Elliptical reflectors are widely used in illumination engineering as a means to concentrate the light of a source in the secondary focal plane. To collect as much light from the source as possible, the reflector must cover a wide angular range of the emitted radiation, hence forward and backward reflections on the reflector must be discussed and conditions for paraxial optics do not apply. We develop a framework of relations that describe the image of an extended source in the secondary focal plane of the ellipsoid. After deriving basic relations between object and image spaces, the propagation of rays depending of their origin and on their direction is investigated. As a result, the function and the aberrations of the reflector can be expressed in terms of an angular dependent magnification for longitudinal and lateral coordinates. With that, one cannot not only calculate luminance and illuminance distributions in the secondary focal plane in a straightforward manner, but can easily understand their properties and their dependence on ellipse parameters. On the basis of the developed relations, implications for the design of "elliptical" short arc high-intensity-discharge (HID) lamps for video projection are discussed.