Proving homotopy of 2 paths

Question

$A$ and $B$ are $2$ points of affixes $i$ and $1$ respectively, $O$ is the origin. $γ_1=[AO]∪[OB]$ and $γ_2=[AB]$ are $2$ paths. I know how to prove that $2$ paths are homotopy but in this case I don't how to deal with $γ_1$ because in order to parametrize the $γ_1$ we have to divide it into two segments. Any help.

Answer 1

Presumably you have written out a formula for $\gamma_1$ which is subdivided into two pieces: $$\gamma_1(t) = \begin{cases} ... &\text{if $0 \le t \le \frac{1}{2}$} \\ ... &\text{if $\frac{1}{2} \le t \le 1$} \end{cases} $$ Divide $\gamma_2$ into two segments as well, by subdividing it at its midpoint, obtaining a formula like this (I'll leave it for you to fill in the missing information): $$\gamma_2(t) = \begin{cases} ... &\text{if $0 \le t \le \frac{1}{2}$} \\ ... &\text{if $\frac{1}{2} \le t \le 1$} \end{cases} $$ And then write the homotopy by subdividing it into two homotopies in the same manner: $$H(t,u) = \begin{cases} ... &\text{if $0 \le t \le \frac{1}{2}$} \\ ... &\text{if $\frac{1}{2} \le t \le 1$} \end{cases} $$