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Perspective Projections of Curves on a Cone (Zoom)

Perspective Projections of Curves on a Cone (Zoom)
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2018-10-31 07:19
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the dynamic process of perspective projections on a cone surface, with an adjustable zoom function to modify the view. Key mathematical concepts include solid geometry (cone, projection transformations, similarity transformations) and analytical geometry (parametric equations). Construction highlights from the summary involve GeoGebra commands/objects like Ray, Point, DynamicCoordinates, Segment, Circle, Intersect, Cone, OrthogonalPlane, IntersectPath, Distance, CurveCartesian, Angle, and Translate. Suitable for high school and above, focusing on spatial geometry, projection transformations, and dynamic geometry demonstrations.

GeoGebra Commands & Parameters

  1. Ray(A, B) → f
  2. Point(f) → G
  3. Point(f) → OO
  4. DynamicCoordinates(OO, If[x(OO) < x(G) + L + deltaa, x(G) + L + deltaa, x(OO)], y(OO), z(OO)) → O
  5. Segment(O, G) → k
  6. Circle(G, L, f) → c
  7. Point(c) → C
  8. Ray(O, C) → g
  9. Intersect(g, a) → E
  10. Circle(D, Distance[D, E], f) → d
  11. Cone(G, O, L) → h, h_2, h_3
  12. Segment(O, H) → i
  13. Point(k) → F
  14. OrthogonalPlane(F, i) → b
  15. IntersectPath(b, h) → e
  16. Distance(D, E) → R
  17. CurveCartesian(t, ((R * (x(O) - t) / (x(O) - xV)) * cos((n * t))), ((R * (x(O) - t) / (x(O) - xV)) * sin((n * t))), t, tMin, tMax) → j
  18. Angle(I, H, O) → α
  19. Point(k) → J
  20. IntersectPath(l, h) → p
  21. Translate(O, vetor) → O'_1
  22. Point(e) → S