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Unwrapping a Cylinder Cut by an Oblique Plane

Unwrapping a Cylinder Cut by an Oblique Plane
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2015-07-22 14:21
Solid GeometryParametric EquationInequality
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 unwrapping of a cylinder cut by an oblique plane, involving core mathematical concepts such as solid geometry, surfaces, inequalities, and parametric equations. The construction utilizes GeoGebra commands like Intersect, Point, Circle, Cylinder, Translate, Plane, IntersectPath, Surface, CurveCartesian, Segment, and CornerThreeD. It is suitable for high school and above, particularly in teaching solid geometry and calculus.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Point(zAxis) → C
  5. Cylinder(A, C, 1) → a, d, e, b
  6. Translate(xOyPlane, Vector[(0, 0, 1.719232508579406)]) → xOyPlane'
  7. Plane((0, 1, z(C) / 2), (0, -1, z(C) / 2), (1, 0, (z(C) / 2 * (1 - t)))) → f
  8. IntersectPath(f, a) → g
  9. Surface((l * (1 - cos((v * φ)))) - 1, (l * sin((v * φ))), ((u * h / 2) * (1 + (t * cos((v * π))))), u, 0, 1, v, -1, 1) → j
  10. Surface(-1, u, ((v * h / 2) * (1 + (t * cos(u)))), u, -3.141592653589793, 3.141592653589793, v, 0, 1) → i
  11. Surface(cos(u), sin(u), h - ((v * h / 2) * (1 + (t * cos(u)))), u, -3.141592653589793, 3.141592653589793, v, 0, 1) → m
  12. CurveCartesian((l * (1 - cos((v * φ)))) - 1, (l * sin((v * φ))), (h / 2 * (1 + (t * cos((v * π))))), v, -1, 1) → k
  13. CurveCartesian(-1, u, (h / 2 * (1 + (t * cos(u)))), u, -3.14159, 3.141592653589793) → n
  14. CurveCartesian((l * (1 - cos((v * φ)))) - 1, (l * sin((v * φ))), 0, v, -1, 1) → o
  15. Segment((-1, -3.141592653589793), (-1, 3.141592653589793)) → p
  16. Segment(D, F) → q
  17. Segment(E, G) → r
  18. Segment((-1, -3.141592653589793, 0), (-1, (-π), (h / 2 * (1 - t)))) → a_1
  19. Segment((-1, 3.141592653589793, 0), (-1, π, (h / 2 * (1 - t)))) → b_1
  20. CurveCartesian(cos(u), sin(u), h, u, 0, 6.283185307179586) → u
  21. CornerThreeD(-1, 11) → C_v