vertices are the points where each optical surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the cardinal points must be known with respect to the vertices to describe the physical system. In anatomy, the surface vertices of the eye's lens are called the anterior and posterior poles of the lens. Modeling optical systems as mathematical transformations In geometrical optics for each ray entering an optical system a single, unique, ray exits. In mathematical terms, the optical system performs a transformation that maps every object ray to an image ray. The object ray and its associated image ray are said to be conjugate to each other. This term also applies to corresponding pairs of object and image points and planes. The object and image rays and points are considered to be in two distinct optical spaces, object space and image space; additional intermediate optical spaces may be used as well. Rotationally symmetric optical systems; Optical axis, axial points, and meridional planes An optical system is rotationally symmetric if its imaging properties are unchanged by any rotation about some axis. This (unique) axis of rotational symmetry is the optical axis of the system. Optical systems can be folded using plane mirrors; the system is still considered to be rotationally symmetric if it possesses rotational symmetry when unfolded. Any point on the optical axis (in any space) is an axial point. Rotational symmetry greatly simplifies the analysis of optical systems, which otherwise must be analyzed in three dimensions. Rotational symmetry allows the system to be analyzed by considering only rays confined to a single transverse plane containing the optical axis. Such a plane is called a meridional plane; it is a cross-section through the system. Ideal, rotationally symmetric, optical imaging system An ideal, rotationally symmetric, optical imaging system must meet three criteria: All rays "originating" from any object point converge to a single image point (Imaging is stigmatic). Object planes perpendicular to the optical axis are conjugate to image planes perpendicular to the axis. The image of an object confined to a plane normal to the axis is geometrically similar to the object. In some optical systems imaging is stigmatic for one or perhaps a few object points, but to be an ideal system imaging must be stigmatic for every object point. Unlike