When P ⊂ ℝ³ is a periodic point set with the period lattice L, an efficient method to determine the quadratic form of L ⊂ ℝ³ (more precisely, its equivalence class over ℤ.) from the average theta series of P has a practical application to the problem known as “powder indexing” in crystallography. By using “topographs” defined in the reduction theory of quadratic forms, we succeeded in developing an algorithm robust against loss and errors of information due to observational problems, suppressing the computation time. We introduce how the topographs were used in the method.

This is an overview of the paper[19], which makes a survey of discrete differential geometry of curves and surfaces. We review a few representatives of discrete curves and surfaces, with emphasis on close connections between both the theories of discrete differential geometry and discrete integrable systems.