PROJECT 1

The most striking property of parabolic mirrors is the focusing. Namely, all light rays parallel to the symmetry axis are reflected to pass a single focal point. (See, for instance, Grossman, ``Calculus", Section 8.2, pp. 550-559).

FIGURE 1

Mirrors of parabolic surfaces are, however, very difficult and expensive to produce. The most common curved mirrors are spherical mirrors, which do not exactly focus light rays at a single point. In other words, an image formed by a spherical mirror is, rigorously speaking, always slightly out of focus. This phenomenon is called spherical aberration.

FIGURE 2

The effect of spherical aberration is more significant for light rays far from the symmetry axis. One method to reduce spherical aberration is to use variable-aperture diaphragm to block off light rays far from the symmetric axis. By allowing only rays near the axis to strike the mirror, a quite satisfactory focused image can be formed by a spherical mirror.

FIGURE 3

All we said in the above can be justfied rigorously by Calculus. Let us study some related problems.

PROBLEM 1. Prove the following theorems:

(i) All light rays parallel to the symmetry axis of a parabolic mirror are reflected through a single focus;

(i) A spherical cap does not bring light rays parallel to the symmetry axis to a single focus.

PROBLEM 2.

(i) Graph the spherical mirror defined by
y=f(x)=-(4-x²)^1/2                           (1) for -2< x< 2.

(ii) Graph the parabolic mirror defined by

y=g(x)=-2 + x²/ 4 for -2< x< 2.

(iii) Combine the graphs of the above two mirrors into a single figure.

(iv) Place a variable-aperture diaphragm of appropriate size at an approriate location to block off light rays far from the y-axis so that a remaining (small) part of the spherical cap can produce pretty good focused images.

PROBLEM 3.

(i) Find the tenth-degree Taylor polynomial of f(x) defined in Equation (1).

(ii) Why can a small part of the spherical mirror do a pretty good job of concentrating vertical light rays at a single point?

PROBLEM 4.

(i) Graph another parabolic mirror defined by
y=h(x)= 5x² for -2< x< 2.

(ii) Find a (small) spherical mirror, that does ``almost" the same job of focusing vertical light rays as the parabolic mirror y=h(x) in Part (i) does.

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