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Geometric Definition of Ellipse by Irina Boyadzhiev

Geometric Definition of Ellipse by Irina Boyadzhiev
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2015-04-19 10:48
Coordinate GeometryGeometric Construction
In collections Conic Sections
This applet demonstrates the geometric definition of an ellipse: an ellipse is the set of all points in the plane, the sum of whose distances to two fixed points (foci) remains constant. Key concepts include geometric construction, ellipse, distance, and vector. Using GeoGebra commands such as Intersect, Ray, Point, etc., the generation process of the ellipse is dynamically illustrated. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → E
  2. Ray((0.01, 0), (1, 0)) → xx
  3. Ray((0.01, 0), (0, 1)) → yy
  4. Point(axis) → F_1
  5. Mirror(F_1, E) → F_2
  6. Line(B, xAxis) → i
  7. Point(i) → D
  8. Point(i) → C
  9. Segment(C, D) → aa
  10. Circle(F_2, a) → e
  11. Circle(F_1, a) → d
  12. Intersect(e, d) → P, G, ,
  13. Ellipse(F_2, F_1, P) → f
  14. Point(f) → R
  15. Distance(E, F_1) → c
  16. Segment(F_2, R) → b
  17. Segment(R, F_1) → g
  18. If(aa > (2 * c), true, false) → sh
  19. Polygon(Corner[1], Corner[2], (x(Corner[2]), y(Corner[2]) + (0.28 * Distance[Corner[3], Corner[2]])), (x(Corner[4]), y(Corner[1]) + (0.28 * Distance[Corner[3], Corner[2]]))) → Bottom, j, k, l, m
  20. Line(F_1, F_2) → n
  21. OrthogonalLine(E, n) → p
  22. Intersect(f, n) → A, K
  23. Intersect(f, p) → L, M
  24. Point(K, Vector[((UnitVector[p] * 1.2) * a)]) → K1
  25. Mirror(K1, p) → K1'
  26. Vector(K1, K) → u
  27. Vector(K1', A) → v
  28. Segment(K1, K1') → q
  29. Midpoint(q) → N
  30. Point(M, Vector[(-(((UnitVector[n] * 1.28) * a)))]) → M1
  31. Mirror(M1, n) → M1'
  32. Segment(M1, M1') → r
  33. Vector(M1', L) → w
  34. Vector(M1, M) → s
  35. Midpoint(r) → Q
  36. If(y(M) > 0, M, L) → PO
  37. If(y(M) < 0, M, L) → NE

AI Drawing Prompt Reference

Draw an ellipse with two fixed points F1 and F2 as foci, ensuring the sum of distances from any point on the ellipse to F1 and F2 is constant. Start by drawing the coordinate axes and locating F1, then mirror F1 to get F2. Construct the ellipse and dynamically illustrate its generation process. Finally, add auxiliary lines and points to ensure the ellipse meets the geometric definition.
The prompt above was compiled from the original drawing commands and is for reference only.