[math-fun] Connecting simple closed curves of the same length

Let C be a rectifiable simple closed curve* (RSCC) in the plane having length L(C) = 2π. Question: --------- Does there always exist a continuous family {C_t} of RSCC such that C_0 = C and C_1 is the unit circle in R^2, such that L(C_t) = 2π for all t in [0,1] ??? —Dan _____ * If C ⊂ R^2 is a RSCC, then C = A([0,1]) for some continuous function A : [0,1] —> R^2 where A(s) = A(t) for s ≠ t if and only if {s,t} = {0,1}, and such that L(C) = sup{∑ ‖A(x_(j+1)) - A(x_j)‖} < oo where the supremum is taken over all 0 = x_0 < x_1 < ... < x_n = 1. The length of C is then defined as L(C).