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自对偶四面体

自对偶四面体
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2023-03-14 00:34
立体几何参数方程几何作图
所属专题 geometriadinamica.es
这个范例演示了四面体的自对偶性质。它涉及四面体的中线、三角形的性质以及参数方程。通过使用GeoGebra命令如OrthogonalLine、Circle、Intersect等，可以构造出四面体的对偶图形。适合高中几何学习者和数学爱好者。

GeoGebra 命令及参数

  1. OrthogonalLine(C, xAxis) → e
  2. Circle(C, 9.5) → d
  3. Intersect(d, e, 2) → H
  4. Point(d) → G
  5. Angle(G, C, H) → ang
  6. If(t < 3, (abs(sgn(t)) * (sgn(t) - 1) / -2), Min[t - 3, 1]) → t4
  7. If(t < 2, 0, Min[t - 2, 1]) → t3
  8. If(t < 1, (abs(sgn(t)) * (sgn(t) - 1) / -2), Min[t - 1, 1]) → t2
  9. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  10. Point(mira2) → B
  11. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  12. Point(mira1) → A
  13. punto3d(A, B, M, O_0, O_r, -1, 1, 1, U, L) → V3
  14. punto3d(A, B, M, O_0, O_r, 1, 1, -1, U, L) → V1
  15. punto3d(A, B, M, O_0, O_r, -1, -1, -1, U, L) → V4
  16. Baricentro1(V1, V4, V3) → C4
  17. punto3d(A, B, M, O_0, O_r, 1, -1, 1, U, L) → V2
  18. Baricentro1(V1, V2, V4) → C2
  19. Midpoint(C2, C4) → S6
  20. Baricentro1(V1, V2, V3) → C1
  21. Midpoint(C1, C4) → S3
  22. Midpoint(C1, C2) → S4
  23. Baricentro1(S4, S3, S6) → S346
  24. If(t5 < 4, 0, t5 - 4) → t6
  25. Midpoint(V1, V4) → octV6'
  26. Midpoint(V1, V2) → octV1'
  27. Midpoint(V1, V3) → octV2'
  28. Midpoint(V4, V3) → octV3'
  29. Midpoint(V2, V4) → octV4'
  30. Midpoint(V2, V3) → octV5'
  31. If(t < 0, t + 5, t) → t5'
  32. If(t > 2, 2, 1) → t32
  33. If(t < 1, 0, If[t > 2, 3 - t, t - 1]) → t2'
  34. If(t > 2, (t2' - 1) / 2, 0) → t31
  35. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  36. Angle(M, O_r, B) → θ
  37. Angle(U, O_r, A) → φ
  38. Vector(O_r, O_0) → u
  39. Translate(Xr, u) → X0
  40. Translate(Zr, u) → Z0
  41. Translate(Yr, u) → Y0
  42. Vector(O_0, Z0) → v
  43. Vector(O_0, Y0) → w
  44. Vector(O_0, X0) → z
  45. Circle(O_0, L) → c
  46. Line(O_0, Y0) → a
  47. Line(O_0, yAxis) → b
  48. Line(O_0, X0) → g
  49. Line(Z0, O_0) → j
  50. Translate(P1r, u) → P1
  51. Segment(Mira3, Mira2) → k
  52. Segment((-1, 0), U) → l
  53. Segment((0, -1), M) → m
  54. If(t1 > 0 ∧ t2 ≟ 0, true, false) → ver1
  55. If(t > 1 ∧ t ≤ 2, true, false) → ver2
  56. If(t3 > 0 ∧ t3 ≤ 1 ∧ t4 ≟ 0, true, false) → ver3
  57. If(t4 > 0 ∧ t6 ≟ 0, true, false) → ver4
  58. If(t6 > 0, true, false) → ver5
  59. Intersect(d, e, 2) → E
  60. Intersect(d, e, 1) → F
  61. Rotate(H, 324°, C) → I
  62. Rotate(H, 252°, C) → J
  63. Rotate(H, 108°, C) → K
  64. Rotate(H, 36°, C) → N
  65. Rotate(H, 210°, C) → O
  66. Rotate(H, 132°, C) → P
  67. Rotate(H, 64.5°, C) → Q
  68. Rotate(H, 277.00000000000006°, C) → R
  69. Angle(I, C, H) → α
  70. Angle(J, C, H) → β
  71. Angle(H, C, K) → γ
  72. Angle(H, C, N) → δ
  73. Angle(O, C, H) → ε
  74. Angle(P, C, H) → ζ
  75. Angle(Q, C, H) → η
  76. Angle(R, C, H) → ι
  77. OrthogonalLine(C, yAxis) → h
  78. CircleArc(C, N, I) → f
  79. CircleArc(C, I, N) → p
  80. Baricentro(V2c, V2b, V3c, V3b, V1b, V1c) → H4
  81. Midpoint(V2, V3) → A1
  82. Baricentro(V2a, V2c, V1c, V1a, V4a, V4b) → H1
  83. Baricentro(V3a, V3c, V2b, V2a, V4b, V4c) → H2
  84. Baricentro(V1a, V1b, V3b, V3a, V4c, V4a) → H3
  85. Midpoint(V1, V4) → A2
  86. Midpoint(V1, V2) → A3
  87. Midpoint(V1, V3) → A4
  88. Midpoint(V2, V4) → A5
  89. Midpoint(V4, V3) → A6
  90. Baricentro1(A2, A4, A3) → B1
  91. Baricentro1(A1, A3, A5) → B2
  92. Baricentro1(A4, A6, A1) → B3
  93. Baricentro1(A1, A3, A4) → B4
  94. Baricentro1(A5, A6, A1) → B5
  95. Baricentro1(A5, A6, A2) → B6
  96. Baricentro1(A3, A5, A2) → B7
  97. Baricentro1(A6, A4, A2) → B8
  98. Baricentro(B8c, B5c, B5b, B7c, B7a, B8a) → HE1
  99. Baricentro(B4a, B5a, B5b, B7c, B7b, B4b) → HE2
  100. Baricentro(B7b, B4b, B4c, B8b, B8a, B7a) → HE3
  101. Baricentro(B8b, B8c, B5c, B5a, B4a, B4c) → HE4
  102. Baricentro1(V2, V3, V4) → C3
  103. Baricentro(C1c, C3c, C4c, C1b, C3a, C4a) → HEX1
  104. Baricentro(C2a, C4a, C1c, C2c, C4b, C3b) → HEX2
  105. Baricentro(C1a, C3b, C2a, C2b, C3a, C1b) → HEX3
  106. Baricentro(C4c, C2b, C3c, C2c, C4b, C3b) → HEX4
  107. Midpoint(C1, C3) → S1
  108. Midpoint(C3, C4) → S2
  109. Midpoint(C2, C3) → S5
  110. Baricentro1(S5, S1, S4) → S145
  111. Baricentro1(S5, S2, S6) → S256
  112. Baricentro1(S1, S2, S3) → S123
  113. Baricentro(P2b, P2a, P1a, P1c, P3a, P3b) → HEXA1
  114. Baricentro(P4c, P3c, P3b, P2b, P2c, P4b) → HEXA2
  115. Baricentro(P4a, P1b, P1c, P3a, P3c, P4c) → HEXA3
  116. Baricentro(P4b, P4a, P1b, P1a, P2a, P2c) → HEXA4
  117. Angle(V4a, H1, V1a) → angH1
  118. Angle(V4b, H2, V2a) → angH2
  119. Angle(V3b, H3, V1b) → angH3
  120. Angle(V1b, H4, V3b) → angH4
  121. Angle(V1a, V1c, V1b) → angV1
  122. Angle(V2c, V2a, V2b) → angV2
  123. Angle(V3c, V3a, V3b) → angV3
  124. Angle(V4a, V4c, V4b) → angV4
  125. Angle(B4c, B4b, B4a) → angTR1
  126. Angle(B7a, B7c, B7b) → angTR2
  127. Angle(B8c, B8a, B8b) → angTR3
  128. Angle(B5a, B5b, B5c) → angTR4
  129. Angle(B7a, HE1, B7c) → angHE1
  130. Angle(B4b, HE2, B4a) → angHE2
  131. Angle(B4c, HE3, B4b) → angHE3
  132. Angle(B8c, HE4, B5c) → angHE4
  133. Angle(C3c, HEX1, C4c) → angHEX1
  134. Angle(C4b, HEX2, C2c) → angHEX2
  135. Angle(C1a, HEX3, C2a) → angHEX3
  136. Angle(C2c, HEX4, C4b) → angHEX4
  137. Angle(C4a, C2a, C3a) → angTRI1
  138. Angle(C4b, C3b, C1a) → angTRI2
  139. Angle(C2c, C1c, C3c) → angTRI3
  140. Angle(C2b, C4c, C1b) → angTRI4
  141. Angle(P3b, HEXA1, P3a) → angHEXA1
  142. Angle(P2b, HEXA2, P2c) → angHEXA2
  143. Angle(P1b, HEXA3, P1c) → angHEXA3
  144. Angle(P1a, HEXA4, P1b) → angHEXA4
  145. Angle(P2b, P2c, P2a) → angTRIA1
  146. Angle(P1b, P1c, P1a) → angTRIA2
  147. Angle(P3a, P3c, P3b) → angTRIA3
  148. Angle(P4a, P4b, P4c) → angTRIA4
  149. Angle(Q2, Q1, Q4) → angPL1
  150. Angle(Q3, Q1, Q2) → angPL2
  151. Angle(Q3, Q4, Q1) → angPL3
  152. Angle(Q2, Q4, Q3) → angPL4
  153. Segment(V4, V3) → v34'
  154. Segment(V2, V1) → v12'
  155. Segment(V1, V3) → v13'
  156. Segment(V4, V1) → v14'
  157. Segment(V2, V4) → v24'
  158. Segment(V2, V3) → v23'
  159. Polygon(B8a, B8c, B5c, B5b, B7c, B7a) → he1, tr3he1', he14', tr4he1', he12', tr2he1', he13'
  160. Polygon(B4a, B5a, B5b, B7c, B7b, B4b) → he2, he24, tr4he2', he12, tr2he2', he23, tr1he2'
  161. Polygon(B7a, B8a, B8b, B4c, B4b, B7b) → he3, he13, tr3he3', he34', tr1he3', he23', tr2he3'
  162. Polygon(B8b, B8c, B5c, B5a, B4a, B4c) → he4, tr3he4', he14, tr4he4', he24', tr1he4', he34
  163. Polygon(C2a, C3a, C4a) → tri1, tri1hex3, tri1hex1, tri1hex2
  164. Polygon(C3b, C1a, C4b) → tri2, tri2hex3, tri2hex2, tri2hex4
  165. Polygon(C2c, C1c, C3c) → tri3, tri3hex2, tri3hex1, tri3hex4
  166. Polygon(C2b, C1b, C4c) → tri4, tri4hex3, tri4hex1, tri4hex4
  167. Polygon(C1c, C3c, C4c, C1b, C3a, C4a) → hex1, tri3hex1', hex14', tri4hex1', hex13', tri1hex1', hex12'
  168. Polygon(C2a, C1a, C4b, C2c, C1c, C4a) → hex2, hex23, tri2hex2', hex24', tri3hex2', hex12, tri1hex2'
  169. Polygon(C1b, C3a, C2a, C1a, C3b, C2b) → hex3, hex13, tri1hex3', hex23', tri2hex3', hex34', tri4hex3'
  170. Polygon(C2b, C3b, C4b, C2c, C3c, C4c) → hex4, hex34, tri2hex4', hex24, tri3hex4', hex14, tri4hex4'
  171. Polygon(P2b, P2a, P2c) → tria1, tria11, tria14, tria12
  172. Polygon(P1c, P1a, P1b) → tria2, tria21, tria24, tria23
  173. Polygon(P3b, P3c, P3a) → tria3, tria32, tria33, tria31
  174. Polygon(P4c, P4b, P4a) → tria4, tria42, tria44, tria43
  175. Polygon(P1a, P1c, P3a, P3b, P2b, P2a) → hexa1, tria21', hexa13, tria31', hexa12, tria11', hexa14
  176. Polygon(P3c, P4c, P4b, P2c, P2b, P3b) → hexa2, hexa23, tria42', hexa24, tria12', hexa12', tria32'
  177. Polygon(P3c, P3a, P1c, P1b, P4a, P4c) → hexa3, tria33', hexa13', tria23', hexa34', tria43', hexa23'
  178. Polygon(P4a, P4b, P2c, P2a, P1a, P1b) → hexa4, tria44', hexa24', tria14', hexa14', tria24', hexa34
  179. Polygon(Q2, Q4, Q1) → poly1, q14, q13, q12
  180. Polygon(Q2, Q3, Q1) → poly2, q24, q23, q12'
  181. Polygon(Q4, Q3, Q1) → poly3, q34, q23', q13'
  182. Polygon(Q2, Q3, Q4) → poly4, q24', q34', q14'
  183. Segment(V2, V1) → v12
  184. Segment(V1, V3) → v13
  185. Segment(V4, V1) → v14
  186. Segment(V2, V3) → v23
  187. Segment(V2, V4) → v24
  188. Segment(V4, V3) → v34
  189. Polygon(V1c, V2c, V2a, V4b, V4a, V1a) → h1, a14, a1t2', a12, a1t4', a13, a1t1'
  190. Polygon(V2a, V2b, V3c, V3a, V4c, V4b) → h2, a2t2', a24, a2t3', a23, a2t4', a12'
  191. Polygon(V1b, V3b, V3a, V4c, V4a, V1a) → h3, a43, a3t3', a23', a3t4', a13', a3t1'
  192. Polygon(V2c, V2b, V3c, V3b, V1b, V1c) → h4, a4t2', a24', a4t3', a43', a4t1', a14'
  193. Polygon(V2c, V2a, V2b) → t02, a1t2, a2t2, a4t2
  194. Polygon(V1a, V1b, V1c) → t01, a3t1, a4t1, a1t1
  195. Polygon(V4a, V4b, V4c) → t04, a1t4, a2t4, a3t4
  196. Polygon(V3a, V3b, V3c) → t03, a3t3, a4t3, a2t3
  197. Polygon(B4b, B4a, B4c) → tr1, tr1he2, tr1he4, tr1he3
  198. Polygon(B5b, B5a, B5c) → t4', tr4he2, tr4he4, tr4he1
  199. Polygon(B7b, B7c, B7a) → tr2, tr2he2, tr2he1, tr2he3
  200. Polygon(B8a, B8c, B8b) → tr3, tr3he1, tr3he4, tr3he3
  201. Segment(B5, B7) → i
  202. Segment(B5, B8) → n
  203. Segment(B5, B4) → q
  204. Segment(B7, B4) → r
  205. Segment(B4, B8) → s
  206. Segment(B8, B7) → a1
  207. Angle(V2, V1, V4) → ang1
  208. Angle(V2, V3, V1) → ang2
  209. Angle(V4, V1, V3) → ang3
  210. Angle(V2, V4, V3) → ang4
  211. Angle(H, O_0, G) → κ

AI 绘图提示语参考

绘制一个自对偶四面体的几何构造图。首先在三维空间中定义坐标原点和坐标轴，以原点为圆心作一个半径较大的辅助圆，并在圆上选取两个动点。通过这两个动点分别构造两个互相垂直的椭圆，取椭圆上的点作为控制参数。利用这些控制点通过自定义的3D点生成函数创建四个顶点，连接这些顶点形成四面体的六条棱。进一步构造四面体各面的重心，以及各棱的中点，找出这些中点形成的几何中心的重心。最后在同一坐标系中保留坐标轴、辅助圆、两个椭圆以及四面体的顶点和棱的构造线条，展示四面体与其对偶图形之间的自对偶关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。