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Geometric Construction: Symmetry of Parabola and Circle

Geometric Construction: Symmetry of Parabola and Circle
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2023-03-15 20:38
CircleGeometric Construction
In collections geometriadinamica.es
This example demonstrates the symmetry of a parabola and a circle. Key mathematical concepts include parabola, circle, and symmetry. It is constructed using GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, etc. Suitable for high school mathematics education and learning.

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((r / 4 * (((((1 - cosh(α)) * sin(α)) * cos(v) / 2) * sin(zAngRot)) + ((((1 - cosh(α)) * sin(α)) * sin(v) / 2) * cos(zAngRot)))), (r / 4 * ((((((-(1 - cosh(α))) * sin(α)) * cos(v) / 2) * cos(zAngRot)) * sin(zAngInc)) + (((((1 - cosh(α)) * sin(α)) * sin(v) / 2) * sin(zAngRot)) * sin(zAngInc)) + (cosh(α) * cos(zAngInc)))) + 6, v, v1, v2) → curva2
  32. CurveCartesian((r / 4 * (((-(1 - cosh(u))) * sin(u) / 2) * sin(zAngRot))), (r / 4 * (((((1 - cosh(u)) * sin(u) / 2) * cos(zAngRot)) * sin(zAngInc)) + (cosh(u) * cos(zAngInc)))) + 6, u, u1, u2) → curva1
  33. CurveCartesian((r / 4 * (((((1 - cosh(u)) * sin(u)) * cos(β) / 2) * sin(zAngRot)) + ((((1 - cosh(u)) * sin(u)) * sin(β) / 2) * cos(zAngRot)))), (r / 4 * ((((((-(1 - cosh(u))) * sin(u)) * cos(β) / 2) * cos(zAngRot)) * sin(zAngInc)) + (((((1 - cosh(u)) * sin(u)) * sin(β) / 2) * sin(zAngRot)) * sin(zAngInc)) + (cosh(u) * cos(zAngInc)))) + 6, u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 4 * (((((1 - cosh(u)) * sin(u)) * cos(v) / 2) * sin(zAngRot)) + ((((1 - cosh(u)) * sin(u)) * sin(v) / 2) * cos(zAngRot)))), (r / 4 * ((((((-(1 - cosh(u))) * sin(u)) * cos(v) / 2) * cos(zAngRot)) * sin(zAngInc)) + (((((1 - cosh(u)) * sin(u)) * sin(v) / 2) * sin(zAngRot)) * sin(zAngInc)) + (cosh(u) * cos(zAngInc)))) + 6, v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 4 * (((((1 - cosh(u)) * sin(u)) * cos(v) / 2) * sin(zAngRot)) + ((((1 - cosh(u)) * sin(u)) * sin(v) / 2) * cos(zAngRot)))), (r / 4 * ((((((-(1 - cosh(u))) * sin(u)) * cos(v) / 2) * cos(zAngRot)) * sin(zAngInc)) + (((((1 - cosh(u)) * sin(u)) * sin(v) / 2) * sin(zAngRot)) * sin(zAngInc)) + (cosh(u) * cos(zAngInc)))) + 6, u, u1, u2], v, v1, v2, (3 * paso)) → list1

AI Drawing Prompt Reference

In the same coordinate plane, draw the x-axis and y-axis, and construct a circle centered at C with radius about 0.62. Draw two lines perpendicular to the axes through C and find their intersections with the circle to obtain key intersection points. Using those points as defining points for major axes, construct two ellipses, arranged to reflect a symmetry between them. Locate intersection points involving one ellipse, the other ellipse, and the axis-related lines, then connect the relevant points with a segment. Determine angle directions from those intersections, and use them to plot two parametric curves with different parameter variables, placed with the same angle settings to visualize the symmetry relationship among the circle, ellipses, and resulting curves.
The prompt above was compiled from the original drawing commands and is for reference only.