Catria
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Homework Statement
Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection of the space curve \vec{X(t)} onto the opsculating plane. Prove that the curvature k(t) is equal to the curvature of the plane curve \vec{γ}.
Homework Equations
k=\frac{\left\|\vec{X'(t)}\times\vec{X''(t)}\right\|}{\left\|\vec{X'(t)}\right\|^{3}} = Curvature
The Attempt at a Solution
I don't even know how to formulate the equation for the orthogonal projection of X onto the osculating plane, so I can't even begin to understand how to solve the problem in question.