Computing the total curvature Let $C$ be the curve in $\Bbb{R}^2$ given by $(t-\sin t,1-\cos t)$ for $0 \le t \le 2 \pi$. I want to find the total curvature of $C$.
I found it brutally by finding the curvature $k(t)$, and then reparametrize it by arc-length $s$, and then $\int_0^Lk(t(s))ds$, where $L=8$ is the lenght of $C$.
I found that the answer is $\pi$. But is there any way to compute it easily? The above computation was somewhat hard, and I guess that there maybe some easy methods(maybe something like Gauss-Bornet, or how much the angle of tangent vector has changed)
 A: At $t = 0$, the limiting value (from the right) of the unit tangent points vertically upward. At $t = 2\pi$, the limiting value (from the left) of the unit tangent points vertically downward. The total curvature is the angle through which the tangent rotates along the curve, namely $\pi$.
A: Length: $s(2 \pi) = \int\limits_0^{2\pi} ||\dot{c}(t)||dt =  \int\limits_0^{2\pi} \sqrt{2(1-cos(t))}dt=8$.
Curvature: A formula to compute the curvature is $\frac{1}{\rho} = ||c''(s)||$ (at least if the used parameter is $s$).
A: If $\gamma (t) $ is a regular space curve, its curvature is
$\kappa =||\gamma'' \times \gamma'||/||\gamma' ||^3 $.
Computation:
$\kappa'= (1-cost,sint,0), \kappa''= (sint,cost,0)$.
$||\gamma''(t) \times \gamma'(t)||= 1-cost$, $||\gamma'(t)||=2(1-cost)$.
$\int \kappa(s) ds = \int \frac{||\gamma''(t) \times \gamma'(t)||}{||\gamma'(t) ||^3}||\gamma'(t) ||dt = \int \frac{||\gamma''(t) \times \gamma'(t)||}{||\gamma'(t) ||^2}dt   =\int^{2 \pi}_{0} \frac{1-cost}{2(1-cost)}= \pi$
A: A simpler answer is as follows: given the natural equation for a curve in the form of $\kappa(s)$, it can be shown that the tangent angle is given by
$$\theta=\int \kappa (s) ds \ \ \ \text{or} \ \ \  \kappa (s)=\frac{d\theta}{ds}$$
Thus from the definition of curvature used here, we obtain
$$K=\int_{s_1}^{s_2}\kappa(s)ds=\int_{\theta_1}^{\theta_2}d\theta=\theta_2-\theta_1$$
In the present example, you will find $K=-\pi$, where the minus sign is due to the curve unwinding in the clockwise direction.
