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Conic Section Demo

Conic Section Demo
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2017-05-23 18:31
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the formation of conic sections as the intersection of a plane and a cone. It involves core concepts like cones, planes, cross-sections, and quadratic curves (ellipse, parabola, hyperbola). Construction highlights include using GeoGebra commands such as OrthogonalLine, Intersect, InfiniteCone, and IntersectPath to create an interactive model. Suitable for high school solid geometry and analytical geometry instruction to visually explore properties of conic sections.

GeoGebra Commands & Parameters

  1. OrthogonalLine(M, xOyPlane) → a
  2. Point(a) → B
  3. Line(A, yAxis) → g
  4. Circle(A, 16, xAxis) → c
  5. Intersect(g, c) → X_1, J
  6. Intersect(zAxis, c) → Z_1, Z_2
  7. CircleArc(A, X_1, Z_1) → f
  8. Point(f) → CC
  9. If(j < 0, A + (0, (16 * cos(0)), (16 * sin(0))), If[j < π / 2 - α - 0.01, A + (0, (16 * cos(j)), (16 * sin(j))), If[j < π / 2 - α + 0.1, A + (0, (16 * cos(π / 2 - α)), (16 * sin(π / 2 - α))), If[j < π / 2 + 0.1, A + (0, (16 * cos(j - 0.1)), (16 * sin(j - 0.1))), If[j > π / 2 + 0.1, A + (0, (16 * cos(π / 2)), (16 * sin(π / 2)))]]]]) → XX
  10. If(anim, XX, CC) → C
  11. Point(zAxis) → H
  12. Intersect(zAxis, xAxis) → O
  13. Line(C, A) → d
  14. Line(A, B) → b
  15. Plane(d, b) → e
  16. Mirror(H, xOyPlane) → H'
  17. Point(zAxis) → I
  18. InfiniteCone(O, H, α) → m
  19. IntersectPath(e, m) → p
  20. Vertex(p) → D, E, F, G
  21. Angle(C, A, Z_1) → β
  22. CircleArc(A, C, Z_1) → h
  23. Circle(A, 20, xAxis) → k
  24. Ray(A, C) → i
  25. Intersect(i, k) → K,
  26. Vector(A, K) → l
  27. Textfield(α) → inputBox1

AI Drawing Prompt Reference

Draw a demonstration of conic sections as the intersection of a plane and a cone, showing the formation of an ellipse, parabola, and hyperbola. Include a cone, a plane, a circle parallel to the x-axis, and a ray. The cone has its vertex at the origin and a base radius of 16. The intersection of the plane with the cone displays different types of quadratic curves depending on the angle. Add a point moving along the circle, connecting it to the cone's vertex to form a line segment. The intersection of this line segment with the circle parallel to the x-axis traces a parabola. A ray emanating from the cone's vertex intersects the circle parallel to the x-axis, showing one branch of a hyperbola.
The prompt above was compiled from the original drawing commands and is for reference only.