What are the Equations For?

These 2 equations (Equation 1 and Equation 2) provide an alternative method for describing and understanding images produced in curved mirrors. They should not be thought of as just "places to plug in numbers", but as very succinct descriptions of the way part of the world works. Equation 2 (D_i = f²/D_o), for instance, tells you that the distance that the image is located from the focus of the mirror (D_i) depends only on the object distance (D_o) and the focal length of the mirror (f) - since only these three quantities are contained in the equation. The size of the object (H_o) has no effect on the location of the image. (It does affect the size of the image - see Equation 1...)

Graph ov Do vs di Equation 2 has a lot to say about the relationship between object distance and image distance, too. For a particular mirror, the focal length, f, is fixed. This means that if D_o (object distance) is large, then D_i (image distance) is small, and vice versa. Also, D_o and D_imust always have the same sign - either both positive or both negative. Physically, this means that the object is placed farther from the mirror than the focus, then the image is located beyond the focus also (both positive), and if the object is between the focus and the mirror, then the image is located on this side of the focus, too (both negative). This is precisely what we found from the ray diagrams.

Also, notice from the graph of equation 2 (at right) that if the object is located far from the focus, (D_o > 20 cm, say), a relatively large movement of the object will produce only a very small movement of the image. When the object moves from D_o = 40 cm to D_o = 20 cm, the image only moves a slight distance away from the focus!

last update October 11, 1999 by JL Stanbrough