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Dupin环面与Steiner链的几何构造

Dupin环面与Steiner链的几何构造
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2025-06-28 23:56
所属专题 Teselaciones regulares
这个范例展示了Dupin环面（通过旋转一个圆得到）、Steiner链以及Pappus链的几何构造。涉及的核心数学概念包括旋转、圆锥曲线、变换和轨迹。作图要点包括使用GeoGebra命令如Circle、Surface、Mirror、Sphere、Conic等来构造这些图形。适合高中或大学数学课程，用于学习几何变换和立体几何。

GeoGebra 命令及参数

  1. Circle((-2, 0), 2) → cir
  2. Surface(r' - R' + (cos(u) * (R' + (r' * cos(v)))), (sin(u) * (R' + (r' * cos(v)))), 6 + (r' * sin(v)), u, 0, (2 * pi), v, 0, (2 * pi)) → toro
  3. Surface(r' + (r' * cos(u)), v, 6 + (r' * sin(u)), u, 0, (2 * pi), v, -50, 50) → cilindro
  4. Sequence(Circle[(r - R, 0) + (R; ((2 * pi) * k / 5)), r], k, 0, 4) → cadena1a
  5. Mirror(cadena1a, cir) → cadena2a
  6. Circle((r - R, 0), R - r) → circun1
  7. Circle((r - R, 0), R + r) → circun2
  8. Mirror(circun1, cir) → c1
  9. Mirror(circun2, cir) → c2
  10. Sphere((-2, 0, 6), 2) → esfera1
  11. Sphere((x(Center[c2]), 0, 6), Radius[c2]) → esfera2
  12. Mirror(recta1, cir) → c1P
  13. Mirror(recta2, cir) → c2P
  14. Sphere((x(Center[c2P]), 0, 6), Radius[c2P]) → esfera2P
  15. Conic(Center[cadena2a(1)], Center[cadena2a(2)], Center[cadena2a(3)], Center[cadena2a(4)], Center[cadena2a(5)]) → elipse
  16. Excentricity(elipse) → c
  17. Translate(Surface[((-((d * (c - ((a * cos(u)) * cos(v)))) + (b^(2) * cos(u))))) / (a - ((c * cos(u)) * cos(v))), (((b * sin(u)) * (a - (d * cos(v))))) / (a - ((c * cos(u)) * cos(v))), (((b * sin(v)) * ((c * cos(u)) - d))) / (a - ((c * cos(u)) * cos(v))), u, 0, (2 * pi), v, 0, (2 * pi)], Vector[(o, 0, 6)]) → dupin
  18. Sequence(Circle[(r - R, 0) + (R; ((k * 2) * pi / n) + α), r], k, 1, n) → cadena1
  19. Mirror(cadena1, cir) → cadena2
  20. Sequence(Circle[(r - R, 0) + (R, ((2 * k) * r) + (5 * α)), r], k, -50, 50) → cadena1p
  21. Mirror(cadena1p, cir) → cadena2p
  22. Zip(Radius[c], c, cadena2) → radios
  23. Zip(Sphere[a, r], a, centros(1)) → esferas1
  24. Zip(Sphere[a, b], a, centros(2), b, radios) → esferas2
  25. Sequence(Circle[(r - R, 0) + (R, ((2 * k) * r)), r], k, -2, 2) → cadena1aP
  26. Mirror(cadena1aP, cir) → cadena2aP
  27. Zip(Radius[c], c, cadena2p) → radiosP
  28. Conic(Center[cadena2aP(1)], Center[cadena2aP(2)], Center[cadena2aP(3)], Center[cadena2aP(4)], Center[cadena2aP(5)]) → elipseP
  29. Excentricity(elipseP) → cP
  30. Zip(Sphere[a, r], a, centros(3)) → esferas1P
  31. Zip(Sphere[a, b], a, centros(4), b, radiosP) → esferas2P
  32. Translate(Surface[((-((dP * (cP - ((aP * cos(u)) * cos(v)))) + (bP^(2) * cos(u))))) / (aP - ((cP * cos(u)) * cos(v))), (((bP * sin(u)) * (aP - (dP * cos(v))))) / (aP - ((cP * cos(u)) * cos(v))), (((bP * sin(v)) * ((cP * cos(u)) - dP))) / (aP - ((cP * cos(u)) * cos(v))), u, 0, (2 * pi), v, 0, (2 * pi)], Vector[(oP, 0, 6)]) → dupinP