ANTICAUSTICS A CORD CONSTRUCTION. AND A GENERAL FOHMULA

Transcription

1 R 96 Philips Res. Rep. 3, , 1948.; ANTICAUSTICS A CORD CONSTRUCTION. AND A GENERAL FOHMULA by C. ZWIKKER : r Summary The author describes a method for constructing antieaustics - if the caustic is given - by generalizing the well-known cord construction for the ellipse. An analytical method of calculation of the antieaustics is developed by means of the geometry of the complex plane. Résumé L'auteur dëcrit une méthode de construction d'anticaustiques - si la caustique est donnëe - en gênëralisant la construction bien connue de l'ellipsc au moyen de la corde, Une méthode analytique de calcul des anticauatiques est dëveloppëe au móyen de la gëomëtrie du plan. complexe. 1. From a constructional and geometrical point of view the design of I' f reflecting surfaces producing a certain desired caustic pattern seems to t present considerable difficulties: The aim of this article is to offer a simple I drawing-board method that will help in many cases, and to give a general -1 formula that will he of use for more accurate constructions. As this problem in mathematiéal analysis is known as the problem of the antieaustic " we 'will denote the Eeflecting surface that we are looking for by this name;. " 2. Suppose a cord is wrapped round part of a polygon c (fig. I) and the I cord kept'stretched by a pencil. Now move the pencil in such a way that I the cord is permanently kept straight. The line a that is described by the i pencil will consist of parts of ellipses, two vertices 'of the polygon acting temporarily as the foci of the ellipse. The curve a therefore shows the,.property that its!angent always makes equal angles with the two parts I I II Fig. 1.

2 ANTICAUSTICS - _A CORn CONSTRUCTION ANP A GENERAL FORMULA 467 of the cord leading to the pencil. When we let the number of sides of the.' polygon c increase this equal-angle property of the pencil curve a will remain, and in the limit We find that, if we wrap the cord round a closed contour c of arbitrary form, the pencil' ~urve a will be such that the tangent at any point will make equal angles with the two tangents drawn from this point of a to the contour C; the curve a may in this case be considered as- the envelope of a set of ellipses. Now place the cord round a closed contour c and a point S lying outside this contour, and execute the same construction, again leading to a curve a (fig. 2). Again' the equal-, ' angle property will occur. Any light ray coming from the source S win be reflected by the curve a (anticaustic) in such a way that the reflected ray will be tangent to the contour c, c therefore acting as the caustic. The length of the cord may be varied leading to different antieaustics for a single choice of source S and caustic c. s Iu practice a light source is never -a mathematical point. In fig. 3 we represent the light source by a contour s. If we wish to prevent reflected light from entering the contour c the cord is now bent round, c and s with a twist. Cases like this occur during the design of incandescent lamps and 5 Fig. 3 5,&7.59.

3 468 C. ZWIKKER fittings where one is anxious to keep light from certain lamp parts that are not allowed to reach high tempèrtures. Other variations of the cord construction may be imagined dependent on the problem at hand., 3. In order to fijid an analytical formula for the curve a we shall treat, the plane as a complex plane, fixing the place of a point with respect to a certain origin by a..vector z:'. -. z = x +jy; '(j2 " -1). The contour c is given by, the dependence of x and y on a paramete~ w, written as z = f(w). We choose the are length, s, m~asured along the curve, as parameter; s contains an arbitrary constant and we shall fix this constant so that s is the length of the cord CAS (fig. 4). Now reflect the point S with respect -to the tangent applied to the point A of the' antieaustic. The image is E. CAE is straight and of length S; E obviously desc~ibes an involute of the contour c if A is moved with the pencil as, described. dz Fig.4,.The "velocity" 'of point C along the contour c is the vector dzj; it has Unit value and is directed along the tangent. The place of point E is there-, fore represented by the vector (fig.4) ze=z-s-, and for point A, situated somewhere on the 'tangent, we have dz'. 'dz za=z-t-,, (1) where t is a real factor still to be determined. On the other hand A is situated on the bisector of the line SE -.For S as the 'origin of coordinates, the vector SE is ze'- Point M is represented by zm == zej2. The bisectör is orthogonal to ze' and point A on it may be represex;tted by the formula (2)

4 - I ANTICAUSTICS - A CORD CONSTRUCTION AND A GENERAL FORMULA 469 where Ä is~ parameter, j indicating the orth~gonal direction of MA with respect to SE. Now tand Ä follow from equating (1) and (2): r- Thus: dz dz za = Z- t - = (!+jä)ze; ZE= Z- S -. -!+jä=--, dz z-t dzz-s- For the conjugate values, denoting these by asterisks, we obtain dz* z*-t ;!-jä==--- z* -sdz* Addition of thes 5two equations eliminates ;. and gives an equation for t,. the solution of which is: S2 -zz* t = - : dz dz* 2s--z*--z since I I _: l.. dz Now zz*is the square of the absolute value of the vect~r z (CSin fig. 4); -we denote it by ri. As furthermore the denominator of t is the derivative of the numerator, one has. (3 ) Plotting the distance t on the tangent of the caustic c will give us the anticaustic, and thus the geometrical problem has been solved. Example: Let c be the unit circle and S on the circle. If we"let sstart at the point S (fig. 5), s is the are length se; the straight 'e -2 sin(s/2); hence..- distance se '. S2-4 sin 2 (s/2) t=,. 2s-2sins and this has to be plotted on the tangent at e for all values of s (fig. 6, curve a). - # - We may makes longer, for example starting the s-scale at the bottom point of the circle (fig. 5b); in this' case

5 470 C. ZWIKKER ~ ~------_.- e 2 = 2 (1 - sin s), S2-2 (1 -. sin s) t =. 2 s + 2 cos s The corresponding curve is shown in fig..6, curve b. The antieaustics have a spiral character; by mirroring an are of the spiral with respect to thc horizontal axis of the circle we may obtain symmetrical shapes. a Fig. 5 5~762 ~. Fig. 6 \ 4. The formula (3) for t is simplified if the light source S moves towar the right to infinity. For this purpose we write: d' d d lit =-ln (S2-- ( 2 ) = -In (s + e) +-In (s - e).. / Now sand e become infinitely large togethe» and as the derivative of a logarithm ten to zero for infinite value of its argument, d/ ~ln(s + eh goes to zero. In the second term we may in the limit substitute --x for e, so that t becomes:,. t= s+ x d _:_ (s + x)

6 ANTICAUST~CS - A CORD CONSTRUCTION AND A GENERAL, FORMULA 471 'and this introduced into (1) will give anticausti?s for parallel rays. We_ may shift back the origin from infinity, leaving (s + x) constant, and as there is an arbitrary constant in s the choice of the origin does not even interfere; it is only the differences in x for the various points of the caustic that influence the shape of the antieaustic.. Example: Let.again c he the unit circle and let us again take the two choices of, s that we made in the last section. We take the origin at the centre of the circle. For s starting in the horizontal axis,,x - cos s. Thus s + cos s t= sin s F~r s starting at the bottom' of the circle, ~;= sin s, (fig. 7, curve a) s + sin s t= ---- (fig. 7, curve b) 1+ cos s Fig.,7

7 472 C. ZWIKKER' The antieaustics 'are spirals that run through the point at. infinity of the horizontal axis. Again symmetrical curves may bé obtained by mirroring of a branch of the antieaustic with respect' to the horizontal axis. 5. The general case is that the caustic is known as z= f(w), where w is different from s. Take for instance the 'cla~sical case of caustics: the cardioid. This is the trajectory of a p-oint ofa circle (say, of unit radius] rolling over an equally big stationary circle. Taking the centre of the stationary circle as origin (fig. 8) the centre M of the rolling circle may be represented by z M = 2 exp (jw) and the point C of the cardioid by. z = 2 exp (jw) + exp (2jw); this formula would be much less simple if we tried to transform w into the are length s. Fig. 8 In this case it is adequate to rewrite (1) as follows: with dz dw za=z-t--' dw dcó S2 _ (22 t-= d -' (S2 _ (22). dw (4) and the difficulties enco~tered 'are not greater than before.. Let u~ work out the case of the cardioid; we take the light source S on the 'cardioid where it cuts the horizontal axis (zs = 3). The cord (! is found by (22~ (z-3) (z*~3)=20-8cosw-12cos2w. The arc-length element is: J/ dz dz* os dw = 4 cos -. dw, dw dw: 2

8 ANTICAUSTICS - A CORD CONSTRUCTIOWAND A GENERAL l 'ORMULA 473 and the total are length: ' Thus, and hence: s = J = 8 sin (w/2) S2 = 32 (1 - cos w) : S2 _: e 2 = 12 (1.; cos w )2 ; dw I-cosco t-"-=---- 2sinw This jntroduced into (4) gives z..4 = 3 e iw, and this is the circle of radius 3 (point A in fig. 8): ' Eindhoven, March 1948 \