- #1
Catria
- 152
- 4
Homework Statement
Let [itex]\vec{X(t)}[/itex]: I [itex]\rightarrow[/itex] ℝ3 be a parametrized curve, and let I [itex]\ni[/itex] t be a fixed point where k(t) [itex]\neq[/itex] 0. Define π: ℝ3 [itex]\rightarrow[/itex] ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to [itex]\vec{X(t)}[/itex] at t. Define γ=π[itex]\circ[/itex][itex]\vec{X(t)}[/itex] as the orthogonal projection of the space curve [itex]\vec{X(t)}[/itex] onto the opsculating plane. Prove that the curvature k(t) is equal to the curvature of the plane curve [itex]\vec{γ}[/itex].
Homework Equations
k=[itex]\frac{\left\|\vec{X'(t)}\times\vec{X''(t)}\right\|}{\left\|\vec{X'(t)}\right\|^{3}}[/itex] = 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.