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Geometric Construction of Wallis Conical Edge

Geometric Construction of Wallis Conical Edge
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2023-03-15 21:40
CircleParametric EquationGeometric ConstructionLocus
In collections geometriadinamica.es
This example demonstrates the geometric construction of the Wallis conical edge (arista cónica de Wallis), a special edge or intersection curve related to conic sections. The core mathematical concepts involve conic sections (ellipse), orthogonal lines, intersections, angles, and parametric curves. The construction uses GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence, creating multiple points (e.g., v20, v10, p, u20, u10), lines, circles, ellipses, and parametric curves, with sequences generating dynamic structures. It is suitable for high school or university courses in analytic geometry, conic sections, or geometric construction, to explore interactions between conics, lines, and circles, as well as parametric curve construction.

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, xAxis) → zz2
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz2, 1) → ZZ4
  12. Intersect(zz0, zz2, 2) → ZZ3
  13. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  14. Point(zElipse2) → Z
  15. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  16. Angle(ZZ8, C, Z) → zAngInc
  17. OrthogonalLine(C, yAxis) → zz1
  18. Intersect(zz0, zz1, 2) → ZZ2
  19. Intersect(zz0, zz1, 1) → ZZ1
  20. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  21. Point(zElipse1) → XY
  22. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  23. Angle(ZZ6, C, XY) → zAngRot
  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(((2 * r) * (((v * cos(α)) * sin(zAngRot)) + ((v * sin(α)) * cos(zAngRot)))) + 2, ((2 * r) * (((((-v) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((3 * sin(α)) * cos(zAngInc)))) - 20, v, v1, v2) → curva2
  32. CurveCartesian(((2 * r) * (((β * cos(u)) * sin(zAngRot)) + ((β * sin(u)) * cos(zAngRot)))) + 2, ((2 * r) * (((((-β) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((β * sin(zAngRot)) * sin(zAngInc)) + ((3 * sin(u)) * cos(zAngInc)))) - 20, u, u1, u2) → a1
  33. CurveCartesian(((2 * r) * (((0.5 * cos(u)) * sin(zAngRot)) + ((0.5 * sin(u)) * cos(zAngRot)))) + 2, ((2 * r) * ((((-0.5 * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((0.5 * sin(zAngRot)) * sin(zAngInc)) + ((3 * sin(u)) * cos(zAngInc)))) - 20, u, u1, u2) → curva1
  34. Sequence(CurveCartesian[((2 * r) * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))) + 2, ((2 * r) * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((3 * sin(u)) * cos(zAngInc)))) - 20, u, u1, u2], v, v1, v2, (2.5 * paso)) → list2
  35. Sequence(CurveCartesian[((2 * r) * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))) + 2, ((2 * r) * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((3 * sin(u)) * cos(zAngInc)))) - 20, v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

The figure consists of ellipses, orthogonal lines, circles, and parametric curves. Ellipses intersect orthogonal lines to yield two points, which together with circle intersections form another ellipse. Ellipse-circle intersection points serve as the starting points for parametric curves, which are generated by rotation and scaling. The figure illustrates the relationships between ellipses, circles, and orthogonal lines, as well as the parametric curve generation process. The construction order is: first draw ellipses and circles, then construct orthogonal lines, find intersections, and finally generate parametric curves.
The prompt above was compiled from the original drawing commands and is for reference only.