  Singularities of the joint density of states (JDOS) and Fourier-transformed
local density of states (FT-LDOS) correspond to the hot spots in quasiparticle
interference patterns. In this paper the singularity of JDOS is analyzed
exactly, with three types of singularities being classified. In particular, the
third type of singularities are found exactly to be envelopes of the contours
of constant energy. Remarkably, we show that JDOS and FT-LDOS have the same
singular points. Approaching to the singular points, both quantities diverge
complementarily in an inverse-square-root manner if the joint curvature is
nonzero. The relative magnitude of divergence is governed by the joint
curvature as well as the product of the quasiparticle velocities. If the joint
curvature of certain singularity is zero, the divergence has a higher order
than -1/2.

Superconductivity;Mesoscale and Nanoscale Physics;Strongly Correlated Electrons