non-negative curvature of space curve

Question

I read in most of the textbook that "curvature of a space curve is always non-negative" but I could not understand the intuition behind this that why is so? Give some nice intuition and proof.

Answer 1

It's a matter of definition. Only for a plane curve can you define signed curvature: You use a right-handed frame $T(s),N(s)$ (tangent, principal normal at $\alpha(s)$) and curvature is positive if the curve is bending towards $N$, negative if it's bending away from $N$. In more than two dimensions, you always define $N(s)$ to be in the direction of $T'(s)$ (assuming it's nonzero), and $\kappa$ is then the magnitude of this vector.