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Dupin (ciclide anular de)

Dupin (ciclide anular de)
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2023-03-15 19:43
CircleIntegralGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric construction and integral calculation of Dupin curves. Key mathematical concepts include circles, angles, rotation, and definite integrals. Construction involves GeoGebra commands such as Segment, OrthogonalLine, Circle, Intersect, Ellipse, Angle, and CurveCartesian. Suitable for high school students or college freshmen learning analytic geometry and calculus fundamentals.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(α))))) / (8 - ((5 * cos(v)) * cos(α))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(α)))) * sin(v) / (8 - ((5 * cos(v)) * cos(α)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(α))))))) / (8 - ((5 * cos(v)) * cos(α))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(α)))) * sin(v) / (8 - ((5 * cos(v)) * cos(α)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(α))) / (8 - ((5 * cos(v)) * cos(α)))) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 10 * ((((5 * (k - (5 * cos(β)))) + ((8 * cos(β)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(β)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(β) / (8 - ((5 * cos(β)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(β)))) + ((8 * cos(β)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(β)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(β) / (8 - ((5 * cos(β)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(β)) * sin(u))) / (8 - ((5 * cos(β)) * cos(u)))) * cos(zAngInc)))), u, u1, u2) → a1
  33. CurveCartesian(((r / 10 * ((5 * (k + 5)) - (8 * (8 - (k * cos(u))))) / (8 + (5 * cos(u)))) * sin(zAngRot)), (r / 10 * (((((-((5 * (k + 5)) - (8 * (8 - (k * cos(u))))))) / (8 + (5 * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k + (5 * sin(u))) / (8 + (5 * cos(u)))) * cos(zAngInc)))), u, u1, u2) → curva1
  34. Sequence(CurveCartesian[(r / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(v)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(v)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(u))) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r / 10 * ((((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))) / (8 - ((5 * cos(v)) * cos(u))) * sin(zAngRot)) + (((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngRot)))), (r / 10 * (((((-((5 * (k - (5 * cos(v)))) + ((8 * cos(v)) * (8 - (k * cos(u))))))) / (8 - ((5 * cos(v)) * cos(u))) * cos(zAngRot)) * sin(zAngInc)) + ((((sqrt(39) * (8 - (k * cos(u)))) * sin(v) / (8 - ((5 * cos(v)) * cos(u)))) * sin(zAngRot)) * sin(zAngInc)) + ((sqrt(39) * (k - ((5 * cos(v)) * sin(u))) / (8 - ((5 * cos(v)) * cos(u)))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

In one coordinate plane, first draw a circle with center C and construct the key points from its intersections with the horizontal and vertical axes. Using the intersection points with each axis, define two ellipses that share the same focal/constraint setup and use the given ellipse scaling value so they are geometrically related. From the circle center and specific ellipse–axis intersection points, measure the angle directions that will act as rotational and incremental angular references. Then generate two parameterized analytic curves whose x–y coordinates are built from those angle directions and the trigonometric expressions involving two parameters (u and v) over their specified ranges. Sample the parameterized curves to obtain a sequence (point set) along each curve, and keep the segment joining the two ellipse intersection points at the chosen parameter values as an additional geometric reference.
The prompt above was compiled from the original drawing commands and is for reference only.