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Da Vinci and Ellipses

Da Vinci and Ellipses
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2022-07-20 17:10
Coordinate GeometryGeometric Construction
In collections circles, ovals, ellipses and architecture
This example explores what Da Vinci and others knew about ellipses and what they didn't. It involves the geometric properties of ellipses, constructing ellipse arcs, lines, and reflections, to help students understand the symmetry and generation of ellipses. Suitable for high school mathematics, particularly in coordinate geometry and conic sections.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. ConstructionStep() → stap
  3. Point(yAxis) → B
  4. Point(yAxis) → C
  5. CircleArc(A, B, C) → c
  6. Segment(B, C) → f
  7. Sequence(Rotate[B, (nn * (90°) / 8)], nn, 1, 8) → l1
  8. Sequence(Segment[A, l1(nn)], nn, 1, 8) → l2
  9. Sequence(Segment[l1(nn), (0, y(l1(nn)))], nn, 1, 8) → l5
  10. Point(xAxis) → D
  11. Ray(B, E) → l
  12. Circle(D, E) → d
  13. Segment(A, D) → m
  14. Segment(B, E) → g
  15. Point(Segment[B + (1, 0), E]) → F
  16. Mirror(F, D) → F'
  17. Segment(F, F') → h
  18. Sequence(Intersect[Ray[l1(nn) + (0.5, 0), l1(nn) + (5, 0)], h], nn, 1, 8) → l3
  19. Sequence(Segment[l1(nn), l3(nn)], nn, 1, 8) → l4
  20. Line(G, h) → i
  21. OrthogonalLine(D, h) → j
  22. Intersect(i, j) → H
  23. Vector(D, H) → u
  24. Sequence(Segment[l3(nn), l3(nn) + u], nn, 1, 8) → l6
  25. Circle(l3(1), l5(1)) → e
  26. Circle(l3(2), l5(2)) → k
  27. Circle(l3(3), l5(3)) → p
  28. Circle(l3(4), l5(4)) → q
  29. Circle(l3(5), l5(5)) → r
  30. Circle(l3(6), l5(6)) → s
  31. Circle(l3(7), l5(7)) → t
  32. Circle(l3(8), l5(8)) → c_1
  33. Intersect(d, c_1, 1) → I
  34. Intersect(d, l6(8)) → J,
  35. Intersect(t, l6(7)) → K,
  36. Intersect(s, l6(6)) → L,
  37. Intersect(r, l6(5)) → M,
  38. Intersect(q, l6(4)) → N,
  39. Intersect(p, l6(3)) → O,
  40. Intersect(k, l6(2)) → P,
  41. Intersect(e, l6(1)) → Q,
  42. Sequence(Segment[l3(nn), l7(nn)], nn, 1, 8) → l8
  43. Conic(K, L, M, N, O) → d_1

AI Drawing Prompt Reference

This is a graphic showing the geometric properties of an ellipse. The graphic includes an ellipse arc constructed from symmetric line segments and reflections. The endpoints of the arc intersect with the coordinate axes, forming a symmetric shape. There are also reflection line segments and circles used to illustrate the symmetry and generation of the ellipse. By these elements, the geometric characteristics and construction methods of the ellipse can be intuitively understood.
The prompt above was compiled from the original drawing commands and is for reference only.