Why curvature of a plane curve is k =d(phi)/ds?

[A330NEO]

Why curvature of a plane curve is k = $$\frac{d\phi }{ds}$$?

I know that curvature of a plane is $$\frac{\left | r'(t) \times r''(t) \right |}{\left | r'(t) \right |^3}$$, and that led to this.

$$k = \frac{\left | \frac{d^2s}{dt^2} \right |sin\phi }{\left | \frac{ds}{dt}\right |^2}$$

But I can't go any further. Any ideas?

 

[Fantini]

That is the definition of curvature. The expression you presented is the expression for the curvature given in terms of the derivatives of an arbitrary parametrization of a curve.

Consider a parametrization by arc length $\varphi: I \subset \mathbb{R} \to \mathbb{R}^3$. Then

$$\left\| \frac{d \varphi}{ds} \right\| = 1 \text{ and } \left\| \frac{d^2 \varphi}{ds^2} \right\| = \kappa,$$

which is the curvature of the curve.

 

[A330NEO]

That is the definition of curvature. The expression you presented is the expression for the curvature given in terms of the derivatives of an arbitrary parametrization of a curve.

Consider a parametrization by arc length $\varphi: I \subset \mathbb{R} \to \mathbb{R}^3$. Then

$$\left\| \frac{d \varphi}{ds} \right\| = 1 \text{ and } \left\| \frac{d^2 \varphi}{ds^2} \right\| = \kappa,$$

which is the curvature of the curve.

uhh... I still don't get it. $$k=\left|\d{T}{s} \right|$$(if length of curve is s and T is for its tangential vector), but how does that become $$\left| \d{\phi}{s} \right|$$? why use $$\phi$$, which is the angle between T and i(1, 0, 0)? Is there an specific characteristic of 'plane' curve that separages it from other 3-dimensional curves?