Let \( \ell_1, \ell_2 : [0,1] \rightarrow \mathbb{C} \) be two loops (the case of general parameter intervals will be addressed momentarily). A continuous deformation from \( \ell_1 \) to \( \ell_2 \) is a continuous function \[ \gamma : [0,1] \times [0,1] \rightarrow \mathbb{C} \] such that \( \gamma(0,t) = \ell_1(t) \), \( \gamma(1,t) = \ell_2(t) \) for all \( t \in [0,1] \), and also \( \gamma(s,0) = \gamma(s,1) \) for all \( s \in [0,1] \).
If \( \ell_1 : [0,a] \rightarrow \mathbb{C} \) and \( \ell_2 : [0,b] \rightarrow \mathbb{C} \) are two loops, then a continuous deformation from \( \ell_1 \) to \( \ell_2 \) is defined to be a continuous deformation from \( \tilde{\ell}_1 \) to \( \tilde{\ell}_2 \), where \( \tilde{\ell}_1, \tilde{\ell}_2 : [0,1] \rightarrow \mathbb{C} \) are the reparametrizations: \( \tilde{\ell}_1(t) := \ell_1(a t) \) and \( \tilde{\ell}_2(t) := \ell_2(b t) \).