  In this paper, we present some novel and intriguing findings on the geometric
structures of Laplacian eigenfunctions and their deep relationship to the
quantitative behaviours of the eigenfunctions in two dimensions. We introduce a
new notion of generalized singular lines of the Laplacian eigenfunctions, and
carefully study these singular lines and the nodal lines. The studies reveal
that the intersecting angle between two of those lines is closely related to
the vanishing order of the eigenfunction at the intersecting point. We
establish an accurate and comprehensive quantitative characterisation of the
relationship. Roughly speaking, the vanishing order is generically infinite if
the intersecting angle is {\it irrational}, and the vanishing order is finite
if the intersecting angle is rational. In fact, in the latter case, the
vanishing order is the degree of the rationality. The theoretical findings are
original and of significant interest in spectral theory. Moreover, they are
applied directly to some physical problems of great importance, including the
inverse obstacle scattering problem and the inverse diffraction grating
problem. 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. Unique identifiability by finitely
many far-field patterns remains to be a highly challenging fundamental
mathematical problem in the inverse scattering theory.
