In this talk, we present some novel findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions. The studies reveal that the intersecting angle between two lines (nodal lines, singular lines and generalized singular lines) is closely related to the vanishing order of the eigenfunction at the intersecting point in R^2. And in R^3, the analytic behaviors of a Laplacian eigenfunction depends on the geometric quantities at the corresponding corner point (edge corner and vertex corner). The theoretical findings can be applied directly to some physical problems including the inverse obstacle scattering problem. Taking two-dimensional case for example, it is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications.