How to deform a curve in specific manner

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I am wondering whether we can deform a path in specific ways continuously i mean if there is a closed piece wise $C^1$ smooth path which has to be deformed to another piece wise $C^1$ smooth path. Let us assume that deformation is possible and let the deformation function be $H(s,t)$. Denote $\gamma_t{(s)}$=$H(s,t)$. So does there exist $H(s,t)$ such that all paths are piece wise $C^1$ smooth paths.