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Ellipse Trace with Two Circles

Ellipse Trace with Two Circles
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2020-06-20 04:29
CircleCoordinate GeometryLocus
This example demonstrates the generation of an ellipse through the locus of intersection points of two circles. Core mathematical concepts include circle, ellipse, locus, and coordinate geometry. Construction highlights involve commands like Line, Circle, Intersect, Point, Rotate, Ray, Segment, CurveCartesian, Polygon, CircleArc from GeoGebra, emphasizing geometric constructions and transformations. Suitable for high school and above students to explore the properties of ellipses and their parametric equations.

GeoGebra Commands & Parameters

  1. Line(A, xAxis) → n
  2. Circle(A, R) → c
  3. Intersect(c, n) → E, F
  4. Point(c) → B
  5. Circle(A, R + r) → d
  6. Rotate(F, (t * w), A) → B'
  7. Ray(A, B') → a
  8. Intersect(d, a) → C,
  9. Circle(B', C) → e
  10. Rotate(C, (t * w1), B') → B''
  11. Ray(C, B'') → b
  12. Segment(C, B'') → i
  13. Segment(A, B') → j
  14. CurveCartesian(x(A) + ((R + r) * (cos((w * i)) + cos((w1 * i)))), y(A) + ((R + r) * (sin((w * i)) + sin((w1 * i)))), i, 0, t) → k
  15. Point(i) → D
  16. Intersect(e, j) → J, K
  17. Segment(A, K) → h
  18. Ray(A, B') → f
  19. Intersect(e, f) → H, I
  20. Rotate(I, (t * w1), B') → I'
  21. Segment(B', I') → g
  22. CurveCartesian(x(A) + (R * cos((w * i))) + (r * cos(((w1 + w - π / 2) * i))), y(A) + (R * sin((w * i))) + (r * sin(((w1 + w - π / 2) * i))), i, 0, t) → l
  23. CurveCartesian(x(A) + (R * cos((w * i))) - (r * cos(((w1 + w) * i) + π)), y(A) + (R * sin((w * i))) - (r * sin(((w1 + w) * i) + π)), i, 0, t) → o
  24. Circle(A, R + r) → r_1
  25. Point(r_1) → L
  26. Rotate(L, 120°, A) → L'
  27. Rotate(L, 240°, A) → L'_1
  28. Polygon(L'_1, L, L') → t1, l', l'_1, l_1
  29. CircleArc(L'_1, L, L') → s
  30. CircleArc(L, L', L'_1) → c_1
  31. CircleArc(L', L'_1, L) → d_1
  32. Point(c) → O
  33. Segment(O, A) → m
  34. Midpoint(O, A) → P
  35. Midpoint(g) → Q
  36. Segment(A, N) → f_1
  37. Segment(A, M) → g_1
  38. Circle(A, I') → e_1
  39. Circle(A, N) → h_1
  40. Circle(A, M) → k_1
  41. Midpoint(g_1) → S
  42. Midpoint(f_1) → T