The Law of Reflection

and Curved Mirrors

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Flat-to-Curved Mirror TransistionWe have already established that the Law of Reflection (angle of reflection = angle of incidence) applies to plane mirrors. If you would place several plane mirrors into a beam of light that contained parallel rays, you would find it relatively easy to arrange the flat mirrors so that they would reflect their portion of the beam through a common spot. (See the diagram at left.) If you were mathematically inclined, you would notice that you had arranged the mirrors to approximate a curve called a parabola. (If you were mathematically inclined, it would be an interesting exercise to prove that this curve is a parabola...)

As the number of plane mirrors increases in this array, the mirrors approximate the parabola more and more closely, and the beam focuses into a smaller and smaller area. For a perfectly-smooth parabolic mirror, the beam will be reflected through a single point, called the focus of the mirror.

Parabola vs. Circle DiagramNotice that the cross-section of this parabolic mirror is almost perfectly circular near the center (the axis of the mirror) but "flattens out" far from the center. This means that a parabolic mirror can be approximated by a circular mirror as long as the objects reflected in the mirror are small compared to the size of the mirror. This is an important observation, since spherical (a 3D circle) mirrors are easier and cheaper to construct that parabolic mirrors. It is also easier to draw a circle than a parabola for a ray diagram!

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last update September 20, 1999 by JL Stanbrough