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PDST Ellipse Locus Demonstration

PDST Ellipse Locus Demonstration
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2019-02-11 01:35
TrigonometryCoordinate GeometryTransformationGeometric ConstructionLocus
This example demonstrates the comprehensive application of ellipse properties including foci, vertices, rotation, etc. Key math concepts involved are ellipse definition, rotation, equilateral triangles, arithmetic mean, correlation coefficient, and coordinate geometry. Construction highlights include using GeoGebra commands such as Focus, Vertex, Segment, Translate, Point, Vector, Dilate, Angle, Rotate, Circle, Intersect, and Center. Suitable for high school mathematics teaching, aiding in the exploration and understanding of ellipse loci.

GeoGebra Commands & Parameters

  1. Focus(c) → F_1, F_2
  2. Vertex(c) → C, D, E, F
  3. Segment(C, D) → d
  4. Translate(C, (0, b + 4)) → C'
  5. Translate(D, (0, b + 4)) → D'
  6. Point(c) → G
  7. Segment(G, F_1) → e
  8. Segment(G, F_2) → f
  9. Translate(C, (b + 4, 0)) → C'_1
  10. Vector(C, C') → u
  11. Vector(D, D') → v
  12. Dilate(C, If[0 < n < 1, 1 - n, 0], C') → C'_2
  13. Dilate(D, If[0 < n < 1, 1 - n, 0], D') → D'_2
  14. Segment(C'_2, D'_2) → l
  15. Dilate(F_1, If[1 < n < 2, 2 - n, 1], C') → A'
  16. Vector(F_1, A') → w
  17. Translate(G, w) → G'
  18. Segment(A', G') → m
  19. Segment(C, D) → p
  20. Angle(F_2, F_1, G) → α
  21. Rotate(A', If[1 < n < 2, ((-(2 - n)) * α)], A') → A''
  22. Rotate(A', If[1 < n < 2, ((-(n - 2)) * α), 0], A'') → A''_1
  23. Rotate(A', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → A''_2
  24. Rotate(G', If[1 < n < 2, ((-(n - 1)) * α), (-α)], A') → G''
  25. Segment(A''_2, G'') → m'
  26. Circle(C', e) → q
  27. Intersect(q, l, 1) → H
  28. Segment(C', H) → r
  29. Segment(F_1, G) → s
  30. Dilate(F_2, If[2 < n < 3, 3 - n, 1], D') → B'
  31. Vector(F_2, B') → t
  32. Translate(G, t) → G'_1
  33. Translate(G, t) → G'_2
  34. Translate(F_2, t) → B'_1
  35. Segment(G'_2, B'_1) → f'
  36. Angle(G, F_2, F_1) → β
  37. Rotate(G'_2, If[2 < n < 3, ((n - 2) * β)], B') → L
  38. Rotate(B'_1, If[2 < n < 3, ((n - 2) * β)], B') → M
  39. Segment(L, M) → f''
  40. Segment(H, D') → a_1
  41. Segment(G, F_2) → b_1
  42. Center(c) → A
  43. Segment(A, G) → c_1
  44. Segment(A, G) → d_1
  45. Dilate(A, If[1 < n < 2, 2 - n, 0], C') → A'_1
  46. Vector(F_1, A'_1) → u_1
  47. Translate(A, u_1) → A'_2
  48. Translate(G, u_1) → G'_3
  49. Segment(A'_2, G'_3) → e_1
  50. Angle(D, F_1, G) → γ
  51. Rotate(A'_2, ((-(n - 1)) * γ), A'_2) → B
  52. Rotate(G'_3, ((-(n - 1)) * γ), A'_2) → I
  53. Segment(B, I) → f_1
  54. Dilate(G, If[2 < n < 3, 3 - n, 0], D') → G'_4
  55. Vector(G, G'_4) → v_1
  56. Translate(A, v_1) → A'_3
  57. Translate(G, v_1) → G'_5
  58. Segment(A'_3, G'_5) → g_1
  59. Rotate(A'_3, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → J
  60. Rotate(G'_5, If[2 < n < 3, ((-(n - 2)) * γ)], G'_4) → K
  61. Segment(J, K) → h_1

AI Drawing Prompt Reference

Draw an ellipse, marking its foci and vertices, to demonstrate rotation. Take points on the ellipse, generate new points and segments through translation and dilation, forming an equilateral triangle. Use vector rotation and scaling to create complex loci. Finally, determine key positions using circles and intersections, connecting the points with lines to fully present the geometric properties and transformation relationships of the ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.