📂Geometry

Definition of a Closed Curve

Definition ¹

A regular curve $\beta (t)$ being a closed curve is equivalent to being a periodic function $\beta$.

Formula: Length of a Closed Curve

If $\alpha (s)$ is the arc length parameterization of a closed curve $\beta (t)$ with period $a>0$, then $\alpha$ is a closed curve with period $L = \int_{0}^{a} |d \beta / dt| dt$. In other words, the length of the closed curve $\beta$ is $L$.

Derivation

$$\begin{align*} s(t+a) =& \int_{0}^{t+a}\left|\frac{d \beta}{d t}\right| d t \\ =& \int_{0}^{a}\left|\frac{d \beta}{d t}\right| d t+\int_{a}^{t+a}\left|\frac{d \beta}{d t}\right| d t \\ =& L+\int_{0}^{t}\left|\frac{d \beta}{d t}\right| d t \\ =& L+s(t) \end{align*}$$ To summarize, $s \left( t + a \right) = s(t) + L$, $$\begin{align*} \alpha (s + L) =& \alpha \left( s (t) + L \right) \\ =& \alpha \left( s (t + a) \right) \\ =& \beta (t + a) \\ =& \beta (t) \\ =& \alpha \left( s(t) \right) \\ =& \alpha (s) \end{align*}$$ Therefore, $\alpha (s)$ is a closed curve. $a > 0$ being $$\beta (t + a) = \beta (t) \qquad , \forall t$$ the smallest positive number satisfying, so $L>0$ also $$\alpha (s + L) = \alpha (s) \qquad , \forall s$$ must be the smallest positive number satisfying. In other words, the length of $\beta$ is $L$.

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