Comment on the Caustic Curve of a Parabola

Henry J. Stalzer, Jr. Grumman Aircraft Engineering Corporation, Bethpage, New York. Received 14 January 1965. In a recent paper,1 Scarborough derived the parametric equa­

tions of the caustic curve for the parabola y2 = 20χ. He stated that the implicit function relating χ and y seemed impossible to obtain.

The purpose of this letter is to give the parametric equations of the caustic curve for the more general parabola y2 = 4ax, and to eliminate the parameter yk/a, thus obtaining an implicit function between χ and y.

For a ray incident on the parabola y2 = 4αx at the point (xk, yk), at an angle θ with the χ axis, Scarborough1 derived the equa­tion of the reflected ray

where m = tan0. The caustic curve is given by the simultaneous solution of Eq.

(1) with

Fig. 1. Coordinate systems.

Fig. 2. Sketch of caustic curve.

Solving, we get

and

These are the parametric equations for the caustic curve, the parameter being yk/a. From these equations we see that

We can rewrite Eq. (3) as

Squaring Eq. (6) we get

and we see that the parameter yk/a appears only as (yk/a)2. Sub­stituting (yk/a)2 = (4/3) [(mx + y)/am] from Eq. (5), we get an implicit function for the caustic curve

Equation (8) can be put into a simpler form by a rotation and translation of the coordinate axis. The rotation is given by the-transformation

where θ is the angle which the incident rays make with the axis of the parabola (χ axis). The equation of the caustic curve in the-primed coordinate system is then

September 1965 / Vol. 4, No. 9 / APPLIED OPTICS 1205.

Fig. 3. Comparison of Scarborough's approximation and the exact caustic.

The translation is given by

which reduces the caustic curve to the form

where b = 9a sinθ. The coordinate systems are shown in Fig. 1. Some general properties of the caustic curve can be obtained

from Eq. (9). The curve is symmetrical about χ" = 0, which in the original coordinate system is the line χ = a + my. The en­tire curve lies above (below) the line y" = 0 for 0 greater than (less than) zero. The line y' = 0 in the original coordinate system is y = - mx. The slope of the curve is

Thus the derivative is zero at y" = 0 and infinite at y" = b/3. From this information the curve can be sketched as shown in Fig. 2.

Scarborough treats the case a = 5, m = tanl00 and finds an approximate equation for the caustic curve,

This is a circle with center (5.118, 0.527) and radius 1.411. The approximate equation and the exact equation [Eq. (8)] are shown together in Fig. 3 for comparison. Scarborough's equation is quite good for that part of the curve below the y axis.

Reference 1. J. B. Scarborough, Appl. Opt. 3, 1445 (1964).

1206 APPLIED OPTICS / Vol. 4, No. 9 / September 1965