P = (f(t), g(t), h(t))
\vec\sigma(t) = f(t)\hat\imath + g(t)\hat\jmath + h(t)\hat k
When ||\vec\sigma(t)|| = \text{const} then \vec\sigma'(t) is perpendicular to \vec\sigma(t) for all t
r = \vec\sigma(t_0) + t\vec\sigma'(t_0)
L = \int_a^b \sqrt{f'(t)^2 + g'(t)^2 + h'(t)^2}dt
A curve is called regular if \vec\sigma'(t) \ne \mathbf 0 for all t
When a curve is regular then T = \frac{\vec\sigma'(t)}{||\vec\sigma'(t)||} is called the unit tangent vector to the curve
If the length of \vec\sigma'(t) is constant and equal to 1, the curve is said to be parametrized by arc length (or unit speed curve)
s = \int_a^t ||\vec\sigma'(u)||du
Let k be a scalar: k = ||\frac{dT}{ds}||. Then k is called the curvature of a curve.
When k \ne 0:
N = \frac{\frac{dT}{ds}}{||\frac{dT}{ds}||}
Then N is called the principle normal vector
Alternative formulae:
k = \frac{||v \times v'||}{||v||^3}
N(t) = \frac{(v \cdot v)v' - (v' \cdot v)v}{||(v \cdot v)v' - (v' \cdot v)v||}