Are these 2 paths homotopic?

Question

Let $X$ be a topological space. Let $a,b$ be points in $X$. Let $f,g$ be paths of length $1$ from $a$ to $b$ such that $Im \,f=Im \,g$.

Does it follow that $f,g$ are homotopic ?

Answer 1

No, it isn't true. Let $X = S^1$, $a=b=1$. Take $f(t) = e^{2 i \pi t}$, $g(t) = e^{4 i \pi t}$. Then $f(0)=f(1)=g(0)=g(1)=1$, $Im \,f = Im \, g = S^1$. They aren't homotopic if we consider homotopy with fixed ends.