Juxtaposition of paths:

Suppose that $\gamma_1,\; \gamma_2$ are two paths such that $\gamma_1(1)=\gamma_2(0),$ that is to say, the end point of $\gamma_1$ is the initial point of $\gamma_2$. The paths $\gamma_1$ and $\gamma_2$ can be juxtaposed to produce a path from $\gamma_1(0)$ to $\gamma_2(1)$ called called the juxtaposition $\gamma_1$ and $\gamma_2,$, denoted by $\gamma_1 \ast \gamma_2$ and defined as :

$$(\gamma_1\ast \gamma_2)(t) = \left\{\begin{array}{lll} \gamma_1(2... ... t \leq 1/2 \\ \gamma_2(2t - 1) & & 1/2 \leq t \leq 1 \\ \end{array} \right.$$