  Optical surfaces represented by second-degree polynomials (quadratic or
conics) are ubiquitous in optics. We revisit the equations of the conic shapes
in the context of grazing incidence optics, gathering together the curves
commonly used in x-ray instruments and synchrotron beamlines. We present the
equations for paraboloids, ellipsoids, and hyperboloids in a common and
consistent notation. We develop the transformations from centered systems that
are commonly used to describe conics and their axes of symmetry, to local
coordinate systems centered on the off-axis mirror surfaces. The equations
presented are directly applicable to ray tracing, fabrication, and metrology
calculations. They can also be used to study misalignments, movement
tolerances, and aberrations of optical surfaces.
