  In this paper, we study an inverse problem of determining the cross section
of an infinitely long cylindrical-like material structure from the transverse
electromagnetic scattering measurement. We establish a sharp logarithmic
stability result in determining a polygonal scatterer by a single far-field
measurement. The argument in establishing the stability result is localized
around a corner and can be as well used to produce two highly intriguing
implications for invisibility and transmission resonance in the wave scattering
theory. In fact, we show that if a generic medium scatterer possesses an
admissible corner on its support, then there exists a positive lower bound of
the $L^2$-norm of the associated far-field pattern. For the transmission
resonance, we discover a quantitative connection between the regularity of the
transmission eigenfunction at a corner and its analytic or Fourier extension.
