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Angles Between Two Triangular Planes

Angles Between Two Triangular Planes
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2019-02-19 10:45
AngleVectorSolid GeometryCalculusGeometric Construction
This example demonstrates the calculation and visualization of the angle between two triangular planes. It involves core mathematical concepts such as angles, planes, vectors, solid geometry, and calculus. The construction uses commands like DynamicCoordinates, Rotate, Point, Plane, and Angle, along with objects like Polygon, Line, and Circle to create a dynamic model. Suitable for high school students and math enthusiasts studying spatial geometry and computational geometry.

GeoGebra Commands & Parameters

  1. DynamicCoordinates(C_0, x(C_0), If[y(C_0) ≥ 1, y(C_0), 1], 0) → C
  2. Rotate(C, theta, xAxis) → C'
  3. Point(xAxis) → A
  4. Point(xAxis) → B
  5. Plane(A, B, C') → d
  6. PointIn(d) → D_0
  7. Polygon(A, B, C) → t1, c, a, b
  8. Line(A, B) → f
  9. DynamicCoordinates(D_0, x(D_0), If[z(D_0) ≥ 0, y(D_0), 0], If[z(D_0) ≥ 0, z(D_0), 0]) → D
  10. Polygon(D, A, B) → t2, b_1, d_1, a_1
  11. Point(f) → P
  12. Angle(C, B, D) → β
  13. Angle(C, A, D) → α
  14. Circle(P, 1, d_1) → g
  15. Intersect(d, g) → E, F
  16. Intersect(xOyPlane, g) → G, H
  17. Ray(P, E) → h
  18. Rotate(D, (-theta), xAxis) → D'
  19. Segment(A, D') → j
  20. Segment(D', B) → k
  21. If(theta ≟ 180°, CircumcircleArc[D', (x(D'), 0, y(D')), D], CircleArc[Q, D, D', xOyPlane]) → e
  22. Ray(P, If[x(B) > x(A), G, H]) → i
  23. Angle(If[x(B) > x(A), G, H], P, E) → ε
  24. Angle(E, P, P + (-1, 0, 0)) → γ
  25. Angle(P + (1, 0, 0), P, If[x(B) > x(A), G, H]) → δ