mirror(M, m) → M'_1
  74. Mirror(M_1, m) → T_1
  75. Mirror(C, m) → C'_1
  76. Mirror(O_1, m) → U_1
  77. Mirror(S, m) → S'_1
  78. Polygon(M'_1, T_1, C'_1, U_1, S'_1) → q1', m'_1, g_2, c'_1, u_1, s'_1
  79. Polygon(T, W_1, Z_1, U) → q2, t_2, w_1, z_1, u_2
  80. Mirror(T, m) → T'
  81. Mirror(W_1, m) → A_2
  82. Mirror(Z_1, m) → B_2
  83. Mirror(U, m) → U'
  84. Polygon(T', A_2, B_2, U') → q2', t', a_2, b_2, u'
  85. LineBisector(V_1, W_1) → h_2
  86. Intersect(h_2, t) → C_2
  87. CircleArc(C_2, V_1, W_1) → d_2
  88. Polygon(W_1, Z_1, D_2, V_1) → q3, l_2, m_2, n_2, v_1
  89. Translate(d_2, Vector[(0.038 * v)]) → e_2
  90. Mirror(e_2, m) → p_2
  91. Mirror(W_1, m) → E_2
  92. Mirror(Z_1, m) → F_2
  93. Mirror(D_2, m) → G_2
  94. Mirror(V_1, m) → H_2
  95. Polygon(E_2, F_2, G_2, H_2) → q3', q_2, r_2, s_2, f_3
  96. Mirror(d_2, m) → c_3
  97. CurveCartesian(((1 - t)^(4) * R_2) + (((4 * (1 - t)^(3)) * t) * U_2) + (((6 * (1 - t)^(2)) * t^(2)) * J_2) + (((4 * (1 - t)) * t^(3)) * M_2) + (t^(4) * N_2), t, 0, 1) → v_2
  98. Mirror(w, m) → w'
  99. Mirror(v_2, m) → w_2
  100. Mirror(Z, m) → Z'
  101. Mirror(J_2, m) → W_2
  102. Mirror(K_2, m) → Z_2
  103. Mirror(I_2, m) → A_3
  104. Mirror(M_2, m) → B_3
  105. Mirror(N_2, m) → C_3
  106. Polygon(Z', W_2, Z_2, A_3, B_3, C_3) → poly4', z', n_3, p_3, a_3, b_3, q_3
  107. Midpoint(D_3, T_2) → E_3
  108. Centroid(poly1) → F_3
  109. Polygon(A_1, D_3, V_2, J_3, L_3) → poly5, r_3, s_3, t_3, d_3, e_3
  110. CurveCartesian(((1 - t)^(4) * E_3) + (((4 * (1 - t)^(3)) * t) * G_3) + (((6 * (1 - t)^(2)) * t^(2)) * H_3) + (((4 * (1 - t)) * t^(3)) * K_3) + (t^(4) * J_3), t, 0, 1) → o_3
  111. OrthogonalLine(N_3, b) → f_4
  112. OrthogonalLine(M_3, b) → g_4
  113. LineBisector(N_3, M_3) → i_4
  114. Intersect(l, i_4) → O_3
  115. Intersect(q, f_4) → P_3
  116. Intersect(q, g_4) → Q_3
  117. OrthogonalLine(R, s) → j_4
  118. Intersect(j_4, i_4) → R_3
  119. Spline({P_3, T_3, Q_3, R}, 4) → u_3
  120. Spline({P_3, S_3, V_3, U_3}, 4) → v_3
  121. OrthogonalLine(W_3, q) → k_4
  122. Intersect(l, g_4) → Z_3
  123. Intersect(k_4, l) → C_4
  124. LineBisector(B_4, U_3) → l_4
  125. Intersect(l_4, p) → D_4
  126. Intersect(l_4, l') → E_4
  127. Vector(D_4, E_4) → w_3
  128. CircleArc(F_4, U_3, B_4) → c_4
  129. Polygon(Q, R, U_3, B_4, O_3, A_4) → poly6, m_4, n_4, p_4, q_4, r_4, s_4
  130. Spline({O_3, A_4, Q}, 3) → d_4
  131. Line(P_3, U_3) → a_4
  132. Intersect(a_4, u_3) → G_4, H_4, F_5
  133. LineBisector(P_3, H_4) → t_4
  134. Intersect(t_4, o_1) → I_4
  135. CircleArc(I_4, P_3, H_4) → e_4
  136. Intersect(a_4, v_3) → J_4, K_4, L_4, M_4, N_4
  137. LineBisector(U_3, K_4) → b_4
  138. Intersect(b_4, d_4) → O_4
  139. CircleArc(O_4, U_3, K_4) → c_5
  140. Mirror(u_3, m) → o_4
  141. Mirror(v_3, m) → u_4
  142. Mirror(Q, m) → Q'
  143. Mirror(R, m) → R'
  144. Mirror(U_3, m) → P_4
  145. Mirror(B_4, m) → Q_4
  146. Mirror(O_3, m) → R_4
  147. Mirror(A_4, m) → S_4
  148. Polygon(Q', R', P_4, Q_4, R_4, S_4) → poly6', q', r', h_5, i_5, j_5, k_5
  149. Mirror(d_4, m) → v_4
  150. Mirror(e_4, m) → d_5
  151. Mirror(c_5, m) → e_5
  152. Mirror(c_4, m) → p_5
  153. Mirror(U_3, m) → T_4
  154. Mirror(K_4, m) → U_4
  155. Segment(T_4, U_4) → l_5
  156. Mirror(H_4, m) → V_4
  157. Mirror(P_3, m) → W_4
  158. Segment(V_4, W_4) → m_5
  159. Intersect(c_5, g_4) → Z_4,
  160. Segment(K_4, Z_4) → n_5
  161. Intersect(e_4, i_4) → A_5,
  162. Segment(H_4, A_5) → q_5
  163. Mirror(K_4, m) → B_5
  164. Mirror(Z_4, m) → C_5
  165. Segment(B_5, C_5) → r_5
  166. Mirror(H_4, m) → D_5
  167. Mirror(A_5, m) → E_5
  168. Segment(D_5, E_5) → s_5
  169. Segment(F_5, P_3) → g_5
  170. Segment(U_3, K_4) → f_5
  171. Intersect(q_5, a_4) → G_5
  172. Segment(G_5, P_3) → t_5
  173. Mirror(G_5, m) → H_5
  174. Mirror(P_3, m) → I_5
  175. Segment(H_5, I_5) → a_5
  176. Mirror(H_4, m) → J_5

AI Drawing Prompt Reference

Draw a complex geometric emblem featuring multiple polygons, circular arcs, and parametric curves, all combined through vector operations, orthogonal vectors, angle bisectors, perpendicular lines, and mirror symmetry. The central part is a square surrounded by arcs and curves, creating a symmetric and layered structure reminiscent of a religious organization's logo.
The prompt above was compiled from the original drawing commands and is for reference only.