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Ellipse Packing

Ellipse Packing
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2014-07-31 03:34
Coordinate GeometryOptimizationInequalityGeometric Construction
This example demonstrates packing ellipses into a unit square, involving geometric concepts like ellipses, similar triangles, rotation, parallelograms, linear equations, and inequalities. It uses GeoGebra commands such as Polygon, Point, and Ellipse for construction, suitable for high school mathematics teaching or geometric construction practice, focusing on spatial filling and optimization problems.

GeoGebra Commands & Parameters

  1. Polygon(P1, P2, 4) → poly1, seg1, seg2, seg3, seg4, P3, P4
  2. Point(Circle[C1, a]) → A1
  3. Ellipse(C1 + (e * (A1 - C1)), C1 - (e * (A1 - C1)), A1) → E1
  4. Point(Circle[C2, a]) → A2
  5. Ellipse(C2 + (e * (A2 - C2)), C2 - (e * (A2 - C2)), A2) → E2
  6. Point(Circle[C3, a]) → A3
  7. Ellipse(C3 + (e * (A3 - C3)), C3 - (e * (A3 - C3)), A3) → E3
  8. Point(Circle[C4, a]) → A4
  9. Ellipse(C4 + (e * (A4 - C4)), C4 - (e * (A4 - C4)), A4) → E4
  10. Point(Circle[C5, a]) → A5
  11. Ellipse(C5 + (e * (A5 - C5)), C5 - (e * (A5 - C5)), A5) → E5
  12. Point(Circle[C6, a]) → A6
  13. Ellipse(C6 + (e * (A6 - C6)), C6 - (e * (A6 - C6)), A6) → E6
  14. Point(Circle[C7, a]) → A7
  15. Ellipse(C7 + (e * (A7 - C7)), C7 - (e * (A7 - C7)), A7) → E7
  16. Point(Circle[C8, a]) → A8
  17. Ellipse(C8 + (e * (A8 - C8)), C8 - (e * (A8 - C8)), A8) → E8
  18. Point(Circle[C9, a]) → A9
  19. Ellipse(C9 + (e * (A9 - C9)), C9 - (e * (A9 - C9)), A9) → E9
  20. Point(Circle[C10, a]) → A10
  21. Ellipse(C10 + (e * (A10 - C10)), C10 - (e * (A10 - C10)), A10) → E10
  22. Point(Circle[C11, a]) → A11
  23. Ellipse(C11 + (e * (A11 - C11)), C11 - (e * (A11 - C11)), A11) → E11
  24. Point(Circle[C12, a]) → A12
  25. Ellipse(C12 + (e * (A12 - C12)), C12 - (e * (A12 - C12)), A12) → E12
  26. Point(Circle[C13, a]) → A13
  27. Ellipse(C13 + (e * (A13 - C13)), C13 - (e * (A13 - C13)), A13) → E13
  28. Point(Circle[C14, a]) → A14
  29. Ellipse(C14 + (e * (A14 - C14)), C14 - (e * (A14 - C14)), A14) → E14
  30. Point(Circle[C15, a]) → A15
  31. Ellipse(C15 + (e * (A15 - C15)), C15 - (e * (A15 - C15)), A15) → E15
  32. Point(Circle[C16, a]) → A16
  33. Ellipse(C16 + (e * (A16 - C16)), C16 - (e * (A16 - C16)), A16) → E16
  34. Point(Circle[C17, a]) → A17
  35. Ellipse(C17 + (e * (A17 - C17)), C17 - (e * (A17 - C17)), A17) → E17
  36. Point(Circle[C18, a]) → A18
  37. Ellipse(C18 + (e * (A18 - C18)), C18 - (e * (A18 - C18)), A18) → E18
  38. Point(Circle[C19, a]) → A19
  39. Ellipse(C19 + (e * (A19 - C19)), C19 - (e * (A19 - C19)), A19) → E19
  40. Point(Circle[C20, a]) → A20
  41. Ellipse(C20 + (e * (A20 - C20)), C20 - (e * (A20 - C20)), A20) → E20
  42. Point(Circle[C21, a]) → A21
  43. Ellipse(C21 + (e * (A21 - C21)), C21 - (e * (A21 - C21)), A21) → E21
  44. Point(Circle[C22, a]) → A22
  45. Ellipse(C22 + (e * (A22 - C22)), C22 - (e * (A22 - C22)), A22) → E22
  46. Point(Circle[C23, a]) → A23
  47. Ellipse(C23 + (e * (A23 - C23)), C23 - (e * (A23 - C23)), A23) → E23
  48. Point(Circle[C24, a]) → A24
  49. Ellipse(C24 + (e * (A24 - C24)), C24 - (e * (A24 - C24)), A24) → E24
  50. Point(Circle[C25, a]) → A25
  51. Ellipse(C25 + (e * (A25 - C25)), C25 - (e * (A25 - C25)), A25) → E25
  52. Point(Circle[C26, a]) → A26
  53. Ellipse(C26 + (e * (A26 - C26)), C26 - (e * (A26 - C26)), A26) → E26
  54. Point(Circle[C27, a]) → A27
  55. Ellipse(C27 + (e * (A27 - C27)), C27 - (e * (A27 - C27)), A27) → E27
  56. Point(Circle[C28, a]) → A28
  57. Ellipse(C28 + (e * (A28 - C28)), C28 - (e * (A28 - C28)), A28) → E28
  58. Point(Circle[C29, a]) → A29
  59. Ellipse(C29 + (e * (A29 - C29)), C29 - (e * (A29 - C29)), A29) → E29
  60. Point(Circle[C30, a]) → A30
  61. Ellipse(C30 + (e * (A30 - C30)), C30 - (e * (A30 - C30)), A30) → E30

AI Drawing Prompt Reference

Draw multiple ellipses packed into a unit square, demonstrating geometric construction and spatial filling. Start by drawing a square, then place several centers randomly inside it. Based on these centers, draw various ellipses of different sizes, arranging them to nest within each other and fill the square as densely as possible, illustrating concepts of geometry and optimization.
The prompt above was compiled from the original drawing commands and is for reference only.