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GeoGebra Geometric Construction: Intersection of Ellipse and Circle

GeoGebra Geometric Construction: Intersection of Ellipse and Circle
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2023-03-15 19:37
CircleGeometric Construction
In collections geometriadinamica.es
This example demonstrates the intersection of an ellipse and a circle, involving concepts such as parametric equations, geometric construction, and locus. The construction uses GeoGebra commands like Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, and Angle. It is suitable for high school students or college freshmen learning analytic geometry and geometric 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((r * ((cos(v) * sin(zAngRot)) + (cos(α) * cos(zAngRot)))), (r * ((((-cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((cos(α) * sin(zAngRot)) * sin(zAngInc)) + (cos(v + α) * cos(zAngInc)))), v, v1, (2 * v2)) → curva2
  32. CurveCartesian((r * ((-sin(zAngRot)) + (cos(u) * cos(zAngRot)))), (r * ((cos(zAngRot) * sin(zAngInc)) + ((cos(u) * sin(zAngRot)) * sin(zAngInc)) + (cos(3.14159 + u) * cos(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r * ((cos(β) * sin(zAngRot)) + (cos(u) * cos(zAngRot)))), (r * ((((-cos(β)) * cos(zAngRot)) * sin(zAngInc)) + ((cos(u) * sin(zAngRot)) * sin(zAngInc)) + (cos(β + u) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r * ((cos(v) * sin(zAngRot)) + (cos(u) * cos(zAngRot)))), (r * ((((-cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((cos(u) * sin(zAngRot)) * sin(zAngInc)) + (cos(v + u) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r * ((cos(v) * sin(zAngRot)) + (cos(u) * cos(zAngRot)))), (r * ((((-cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((cos(u) * sin(zAngRot)) * sin(zAngInc)) + (cos(v + u) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw an ellipse and a circle to show their intersections. Start by drawing two line segments and two points. Then, using these line segments and the center, draw a circle. Find the two intersection points of the circle with a vertical line. Next, draw an ellipse using these two points. Then, find the two intersection points of this ellipse with the vertical line and draw a line segment between them. Finally, draw two curves representing the parametric equations of the two ellipses.
The prompt above was compiled from the original drawing commands and is for reference only.