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Osculating Plane as Intersection of Normal Planes

Osculating Plane as Intersection of Normal Planes
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2015-10-16 03:53
VectorSolid GeometryCalculusDerivative
This example demonstrates how the osculating plane can be constructed as the intersection of normal planes. It involves core mathematical concepts like normal planes, derivatives, and vector operations. GeoGebra commands and objects such as CurveCartesian, Vector, Line, Intersect, and Circle are used in the construction. Suitable for high school and above students learning advanced concepts in spatial geometry and calculus.

GeoGebra Commands & Parameters

  1. CurveCartesian(cos(t), sin(t), t, t, 0, 20) → h
  2. Vector(h(a), h(a) + ((-sin(a)), cos(a), 1)) → TangentA
  3. Vector(h(b), h(b) + ((-sin(b)), cos(b), 1)) → TangentB
  4. Line(C, D) → c
  5. Vector(h(a), h(a) + ((-sin(a)), cos(a), -1)) → u
  6. Intersect(c, d) → E
  7. Circle(E, A, c) → e
  8. Vector(A, A - ((cos(a) - cos(b)) / (b - a), (sin(a) - sin(b)) / (b - a), sin(b - a) / (b - a))) → intersect