$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$
The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and $x_2-x_1$ is the length of the base of the rectangle. The geodesic is the shortest length curve parametrized by this on the condition that the area = c. The solution isn't always a strait line but could be some curved function. How do I find the function ? I know this is a variational problem involving a length integral and an area integral.
Do I need to find the 2x2 metric tensor for the 2-d parameterization curve and solve a pair of differential equations like here ?