### Ray Tracing

Often it will be useful to determine the approximate position of an image, given the position of the object and the focal length in a lens or mirror system without resorting to the lens equation. We can do this by drawing diagrams and mapping out the path of the light rays. This process is known as analytical ray tracing. The basic strategy is to select a significant point on the object (like the top) and to draw several principal rays from that point. For a mirror, the three principle rays, shown in , are: i) parallel to the axis, returning through the focal point; ii) to the midpoint of the mirror, reflecting at an equal angle on the opposite side to the central axis; and iii) through the center of the sphere of which the mirror is a part, returning along the same path.

The principal rays for a lens are similar: i) parallel to the central axis, refracting through the focal point; ii) a straight ray through the center of the lens; and iii) through the focal point on the near side, refracting parallel to the axis. At least two principal rays must be drawn from the object; the point where the principal rays cross (or appear to cross, for a virtual image) is the location of the image. To determine whether an object is magnified or diminished, it would be necessary to choose one other point (such as the base) and compare this distance between the two points in the image to their positions in the object. To find the locations of virtual images it is necessary to trace the light rays backwards behind the mirror or lens.

Ray tracing is especially useful when complicated systems of mirrors and/or lenses need to be analyzed. Ray tracing can give a rough but quick idea of how the system will behave. For example, it can fairly quickly be determined that a single concave lens will always produce virtual, diminished, upright images, irrespective of the position of the object. However, for a convex lens, the location of the image depends on the location of the object. Recall that real objects and real images have so > 0, or sI > 0, while these distances are negative for virtual objects and images (virtual objects can arise when the image for one lens becomes the object for another in a lens system). f > 0 corresponds to converging lenses or mirrors and f < 0 corresponds to diverging lenses or mirrors. Positive yo or yi correspond to upright objects and images respectively. A negative magnification corresponds to an inverted image.