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Plücker Conic Construction and Analysis

Plücker Conic Construction and Analysis
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2023-03-15 21:20
Coordinate GeometryGeometric ConstructionLocus
In collections geometriadinamica.es
This example demonstrates the construction and analysis of the Plücker conic. Key mathematical concepts include conic sections, geometric construction, coordinate geometry, and locus theory. The construction involves GeoGebra commands such as Segment, OrthogonalLine, Circle, Ellipse, and Angle, along with CurveCartesian for dynamic visualization. Suitable for high school students or undergraduate learners in geometry and coordinate geometry.

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. CurveCartesian(((r * u) * sin(zAngRot)), (r * (((-u) * cos(zAngRot)) * sin(zAngInc))), u, u1, u2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r * ((α * sin(zAngRot)) + (v * cos(zAngRot)))), (r * ((((-α) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (((2 * α) * v / (α^(2) + v^(2))) * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r * ((u * sin(zAngRot)) + (β * cos(zAngRot)))), (r * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((β * sin(zAngRot)) * sin(zAngInc)) + (((2 * u) * β / (u^(2) + β^(2))) * cos(zAngInc)))), u, u1, u2) → a
  34. Sequence(CurveCartesian[(r * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (((2 * u) * v / (u^(2) + v^(2))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + (((2 * u) * v / (u^(2) + v^(2))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Start by defining a center point and drawing a circle centered at it. Construct a line perpendicular to the y-axis and another line perpendicular to the x-axis, both intersecting the circle to produce four intersection points. Use each pair of opposite points as foci to draw two ellipses of equal size but oriented perpendicularly to each other. Connect the intersection points of each ellipse with the respective perpendicular lines to form line segments. Select a point on the first ellipse and create an angle using the center point and another point on the ellipse; do the same on the second ellipse to create a second angle. Using these two angles as rotation parameters and the center point as origin, based on the ellipse intersection segments and angles, draw several parametric curves forming a family of conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.