- Introduction
- Observations With a Concave Mirror
- Ray Tracing
- Paraxial Imaging
- Concave Mirrors and Projective Geometry
- References

The concave spherical mirror is the simplest image-forming optical system, and it has many applications. It is an example of a *catoptric* system, which involves reflection. The law of reflection is particulary simple, so the geometrical optics of the concave mirror is easy. Since the light does not enter any medium other than the air, there is no problem with chromatic aberration, and colors do not appear. The mirror produces a caustic surface, since it does not bring rays of finite height to a point focus. In this article, I shall review the paraxial optics of the concave and convex mirrors, and then treat the problem with projective geometry, a rather unusual extension of the theory, that is of some mathematical interest. Concave mirrors produce both real and virtual images, which are magnified or reduced. Convex mirrors produce only reduced virtual images, so they are of less interest. The reader is urged to obtain a concave mirror and experiment with it, both in sunlight and artificial light.

Aspheric mirrors are relatively common. The most important is the parabolic mirror, which eliminates spherical aberration for collimated on-axis rays. They are used in reflecting telescopes and searchlights, in which the entering or leaving light is always of this type. For rays not parallel to the axis, the parabolic mirror exhibits very strong astigmatism and coma. Glass correcting plates can correct the spherical aberration of a spherical mirror, and this is often a better solution than the use of a parabolic mirror, since it is very difficult to produce a good parabolic surface.

I bought a concave shaving mirror about 145 mm in diameter, and with a focal length of about 180 mm. It is a polished metal mirror, and the optical quality is not high, but sufficient for its intended use. The first thing to do is to examine images formed by the mirror. If you hold it up close to your face, an erect, magnified virtual image is seen that appears to be behind the mirror. This is how the mirror is intended to be used, and the modest magnification of about 2 is appropriate. Now prop the mirror up and stand about a metre away from it. You'll see an inverted, shrunk real image of your face. Now bring your hand up from below a bit closer to the mirror, and an image of a hand will come down from above. Your stereo depth perception should tell you that the image of the hand is in front of the mirror. A position of your hand can be found in which the image is the same size, and at the same position. By moving your head slightly to the right and left, you can verify that there is no parallax between hand and image at this position. The image is real because it is on this side of the mirror and light rays pass through it. Your hand is now at the centre of curvature of the mirror. This is the best way to estimate the focal length of the mirror, which is half its radius.

Next, take the mirror out into the sunlight and direct it toward the sun with one hand while you hold a white paper screen in the other. The light can be dazzling, so neutral sunglasses may be used. The lack of perfection in the mirror's form will be quite evident in these observations. Let the light from the mirror fall on the screen, but as close to normal incidence as possible. Move the screen to find the focal point of the mirror. This will not be a point, but a rather confused tangle of light. I have not been able to ignite a piece of paper, but the focused sunlight is quite warm on the skin. When the screen is closer to the mirror than the focal point, a bright ring of light will be seen that is a cross-section of a *caustic* (i.e., "burning") surface. There is light inside the caustic, but it is not nearly as strong as the bright circle. If the screen is held beyond the focal point, a disc of apparently uniform intensity is seen, surrounded by a narrow bright ring. This bright ring is the result of diffraction at the edge of the mirror, though no fringes are evident except the first bright one.

Ray tracing for a spherical mirror is very easy. The geometry for a concave mirror is shown at the left, and the formulas for finding the reflection angle i in terms of the ray height h, and the point at which a ray of angle i crosses the axis; the distance from this point to the vertex V is x. For small h, we can set sin i = i, so that h = ri and x = r/2. Since x is independent of h in this approximation, the point at x = r/2, F, is the focal point of the mirror. The next approximation in i gives x = (r/2)(1 - i^{2}/2). It is instructive to make an accurate plot of the rays for i = 10°, 15°, 20°, 25° and 30°. This plot will make evident the *caustic surface* on which neighboring rays intersect. That is, near the caustic, two rays interfere to produce increased intensity. The intensity near a caustic was first studied by Airy, and his results are embodied in the theory of the rainbow, for which the reader should consult The Rainbow.

The caustic is one of the interesting features of the concave mirror. This curve can often be seen on the surface of coffee in a cylindrical coffee cup, where the cup is the reflecting surface, and in many similar arrangements. The refraction in certain bathroom windows also shows caustics, in the light of distant point sources. The increase of intensity at a caustic, and the appearance of interference fringes parallel to it, are its principal features.

The fact that x was independent of h to a good approximation for small h is a remarkable property. It means that a finite bundle of rays, enough to represent a reasonable intensity, which come from a point will again be focused to a point by the mirror. That is, the mirror is capable of *stigmatic imaging*. The point from which the rays begin is called the *object*, and the point where they again come together is called the *image*. This is a one-to-one *mapping* from an *object space* onto an *image space*. When the rays are all close to a line called the *optical axis*, this mapping is *linear*. If the distance normal to the axis between two neighboring image points is y', and the similar distance for the object points is y, then the (transverse) *magnification* is the ratio y'/y. All these statements can be proved by considering *paraxial* rays, which do not depart far from the axis and are at small inclinations such that the angle of inclination can be set equal to its sine, and its cosine equal to unity.

What will be said here applies not only to concave mirrors, but also to convex mirrors and thin lenses, and, with proper definitions and conventions, to general paraxial systems. If this material is not familiar to you, then convex mirrors are an excellent example to begin with. The object point on the axis from which light is rendered parallel by the mirror is the *primary focal point* F. The image point on the axis to which parallel incident light is focused is the *secondary focal point* F'. We have already determined that these points are the same, and are a distance r/2 from the mirror. The location of the mirror itself is the vertex V, and its centre of curvature is at C. The mirror is represented by a plane normal to the axis at V. A ray that passes through V is reflected there so that its slope is multiplied by -1. A ray that passes through C is reflected parallel to itself at the mirror.The distance between these points is the radius of the mirror, r.

The method of paraxial ray tracing is shown at the right. The lines that are drawn suggest the actual rays, but the transverse scale of the drawing is greatly expanded. This is the reason why the concave mirror is represented as a straight line, even though its centre of curvature C seems nearby. The object point P is imaged at the image point Q. The object height is represented by y, and the image height by -y'. The minus sign means that y' is negative as shown. Three rays are shown that can easily be drawn, one parallel to the axis and reflected through F, another passing through F and reflected parallel to the axis, and the third through the centre of curvature, which is reflected onto itself. A fourth ray can be drawn to V, which is reflected at equal angles. Only two rays are necessary to locate the image of a given object point, but the other two provide checks. A thin lens has all these rays except for the one through the centre of curvature.

The object distance is represented by s, the image distance by s', and the focal length by f = r/2. Similar triangles give -M = -y'/y = (s' - f)/f = f/(s - f) = s'/s, the last from the ray through V that is not shown. From these expressions, we can get, for example, (s - f)(s' - f) = f^{2}. From this, ss' - f(s + s') = 0, or 1/f = 1/s + 1/s', which is the Gaussian lens equation. If we set x = s - f and x' = s' - f', then we get the Newtonian lens equations xx' = f^{2}, M = -x'/f = -f/x. For the convex mirror, x is measured to the left of F and x' to the right.

The reader is encouraged to draw the ray constructions for object points at C, between C and F, and to the right of F, and to verify using the Gaussian and Newtonian lens formulas. The results should be correlated with the observations of images in the mirrors.

The paraxial ray construction for a convex mirror is shown at the left. The focal point F is now to the right of the vertex, and the focal length f = -r/2. Using this focal length, the Gaussian formula 1/s + 1/s' = 1/f and the Newtonian formula xx' = f^{2} still hold, with the signs of the quantities properly interpreted. The same four rays are available for the construction. Note that for any object distance, the virtual image is between F and V and reduced in height. As the object approaches the mirror, so does the image. The convex mirror has its uses, but it is far less versatile than the concave mirror that produces a real image. Concave mirrors are often used as right-side rear vision mirrors in cars because of the large field of view. However, the reduction in image size is a serious hazard. The image of the sun in a convex mirror is virtual, and very small, so it can be used as a point source.

A plot of image position s' as a function of object position s is shown at the right. The Gaussian lens formula can be expressed as s' = s/(s/f - 1), showing that s' is a rational linear function of s, a ratio of linear expressions. The curve is an equilateral hyperbola. If the axes are shifted by a distance f horizontally and vertically, the hyperbola is symmetric with respect to the axes, and we have the Newtonian lens formula xx' = f^{2}. Notice that the object distance s can be negative. This corresponds to a virtual object, to which the rays converge, but strike the mirror before reaching the object. The image and object can be at any point in image or object space. This same plot can be used for the thin lens. For the concave mirror, f = r/2.

The relation between paraxial object and image for a concave mirror can also be expressed in projective geometry. Although the reader is undoubtedly familiar with the paraxial ray constructions that have just been explained, the use of projective geometry will probably be new. We will restrict ourselves to the principles here, explained as simply as possible. This is, of course, interesting geometric mathematics more than an aid to optical design.

The use of ratios to express the division of line segments is shown at the left. The top line segment AB is divided internally at C in the ratio AC/BC. Line segments are considered positive in the direction from left to right, negative in the opposite direction. In this case, BC = -CB and is a negative number, while AC is positive, so the ratio is negative or zero. The middle line segment AB is divided externally at D in the ratio AD/BD, which is positive or zero. Considering all points of division, internal and external, the ratio can vary from +∞ to -∞. The bottom line segment AB is divided both internally and externally by points C and D. These points could both be internal or external, but one of each is shown in the diagram. The division is expressed by the *double ratio* (ABCD) = (AC/BC)(BD/AD).

From the definition of the double ratio, it is clear that (ABCD) = (CDAB) = (BADC) = (DCBA), and that (BACD) = (ABDC) = 1/(ABCD). The diagram at the right shows a line x with points of division A, B, C and D. The point O is the *centre of projection*, and rays from it define the points of division. We consider the ray y defining point D to be variable, and to rotate about O, making a full circle. This changes the ratio BD/AD, while AC/BC remains unchanged. If ray y begins at B, then BD = 0, and (ABCD) = 0. As y rotates anticlockwise, BD/AD remains positive, so that (ABCD) is increasingly negative. When y is just about horizontal, in position y', BD/AD is unity. As it passes horizontal, D moves to -∞, but BD/AD is still positive, since both numerator and denominator are negative. When it nears A, AD becomes small and negative, so (ABCD) is near -∞. As it passes A, the sign of AD changes but BD remains the same, so (ABCD) flips from -∞ to +∞. When D coincides with C, (ABCD) = (AC/BC)(BC/AC) = +1. As D moves toward B, (ABCD) decreases from +1 to 0. We have been through a complete cycle of values of (ABCD), which should give a better feeling for its meaning. To any value of this ratio corresponds a certain position of D, the other points remaining fixed.

We now introduce the projective property shown in the diagram at the left. Rays from a centre O divide line x at points A, B, C, D in the double ratio (ABCD). These rays also divide line y from a second point of projection G in the same double ratio. The proof of this is somewhat intricate and not very illuminating, so I shall omit it here. The truth of this proposition is not hard to believe, so I hope any misgivings will be calmed.

There is a very special division of a line that has the property (ABCD) = -1, called a *harmonic division*. We see from the definition of double ratio that it means that AD/BD = -AC/BC, or that the line is divided in the same ratio internally and externally. If we let the points of division of the optical axis be the object, center of curvature, image and vertex, then this is the condition relating the object and image positions, as we shall prove.

The construction of a harmonic division is shown at the right. We begin with points A, C, D on a line, and want to find the point B so that (ABCD) = -1. To do this, choose any point Q and join QA, QC and QD. The line QA is only needed for the proof, not to find D. From C draw CR in any convenient direction, then join AR, locating P on QC. Now draw DP, intersecting CR at S. Now draw QS and extend it to D. Now (ABCD) = -1. To demonstrate this, with Q as a projection centre, (ABCD) = (AOPR) by the projective property. If we now take S as the projection centre, projection on the same two lines gives (AOPR) = (ABDC), since D and C are interchanged in this case, since S lies between the lines. We have already shown that (ABDC) = 1/(ABCD) from the definition of the double ratio. This means that (ABCD)^{2} = 1. We have shown that (ABCD) = 1 only if D and C coincide, which they do not. The only other possibility is (ABCD) = -1, so A, C, B, D divide the line harmonically, which we had to prove.

To relate all this to the concave mirror, let A = object, C = centre of curvature, B = image and D = vertex. Then, AD = s, BD = s', AC = s - r, BC = s' - r, and (ABCD) = (AC/BC)(BD/AD) = -1, or (s - r)/(r - s') = s/s', from which ss' - rs' = rs - ss', and 1/s + 1/s' = 2/r, the Gaussian lens formula. This shows that the four points mentioned divide the optic axis harmonically, and we can use the construction to find the relation between object and image.

Suppose we are given points C and D, the center of curvature and the vertex, and the location of the image at some point B between them. To find the object, proceed as follows. Choose some point Q and draw QC, QB and QD. From C, draw a line intersecting QD at some point R, and QB at S. Now draw DS and extend it to cut QC at point P. The line RP, extended, intersects the axis at the object point A. This is an easy construction that can be carried out with only a straightedge.

Other cases can be solved in a similar way. For example, if object, image and vertex are known, and the centre of curvature is to be found, proceed as follows. From a convenient point T draw TA and TB. From D, draw a line intersecting TA at point R and TB at point P. Now join BR and AP, intersecting at S. Then TS extended cuts the axis at C. The result can be checked by finding the image position B now that A, C and D are known. The general rule is to draw rays from an external point to the two points on either side of the unknown, and from the third point cutting these rays. The construction is based on the quadrilateral PQRS with no three points on the same line, as in the figure for the proof.

J. P. C. Southall, *Mirrors, Prisms and Lenses*, 3rd ed. (New York: Dover, 1964). The projective geometry method is on pp. 156-168.

F. A. Jenkins and H. E. White, *Fundamentals of Optics*, 2nd ed. (New York: McGraw-Hill, 1950). There are later editions, which are not a great improvement over the classic second edition. Chapter 6 treats spherical mirrors. Any good undergraduate optics text will do as well, though many modern texts do not do a good job with geometrical optics.

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Composed by J. B. Calvert

Created 9 August 2003

Last revised 10 August 2003