Equation of Lightpath in fractal spaces

The fractal dimension of Koch and Cesàro curves has different values. These two curves are not differentiable and inte- grable in the sense of ordinary calculus. This paper shows how fractal calculus can be applied to these two curves. To do this, we need to obtain the corresponding staircase function. Calculating the path integral on these two fractal curves is the starting point for formulating and extracting Lagrange equations. Fractal integration and fractal derivation on these two fractal paths are performed analytically, and the results are plotted in a MATLAB environment. In addition, we present an application by finding light path equations in this fractal space using the generalized Fermat’s principle in fractal calculus.