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Spiral Screw Bisection of a Cone

Spiral Screw Bisection of a Cone
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2018-12-09 01:18
Coordinate GeometrySolid GeometryOptimizationGeometric Construction
This example demonstrates the spiral screw bisection of a cone. It involves core mathematical concepts such as cones, curve sketching, linear programming or optimization, similar triangles, congruence, and isosceles triangles. The construction utilizes GeoGebra commands and objects like Intersect, Circle, Cone, Surface, CurveCartesian, Rotate, and Translate. It is suitable for high school and above, particularly for teaching and exploring solid geometry, coordinate geometry, and optimization problems.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xAxis) → A
  2. Circle(A, π, zAxis) → c
  3. Cone(c, π) → a, a_2, a_3
  4. Surface((u * cos(t)), (u * sin(t)), t, t, 0, π, u, -4, 4) → b
  5. CurveCartesian(((π - t) * cos(t)), ((π - t) * sin(t)), t, t, 0, π) → d
  6. Rotate(d, π) → d'
  7. Surface((u * x(d(t))) + ((1 - u) * x(d'(t))), (u * y(d(t))) + ((1 - u) * y(d'(t))), (u * z(d(t))) + ((1 - u) * z(d'(t))), t, 0, π, u, 0, 1) → e
  8. CurveCartesian((π * cos(t)), (π * sin(t)), 0, t, 0, π) → f
  9. Rotate(f, π, zAxis) → g
  10. Translate(Rotate[e, n], Vector[(0, 0, n)]) → h
  11. Translate(Rotate[d, n], Vector[(0, 0, n)]) → i
  12. Translate(Rotate[d', n], Vector[(0, 0, n)]) → j
  13. Translate(Rotate[f, n], Vector[(0, 0, n)]) → k
  14. Translate(Rotate[CurveCartesian[t, 0, 0, t, (-π), π], n], Vector[(0, 0, n)]) → l
  15. CurveCartesian(t, 0, 0, t, (-π), π) → m
  16. Translate(Rotate[Surface[(u * x(f(t))), (u * y(f(t))), 0, t, 0, π, u, 0, 1], n], Vector[(0, 0, n)]) → o
  17. Surface((u * x(g(t))), (u * y(g(t))), 0, t, 0, π, u, 0, 1) → p
  18. Surface((u * x(i(t))) + ((1 - u) * x(k(t))), (u * y(i(t))) + ((1 - u) * y(k(t))), (u * z(i(t))) + ((1 - u) * z(k(t))), t, 0, π, u, 0, 1) → q
  19. Surface((u * x(j(t))), (u * y(j(t))), (u * z(j(t))) + ((1 - u) * (π + n)), t, 0, π, u, 0, 1) → r
  20. Surface((u * x(g(t))) + ((1 - u) * x(d'(t))), (u * y(g(t))) + ((1 - u) * y(d'(t))), (u * z(g(t))) + ((1 - u) * z(d'(t))), t, 0, π, u, 0, 1) → s
  21. Surface((u * x(d(t))), (u * y(d(t))), (u * z(d(t))) + ((1 - u) * π), t, 0, π, u, 0, 1) → t