Ok, So I am trying to look for the branch of mathametics that deals with scalar property such as surface which shows vector like qualities. So for analogy I am looking for a scalar area such that any path (vector) taken inside that area will result in closed circuit path much like Nescar racing circuit as well as satisfy the path length constraint. So the following is the objective:
$$\textrm{minimize}\ A\ \ \ \ \ \textrm{s.t}\\ \Sigma_{i=0}^{k}\ \sqrt{{(x_i-x_{i+1})}^2+{(y_i-y_{i+1})}^2}\ \le N \\ x_0 =x_k \\ y_0 =y_k$$
So basically the area must be something like a road layout (not necessarily long rectangular surface but arbitrary shape) that makes a circuit such that any path within that road is closed (arrive same destination where you started) . So is there any such thing that determines such surface such that the surface acts as a bounding box inside which any path that is constructed is a closed circuit. Again the notion of the path is that U-turns are not allowed and path must be smooth otherwise we can make zig-zag path and U-turns resulting in tiny surface. Illustration of such surface is as follows:
In summary instead of doing this the hardway as i mentioned is there any functional or anything branch of mathematics that solves this in much comprehensive way such as dealing with the surface directly so surface acts as a path (instead of explicitly describing a path within a surface as I did in my example)?