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Spiral Screw Bisection of a Sphere

Spiral Screw Bisection of a Sphere
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2018-12-09 05:22
Coordinate GeometrySolid GeometryCalculusParametric EquationGeometric Construction
This example demonstrates the geometric construction of bisecting a sphere using a spiral screw motion. It involves core mathematical concepts such as sphere geometry, coordinate geometry, and integral calculus. The construction utilizes commands like Surface, CurveCartesian, Rotate, and Translate in GeoGebra. Suitable for high school and above students to understand the application of spatial geometry and calculus in geometric problems.

GeoGebra Commands & Parameters

  1. Surface((cos(t) * sin(u)), (sin(t) * sin(u)), cos(u), t, 0, (2 * π), u, 0, π) → a
  2. CurveCartesian((r(t) * cos(t)), (r(t) * sin(t)), (2 / π * t) - 1, t, 0, π) → c
  3. Rotate(c, π, zAxis) → d
  4. Surface((u * x(c(t))) + ((1 - u) * x(d(t))), (u * y(c(t))) + ((1 - u) * y(d(t))), (u * z(c(t))) + ((1 - u) * z(d(t))), t, 0, π, u, 0, 1) → b
  5. Translate(Rotate[b, n, zAxis], Vector[(0, 0, (2 / π * n))]) → e
  6. Translate(Rotate[c, n, zAxis], Vector[(0, 0, (2 / π * n))]) → f
  7. Translate(Rotate[d, n, zAxis], Vector[(0, 0, (2 / π * n))]) → g
  8. Surface(c, π, zAxis) → h
  9. Surface(g, π, zAxis) → i