  We initiate a systematic study of one-loop integrals by investigating the
connection between their singularity structures and geometric configurations in
the projective space associated to their Feynman parametrization. We analyze
these integrals by two recursive methods, which leads to two independent
algebraic algorithms that determine the symbols of any one-loop integrals in
arbitrary spacetime dimensions. The discontinuities of Feynman diagrams are
shown to arise from taking certain "spherical contour" residues in Feynman
parameter space, which is geometrically interpreted as a projection of the
quadric surface (associated to the Symanzik polynomial at one loop) through
faces of the integration region (which is a simplex). This geometry also leads
to a manifestly Lorentz-invariant understanding for perturbative unitarity at
one loop.
