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Net of a Cone by Norm Prokup

Net of a Cone by Norm Prokup
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2017-11-08 12:34
CircleVectorSolid GeometryParametric EquationGeometric Construction
This example demonstrates the net of a cone, illustrating how a conical surface can be unrolled into a flat geometric shape. Core math concepts include the cone, circle, polygon, vector, rotation, and parametric equations. The construction utilizes GeoGebra commands and objects such as CurveCartesian, Segment, Line, Rotate, Vector, Surface, Circle, and Angle. It is suitable for high school students studying solid geometry, helping them understand the geometric properties and unrolling process of a cone.

GeoGebra Commands & Parameters

  1. CurveCartesian(((4 * cos(t)) * sin(t)), 0, ((4 * cos(t)) * cos(t)), t, 0.01, (0.4 * π)) → a
  2. Segment(A, C) → f
  3. Line(A, C) → g
  4. Segment(C, B) → r
  5. Segment(B, D) → rr
  6. Rotate(B, (-θ), g) → B_2
  7. Rotate(B, θ, g) → B_1
  8. Vector(C, B) → u
  9. Rotate(u, 90°, g) → u'
  10. Surface(((1 - t) * x(A)) + (t * (x(C) + (x(u) * cos(s)) + (x(u') * sin(s)))), ((1 - t) * y(A)) + (t * (y(C) + (y(u) * cos(s)) + (y(u') * sin(s)))), ((1 - t) * z(A)) + (t * (z(C) + (z(u) * cos(s)) + (z(u') * sin(s)))), s, (-θ), θ, t, 0, 1) → b
  11. Segment(A, B_1) → h
  12. Segment(B_2, A) → i
  13. Circle(D, abs(B - D), Vector[D, A]) → c
  14. Angle(F, B, D) → α
  15. Rotate(c, (d * (π / 2 + β)), yAxis) → c'
  16. Rotate(D, (d * (π / 2 + β)), yAxis) → D'
  17. CurveCartesian(x(C) + (x(u) * cos(s)) + (x(u') * sin(s)), y(C) + (y(u) * cos(s)) + (y(u') * sin(s)), z(C) + (z(u) * cos(s)) + (z(u') * sin(s)), s, (-θ), θ) → e
  18. Vector(D, D - (sqrt(3), 0, 1)) → v
  19. Rotate(c, (d * (π / 2 + β)), yAxis) → c''