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r=0: Limit Construction from Circle to Point and Geometric Exploration

r=0: Limit Construction from Circle to Point and Geometric Exploration
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2017-07-09 07:22
TriangleCircleGeometric ConstructionSlider AnimationLocus
In collections ATC Cobwebs and Linear Difference Equations
This example demonstrates a geometric construction with radius r=0, where dynamic adjustment of the radius parameter causes circles, triangles, and cones to degenerate into points or lines. Core mathematical concepts include the equation of a circle, triangle sides and intersections, and the connection to indefinite integrals and cone volume. The construction uses GeoGebra commands such as Point, Segment, and Intersect to create points, segments, and intersection points, with multiple auxiliary objects (A1 to A22, B1, C1, etc.) tracking geometric relations as the radius changes. Suitable for high school or early college students to understand limit concepts, geometric degeneration, and parametric construction in dynamic geometry software.

GeoGebra Commands & Parameters

  1. Point(yAxis) → C
  2. Point(xAxis) → A
  3. Segment(A, (x(A), f(x(A)))) → c
  4. Segment((A1, A2), (A2, A2)) → B1
  5. Segment((A2, A2), (A2, A3)) → C1
  6. Segment((A2, A3), (A3, A3)) → B2
  7. Segment((A3, A3), (A3, A4)) → C2
  8. Segment((A3, A4), (A4, A4)) → B3
  9. Segment((A4, A4), (A4, A5)) → C3
  10. Segment((A4, A5), (A5, A5)) → B4
  11. Segment((A5, A5), (A5, A6)) → C4
  12. Segment((A5, A6), (A6, A6)) → B5
  13. Segment((A6, A6), (A6, A7)) → C5
  14. Segment((A6, A7), (A7, A7)) → B6
  15. Segment((A7, A7), (A7, A8)) → C6
  16. Segment((A7, A8), (A8, A8)) → B7
  17. Segment((A8, A8), (A8, A9)) → C7
  18. Segment((A8, A9), (A9, A9)) → B8
  19. Segment((A9, A9), (A9, A10)) → C8
  20. Segment((A9, A10), (A10, A10)) → B9
  21. Segment((A10, A10), (A10, A11)) → C9
  22. Segment((A10, A11), (A11, A11)) → B10
  23. Segment((A11, A11), (A11, A12)) → C10
  24. Segment((A11, A12), (A12, A12)) → B11
  25. Segment((A12, A12), (A12, A13)) → C11
  26. Segment((A12, A13), (A13, A13)) → B12
  27. Segment((A13, A13), (A13, A14)) → C12
  28. Segment((A13, A14), (A14, A14)) → B13
  29. Segment((A14, A14), (A14, A15)) → C13
  30. Segment((A14, A15), (A15, A15)) → B14
  31. Segment((A15, A15), (A15, A16)) → C14
  32. Segment((A15, A16), (A16, A16)) → B15
  33. Segment((A16, A16), (A16, A17)) → C15
  34. Segment((A16, A17), (A17, A17)) → B16
  35. Segment((A17, A17), (A17, A18)) → C16
  36. Segment((A17, A18), (A18, A18)) → B17
  37. Segment((A18, A18), (A18, A19)) → C17
  38. Segment((A18, A19), (A19, A19)) → B18
  39. Segment((A19, A19), (A19, A20)) → C18
  40. Segment((A19, A20), (A20, A20)) → B19
  41. Segment((A20, A20), (A20, A21)) → C19
  42. Segment((A20, A21), (A21, A21)) → B20
  43. Segment((A21, A21), (A21, A22)) → C20
  44. Segment((A21, A22), (A22, A22)) → B21
  45. Segment((A22, A22), (A22, A23)) → C21
  46. Segment((A22, A23), (A23, A23)) → B22
  47. Segment((A23, A23), (A23, A24)) → C22
  48. Segment((A23, A24), (A24, A24)) → B23
  49. Segment((A24, A24), (A24, A25)) → C23
  50. Segment((A24, A25), (A25, A25)) → B24
  51. Segment((A25, A25), (A25, A26)) → C24
  52. Segment((A25, A26), (A26, A26)) → B25
  53. Segment((A26, A26), (A26, A27)) → C25
  54. Segment((A26, A27), (A27, A27)) → B26
  55. Segment((A27, A27), (A27, A28)) → C26
  56. Segment((A27, A28), (A28, A28)) → B27
  57. Segment((A28, A28), (A28, A29)) → C27
  58. Segment((A28, A29), (A29, A29)) → B28
  59. Segment((A29, A29), (A29, A30)) → C28
  60. Segment((A29, A30), (A30, A30)) → B29
  61. Segment((A30, A30), (A30, A31)) → C29
  62. Segment((A30, A31), (A31, A31)) → B30
  63. Segment((A31, A31), (A31, A32)) → C30
  64. Segment((A31, A32), (A32, A32)) → B31
  65. Segment((A32, A32), (A32, A33)) → C31
  66. Segment((A32, A33), (A33, A33)) → B32
  67. Segment((A33, A33), (A33, A34)) → C32
  68. Segment((A33, A34), (A34, A34)) → B33
  69. Segment((A34, A34), (A34, A35)) → C33
  70. Segment((A34, A35), (A35, A35)) → B34
  71. Segment((A35, A35), (A35, A36)) → C34
  72. Segment((A35, A36), (A36, A36)) → B35
  73. Segment((A36, A36), (A36, A37)) → C35
  74. Segment((A36, A37), (A37, A37)) → B36
  75. Segment((A37, A37), (A37, A38)) → C36
  76. Segment((A37, A38), (A38, A38)) → B37
  77. Segment((A38, A38), (A38, A39)) → C37
  78. Segment((A38, A39), (A39, A39)) → B38
  79. Segment((A39, A39), (A39, A40)) → C38
  80. Intersect(f, b) → B

AI Drawing Prompt Reference

Draw a circle with its center on the y-axis and a point on the x-axis lying on the circle. Then mark a sequence of points along the circle and connect them with straight segments to form a polygonal path that spirals inward toward the center. Also construct a right triangle using the center and the x-axis point, and draw its hypotenuse. Use a slider to control the radius; as the radius shrinks to zero, the circle, triangle, and polygonal path degenerate into points or lines, illustrating the limiting behavior of the geometric shapes.
The prompt above was compiled from the original drawing commands and is for reference only.