OPTICAL COLLINEATION, INDEPENDENT OF METRICS

BY LUDWIK SILBERSTEIN'-

The purpose of the present paper, of which an account was sometimeago communicated through Messrs. Adam Hilger, Ltd., to a Florencemonthly,2 is to show how the optical collineation of M6bius-Abbe,familiar from textbooks on geometrical optics, can be set up and de-veloped without any reference to metrical concepts such as length orangle, and therefore also parallelism or orthogonality. In fact, althoughcommonly treated with the aid of Cartesian co-ordinates and other con-cepts of Euclidean geometry, the optical collineation may be consideredas a purely projective transformation of the object space into the imagespace, and its properties should, therefore, be independent of the usualmeasuring processes and of the particular metrical nature of the con-templated space, Euclidean, Riemannian (elliptic) or Lobatchevskyan(hyperbolic). It can be based entirely on the axioms of projectivegeometry, and when thus treated, may be designated as the non-metrical generalization of the familiar optical transformation.

To follow the investigation, scarcely more than the elementary knowl-edge of non-metrical vector addition is needed, as explained in theauthor's Projective Vector Algebra (London, Bell, 1919). To avoiddigressions, the reader may be referred particularly to the few sectionswhich treat of vector addition with respect to a conventional referenceframe, "the T-plane" which may be any plane, in the case of three, and"the T-line" (any straight line) in the case of two dimensions.

It is enough to consider in detail the relations in two dimensions, inany plane passing through the optical axis; the generalization to threedimensions being perfectly obvious.

Thus, let Fi, F2, be two fixed points and pi, P2, two fixed straight lineswhich meet in a point, Tp, of the T-line itself, the latter being any con-ventionallychosenstraightline. The points F,, F2, will thereafter playthe part of the foci, and the lines PI, P2, that of the principal lines of thesystem. Let the join Fi, F2, to be called axis, intersect the T-line in TF,as shown in the figure. The two points F1 , F2, and the two lines pi,

P2, together with the conventional T-line, will henceforth be our fixedelements or data.

I Paper read October 26, 1923, at the Cleveland annual meeting of the Optical Society of

America.2 Rivista d'Ottica e Meccanica, 1; Sept., 1919.

675

LUDWIK SILBERSTEIN

Now, given any point R of the plane as object, let us establish thefollowing purely projective correspondence leading to a definite pointR', the image of R--

Draw RF1 crossing pl in A,, and RTF, crossing P2 in B2; next drawthe lines B2F2 and A 1TF. Their cross R' will be considered as the imageof R.

This one-to-one correspondence of points can be characterized brieflyin terms of the involved graphical, i.e. straight-edge operations bywriting RF -pi for the cross of the lines RF1 and pi, etc. Thus,

R'=[(RFl.pl)TF] [(RTF.P2 )F2 ]. (1)If the space be Euclidean, if T were "the line at infinity," and pi, P2

perpendicular to F1F2, this construction would at once reduce to theusual optical collineation, well known from treatises on geometricaloptics. It is precisely a non-metrical generalization of the latter andrequires, therefore, the use of a straight-edge only. The figure corre-sponds to the case of what is technically called a collective dioptricalsystem. The remaining optical systems will be covered by differentarrangements of the four elements F, F2, Pi, P2. The reasoning andthe construction will always be the same, viz. as in (1).

The important thing, however, is that also the usual formulae of theoptical collineation continue to hold for this generalized correspondenceof objects and images, provided all lengths and heights, in fine, alldistances, instead of being measured by a rigid rod, in inches orcentimeters, are expressed in projective steps (differences of projectiveco-ordinates) relative to the conventional T-line and wholly independentof the use of rigid bodies.

To see this, it is enough to put the graphical construction (1) in termsof non-metrical vector algebra. Let, therefore, R, F, and so on, benon-metrical vectors, whose end-points are R, F, etc. respectively, andwhose common origin is any point 0 chosen arbitrarily on the axis (notshown on the figure). The choice of 0 will, of course, have no influenceupon our results. Thus, Al being the cross of RF1 and p, and P thecross of p and the axis, we have

A= aR+ (1-a)F = P1+0p, (2)where p is a vector along the principal line pi, while a and fl denote scalaror ordinary numbers to be determined presently. The equationexpresses simply that the point A is collinear with R, F and that itlies upon the principal line p. Notice that since pi and 2 meet on theT-line, every vector along P2, as well as along p, will be of the form p,

676 [J.O.S.A. & R.S.I., 8

OPTICAL COLLINEATION

where X is some scalar. (See Projective Vector Algebra'.) Next, leta be a conventionally chosen standard vector along the axis FF 2.Then, since a and p are linearly independent (their termini TF and Tpare, by assumption, distinct points), any vector can be written

R=xa+yp, (3)so that x, y will be the projective co-ordinates of the point R, the end-point of R. Substitute (3) in (2) and write

F, =opa, P1=i ra,

Graphical definition of projective optical collincationt.

and similarly F 2 = 2a, P 2 = r2 a; then (2) will giveax+(1-a)oi=7,rl, ay=,

whence a and are readily determined and convert (2) into

A1 = P+ VIl P. (4)X- pi1

where f = 7r- , the number of projective steps leading from the focusF1 to the principal point pi. Similarly, the vector whose end-point isB2 ,

B 2 =P 2 +YP=R+(7r 2 -x)a, (5)and finally, noticing that every vector along ATF is of the form yua,

R'=Ai+uai=vF 2 + (1-v)B 2 ,where is a new scalar. Substituting here Al and B2 from (4) and (5),and writing f2 = 2 - r2, it will be found at once that

R/= [(x - 1)52 -fif 2 ]a+fiyp (6)x -This is the required formula expressing the correspondence between theobject and the image point.

May, 1924] 677

LUDWIK SILBERSTEIN

If we write R'= x'a+y'p, so that x', y' will be the projective co-

ordinates of R', referred as before to the axis and either principal line,the last formula splits into

fif2 , _fy (7)X=02-X - (i, X (P1. 7

or, putting =x- p, ('= P 2 -x', so that I, i' will be the axial projec-tive co-ordinates of the object and the image counted from the foci Fi,and F2 respectively,

t'=fif2=const., } (8)Y'1Y =fil = '/f2. }8

These equations are exactly of the form of those known from Abbe'stheory. The constants, f, = 7r, - oi and f2= 9P2 - r2 are the non-metricalgeneralizations of the familiar "focal distances," counted from the prin-cipal points P1, P2 . The only difference is that our fi, f2, as well as t

and so on, are not measured in rigid steps, such as cms or inches, butin projective unit steps or in staudtians, so to call the steps of vonStaudt's projective scale, whose construction is widely known. 3 The

formulae of optical collineation just derived will become identical withthe usual ones with respect to their contents, and not merely their form,Frovided space is Euclidean, and the T-line or, in case of three dimen-

sions, the T-plane is relegated to infinity. It may be interesting to notethat for practical purposes the part of the T-plane is often taken overwith sufficient accuracy by a distant screen or wall of the laboratory.

The special case of a telescopic system, which in the usual treatment is

made to correspond to infinite fl and f2 but bearing a finite ratio toeach other, can be defined more simply by stating that to every finitecouple x, y corresponds a finite couple x', y', that is to say, in the present

treatment, that F1 and F2 coalesce into a single point of the T-line.In short symbols, F=F 2 TF. Since in this case Al, B2, TF are col-

linear, the construction defined by the figure or by the formula (1)fails, as is also the case with the usual construction. For R', being thecross of ATF with B2TF, becomes an undetermined point of the line

RTF. In this case, the image of any point R can be found analyticallyby the familiar artifice. If 4o, yo and 4O'yo' be an arbitrary pair of cor-responding points, substitute in (8)

Readers unacquainted with it may consult Non-Eitclidea Geowelry, by J. Coolidge, or

the booklet already quoted.

[J-0-S.A. & R.S.I., 8678

OPTICAL COLLINEATION

and let F, and F2 tend to TF, i.e. fi, f2 to o . Then, denoting the con-stants yo'/yo andf 2/f' by m0 and a respectively, the result will be4

X' -amo2 X, y'= moy, (8a)which are the familiar formulae for a telescopic system.

The special case of coinciding principal points, in which also pi, p2 arefused into one line, offers no particular interest. In this case A, andB2 become collinear with T,. This, however, does not give rise to anynew questions. Turning once more to the general case, represented bythe figure or by the formulae (8), notice that for x = or = 0 we have

= oo, that is to say, the image of the focus F1 is on the T-line at TF.Similarly if the object be placed at F2, the image will be at TF. More-over, is zero for every point of the line FIT,, and similarly '=0 allalong F2T,, so that F1T, and F2T, play the part of focal lines (and inthe case of three dimensions, focal planes); their image is the T-lineitself. The two focal lines and the two principal lines all intersect onthe T-line at Tp; this is the non-metrical generalization of their parallel-ism in the usual treatment.

Next, let the object R be on the principal line pi. Then the crossA, =RFi Pl will be R itself and the image R' = (AlTF) (B2F2) will be-come

R'= (RTp) (B2F2)=B2,i.e. a point on p2. Thus, the image of the first principal line (plane) isthe second principal line (plane), a familiar result. In fine, all the funda-mental properties reappear in this generalized optical collineation.

The second of (8) is the non-metrical formula for the lateral magni-fication, y'/y =fi/. It has an intrinsic meaning, for although the stand-ard vectors a, p, were arbitrary, yet y'/y is independent of the choiceof the unit, this being common to y' and y; similarly for f,/ . The"magnification" stands here, of course, for the ratio of the numbers ofstaudtians contained in y' and in y. As long as the T-line (T-plane)is kept fixed, this ratio is fixed for every point y. The same remarkapplies to the longitudinal or axial magnification which is, by (8),

d~'Id = -fif2l42-To close the paper, a few words may be said about. the non-metrical

generalization of the so-called optical center of a system. This is, in theordinary treatment, a point C, such that the image 1' of every ray

4 The simplest way of passing to the limit seems to be the following one. From the secondof (8) we have t0 /-O=amo2 and 0o0o'=fjf2 = a 2

tF. Thus the first of (8) gives X'+amo 2 X+XX'/1 0 = 0, and since o tends (together with f, A2) to infinity, X'+amno 2

X =0, and so on.

May, 124] 679

LUDWIK SILBERSTEIN

passing through it is "parallel" to 1. This is now manifestly to be re-

placed by the requirement that every I passing through C should meetits image 1' on the T-line (T-plane), i.e. if R1, R2 , be any two points on 1,

R1'-R2' =X(R1-R 2) .

This gives at once, by (7) or (8), for the ray 1,

y: (f2+ )=a= const.Two such rays, say,

y-al =alf2, y-a2 4 = a2f2,

intersect manifestly inY = , S.=f2, (9)

independently of the values a,, a2, of the constant a. Thus, all rays I

of the contemplated kind pass through the point (9), which will thus be

the required optical centre C. It is a point on the optical axis. Ac-cording to (9), its position on the axis is determined by

=X - (p1= -f2=72- V2;

in vector symbols,C-F1 = P2-F2, (10)

> ->

or FiC = F2P2 , which reads: the step from the first focus to the opticalcentre is projectively equal to that from the second focus to the secondprincipal point. Thus, the T-line, or in three dimensions the T-plane,being fixed, and F1, F2, P2, being given, the optical centre can readilybe constructed by the rules explained in Projective Vector Algebra.To sum up the last result, all rays I passing through C meet their imagesI' on the T-plane, and no other ray has this property.

COMMUNICATION No. 198,RESEARCH LABORATORY, EASTMAN KODAK CO.,

ROCHESTER, N. Y., NOVEMBER 19, 1923.

[J.O.S.A. & R.S.I., 8680