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Arbitrary Cone Sections Compared: Parabola vs Circle Formation

Arbitrary Cone Sections Compared: Parabola vs Circle Formation
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2015-11-01 23:23
Coordinate GeometrySolid GeometryGeometric ConstructionLocusGraphing
This GeoGebra example demonstrates various curves formed when a plane intersects a cone. A parabola occurs when the plane is parallel to a generating line of the cone; otherwise, circles or other conic sections appear. Key mathematical concepts include conic section definitions, parabola focus and directrix properties, and circle geometry. Construction highlights: Using Point, Line, Intersect to create intersection curves; applying AngularBisector and OrthogonalLine to locate parabola focus; employing Rotate and Mirror for transformations; generating conics via Parabola, Ellipse, Circle commands. Suitable for high school and college students studying coordinate geometry and conic sections, helping understand the unified properties and geometric construction methods of conic sections.

GeoGebra Commands & Parameters

  1. Point(b) → A
  2. Line(A, P) → c
  3. Intersect(a, b) → O
  4. AngularBisector(O, A, P) → d
  5. Intersect(d, yAxis) → C_1
  6. OrthogonalLine(C_1, c) → e
  7. Intersect(c, e) → F_1
  8. Circle(C_1, F_1) → f
  9. Intersect(a, c) → E
  10. AngularBisector(b, c) → g, h
  11. Intersect(h, yAxis) → C_2
  12. OrthogonalLine(C_2, c) → i
  13. Intersect(c, i) → F_2
  14. Circle(C_2, F_2) → k
  15. Segment(A, E) → j
  16. Line(A, a) → l
  17. OrthogonalLine(A, yAxis) → m
  18. Intersect(m, yAxis) → A'
  19. Angle(A', A, P) → α
  20. Vector(A, O) → u
  21. Rotate(Translate[F_1, u], (-α)) → F_1'
  22. Rotate(Translate[F_2, u], (-α)) → F_2'
  23. Ellipse(F_1', F_2', j / 2) → p
  24. OrthogonalLine(C_3, l) → q
  25. Intersect(l, q) → F_3
  26. Rotate(Translate[F_3, u], -45*°) → F_3'
  27. Mirror(F_3', yAxis) → B
  28. OrthogonalLine(B, xAxis) → r
  29. Parabola(F_3', r) → s
  30. Line(A, E) → t
  31. Point(t) → R_1
  32. Point(t) → R_2
  33. Ray(E, R_1) → r_1
  34. Ray(A, R_2) → r_2

AI Drawing Prompt Reference

Draw a cone intersected by a plane. When the plane is parallel to a generating line of the cone, the intersection is a parabola, showing its focus and directrix. When the plane is not parallel to the generating line, the intersection becomes a circle or ellipse. Mark the foci and directrices for both the parabola and ellipse, and draw line segments from the cone's vertex to the intersection curves representing the generating lines. Also show the special angular relationships when forming the parabola, and the elliptical trajectory obtained through rotation and translation.
The prompt above was compiled from the original drawing commands and is for reference only.