﻿ Mirror

## Ray tracing with spherical mirrors in the paraxial approximation

The focal length f and the radius of curvature R = 2f are positive for a concave mirror and negative for a convex mirror.  The perpendicular distance of the object from the mirror surface is xo, and the perpendicular distance of the image from this surface is  xi.  Distances in front of the mirror are positive and distances behind the mirror are negative negative.
In the paraxial approximation  xo and xi satisfy the mirror equation, 1/xo + 1/xi = 1/f = 2/R.  The magnification M is the ratio of the height of the image hi to the height of the object ho.  M = hi/ho= -xi /xo.  If the magnification is negative, the image is inverted.

We can determine the positions and sizes of images of points formed by spherical mirrors geometrically by drawing ray diagrams.  Only two incident rays and their reflections must be drawn.  The intersection of the two reflected rays, or, for divergent rays, the intersection of their backward extensions, marks the position of the image of your chosen point on the object.  Choose two or more of the rays listed below.

• Draw the object in front of the mirror surface.
Draw an incident ray parallel to the optical axis from a point on the object to the mirror line, and a reflected ray from the mirror line through f.
(The reflected ray, or an extension of the reflected ray must pass through f.)
• Draw an incident ray through f, and a reflected ray parallel to the optical axis.
(The incident ray, or an extension of the incident ray must pass through f.  This can be done if the object is not located a distance f in front of the mirror.)
• Draw a ray through the center of curvature R.  This ray reflects back onto itself.
(The ray or its extension must pass through R.  This can be done if the object is not located a distance r in front of the mirror.)
• If you have a protractor, draw a line from the object point to the point where optical axis intersects the mirror.  The reflected ray makes the same angle with the optical axis as the incident ray.