If you want a more mathematical description or construction, the equidistant curves for a smooth curve $\gamma$ depend on the cut locus for $\gamma$. The cut locus is the set of points where there is more than one closest point on $\gamma$, and it is closely related to the whole theory of Voronoi diagrams, and you . You can compute a good approximation of it from a Voronoi diagram program or a convex hull program, if you lift the curve to the paraboloid $z = x^2 L+ + y^2 \subset \mathbb R^3$. In the complement of the cut locus, there is a smooth map $(x, y) \to C(x, y)$ where $C(x,y)$ is the closest point on $\gamma$; it can be traced out implicitly, it's the inverse function to what you're already doing. For a generic smooth curve, the cut locus is a piecewise smooth tree, whose endpoints are centers of osculating circles where the curvature of $\gamma$ has a local maximum. (However, in general, the cut locus can be quite complicated and have infinitely many branches, even for a $C^\infty$ curve). The edges of the cut locus can be traced from these endpoints, using the implicit function theorem; the main difficulty is keeping track of enough information to get the correct combinatorics for the graph. It's equivalent to the problem of constructing the convex hull of a simple curve on the paraboloid above.
If you want a more mathematical description or construction, the equidistant curves for a smooth curve $\gamma$ depend on the cut locus for $\gamma$. The cut locus is closely related to the whole theory of Voronoi diagrams, and you can compute a good approximation of it from a Voronoi diagram program or a convex hull program, if you lift the curve to the paraboloid $z = x^2 L+ y^2 \subset \mathbb R^3$. In the complement of the cut locus, there is a smooth map $(x, y) \to C(x, y)$ where $C(x,y)$ is the closest point on $\gamma$; it can be traced out implicitly, it's the inverse function to what you're already doing. For a generic smooth curve, the cut locus is a piecewise smooth tree, whose endpoints are centers of osculating circles where the curvature of $\gamma$ has a local maximum. The edges of the cut locus can be traced from these endpoints, using the implicit function theorem; the main difficulty is keeping track of enough information to get the correct combinatorics for the graph. It's equivalent to the problem of constructing the convex hull of a simple curve on the paraboloid above.