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Ellipse Polar Connection

Ellipse Polar Connection
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2016-11-27 05:53
VectorTrigonometryCoordinate GeometryGeometric Construction
In collections Ellipse Analysis
This example demonstrates the polar connection of an ellipse and its geometric properties. It involves core math concepts like ellipse, polar coordinates, similar triangles, vectors, and transformations. Construction uses commands such as Circle, Point, Line, Midpoint, OrthogonalLine, Intersect, Locus, Segment, Distance, Angle, Vector, CurveCartesian, Length. Suitable for high school students and math enthusiasts to understand analytical geometry and geometric transformations.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Line(A, C) → f
  4. Point(xAxis) → D
  5. Line(C, D) → g
  6. Midpoint(D, C) → E
  7. OrthogonalLine(E, g) → h
  8. Intersect(f, h) → F
  9. Line(D, F) → i
  10. Locus(F, C) → loc1
  11. Midpoint(A, D) → G
  12. OrthogonalLine(G, xAxis) → j
  13. Segment(D, F) → k
  14. Line(F, xAxis) → l
  15. Segment(A, F) → m
  16. OrthogonalLine(E_d, l) → p
  17. Intersect(l, p) → H
  18. Segment(F, H) → q
  19. Distance(D, J) → distanceDJ
  20. Distance(G, J) → distanceGJ
  21. Distance(G, D) → distanceGD
  22. Angle(D, A, F) → α
  23. Vector(A, F) → u
  24. Circle(I, 1) → d
  25. Circle(I, 2) → r
  26. Circle(I, 3) → c_1
  27. Circle(I, 4) → d_1
  28. Circle(I, 5) → e_1
  29. Circle(I, 6) → f_1
  30. Circle(I, 7) → g_1
  31. Circle(I, 8) → h_1
  32. Circle(I, 9) → k_1
  33. Circle(I, 10) → p_1
  34. Circle(I, 11) → q_1
  35. Circle(I, 12) → r_1
  36. Circle(I, 13) → s_1
  37. Circle(I, 14) → t_1
  38. Circle(I, 15) → c_2
  39. Segment(G, K) → b
  40. Segment(G, J) → i_1
  41. Segment(R, S) → j_1
  42. Segment(D, K) → l_1
  43. Midpoint(I, D) → T
  44. Intersect(p, q) → U
  45. Angle(D, I, F) → β
  46. CurveCartesian((cos(t) * E_b^(2) / E_a / (1 - (E_c / E_a * cos(t)))), (sin(t) * E_b^(2) / E_a / (1 - (E_c / E_a * cos(t)))), t, 0, (2 * π)) → w
  47. Distance(D, p) → distanceDp
  48. Intersect(p, xAxis) → V
  49. Segment(S, W) → m_1
  50. Length(u) → a_1
  51. Distance(I, F) → distanceIF
  52. TableText({{A2, B2}, {A3, B3}, {A4, B4}, {A5, B5}, {A6, B6}, {A7, B7}, {A8, B8}, {A9, B9}, {A10, B10}, {A11, B11}, {A12, B12}, {A13, B13}}, "|_") → text1

AI Drawing Prompt Reference

Draw an ellipse and connect points on the ellipse using polar coordinates to demonstrate its geometric properties. First, draw a circle and pick a point C on it, then draw a line f through C that intersects the x-axis at D. Find the midpoint E of D and C, draw a perpendicular line h from E to f, and find the intersection F of h and f. Draw a line i through F, and use Locus(F, C) to get the locus loc1. Next, find the midpoint G of D and F, draw a perpendicular line j from G to the x-axis, draw the segment DK from D to F, draw the segment LF from F to the x-axis, draw the segment AF from A to F, draw a perpendicular line p from F, find the intersection H of p and l, draw the segment FH, calculate the distance DJ, GJ, GD, and the angle DAF as α, draw vector AF, draw circles d, r, c1, d1, e1, f1, g1, h1, k1, p1, q1, r1, s1, t1, c2, draw segments GK, GJ, RS, DK, find the midpoint T of I, find the intersection U of p and q, calculate the angle DAF as β, draw the curve w, calculate the distance Dp, and find the intersection V of p and the x-axis.
The prompt above was compiled from the original drawing commands and is for reference only.