1. The problem statement, all variables and given/known data Provide a complete proof that a regular plane curve γ : I → R2 can near each point γ(t0) be written as a graph over the tangent line: more precisely, there exists a smooth real valued map x → f(x) for small x with f(0) = 0 so that x → xT(t0) + f(x)JT(t0) parametrizes γ near γ(t0). Here T = γ'/||γ'|| is the unit length tangent vector. 2. Relevant equations 3. The attempt at a solution So I don't know what exactly they mean as a graph over the tangent line (like an xy-coordinate system where the tangent line is one of the axes of the graph?) but the further explanation kind of clears it up a little bit in that it seems like the map x → xT(t0) + f(x)JT(t0) basically lines up the x direction along the tangent line at the point t0 and since JT is orthogonal to T we can see that the map also lines up the y (or f(x)) direction along the normal to the tangent line, thus creating an orthonormal basis at every point t0 on the curve. Can anyone tell me if this thinking is correct and if so, how I could potentially put this down in a mathematical argument?