# Cardinal Point (optics) - Modeling Optical Systems As Mathematical Transformations - Ideal, Rotationally Symmetric, Optical Imaging System

Ideal, Rotationally Symmetric, Optical Imaging System

An ideal, rotationally symmetric, optical imaging system must meet three criteria:

1. All rays "originating" from any object point converge to a single image point (Imaging is stigmatic).
2. Object planes perpendicular to the optical axis are conjugate to image planes perpendicular to the axis.
3. 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. The word originating is in quotes because in geometrical optics the term ray is not used in quite the same way as mathematicians typically use the term in geometry. In mathematics a line extends infinitely in both directions but a ray extends infinitely in one direction with a definite terminus in the other direction. In optics a ray extends infinitely in both directions so it is analogous to a geometrical line. In stigmatic imaging an object ray intersecting any specific point, A, in object space must be conjugate to an image ray intersecting the conjugate point A' in image space. A consequence is that every point on an object ray is necessarily conjugate to some point on the conjugate image ray.

Geometrical similarity implies the image is a scale model of the object. There is no restriction on the image's orientation. The image may be inverted or otherwise rotated with respect to the object.