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Intersection of Torus and Plane

Intersection of Torus and Plane
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2017-07-24 19:16
Coordinate GeometrySolid GeometryGeometric Construction
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the intersection of a torus and a plane. Key mathematical concepts include solid geometry and coordinate geometry. Using GeoGebra commands such as Surface, CurveCartesian, and IntersectPath, the intersection line can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(O, R, xOyPlane) → midtor
  2. Surface(((R + (r * cos(u))) * cos(v)), ((R + (r * cos(u))) * sin(v)), (r * sin(u)), u, 0, (2 * π), v, 0, (2 * π)) → Torus
  3. CurveCartesian(x(Q) - (sin(α) * u), y(Q) + (cos(α) * u), u, -10, 10) → αline
  4. Surface(x(Q) + (sin(α) * t) - ((sin(ϕ) * cos(α)) * w), y(Q) - (cos(α) * t) - ((sin(ϕ) * sin(α)) * w), z(Q) + (w * cos(ϕ)), t, -10, 10, w, -10, 10) → s
  5. Vector(Q, Q + (sin(α), (-cos(α)), 0)) → t_{axis}
  6. Vector(Q, Q + (((-cos(α)) * sin(ϕ)), ((-sin(α)) * sin(ϕ)), cos(ϕ))) → w_{axis}
  7. Circle(O, R + r, xOyPlane) → exttor
  8. Circle(O, R - r, xOyPlane) → intttor
  9. Line(Q, t_{axis}) → tline
  10. Line(Q, w_{axis}) → wline
  11. IntersectPath(xypla, αϕQpla) → f
  12. ImplicitCurve(torus(x, (ρ * cos(ϕ)) - (y * sin(ϕ)), z(Q) + (y * cos(ϕ)))) → ftw