FEBRUARY 1937 MONTHLY WEATHER REVIEW
THE GEOMETRICAL THEORY OF HALOS-I11
By EDGAR W. WOOLARD
[Weather Bureau, Washington, D. C., February 19371
REFLECTION
The calculation of the image produced by external reflection is easily accomplished as indicated in figure 6.'
Internal reflection (which occurs whenever an internal
ray meets an interface at an angle with the normal
greater than y=arc sin -7 p>1) conforms to the same law,
but may be associated with refraction a t some other point
in the complete path of the ray; when the ray is incident
1
P
FIGURE fi.-Simplc Reflection: 6 , luminous source: SC, iucideut ray; C!N, normal to inter- face nt point of incidence; i. angle of incideuce; (261, reflected ray; R, angle of reflection; S', virtual image; D, deviation.
on, and also emerges from, a face parallel to that from which it is int,ernally reflected, the emergent ray is parallel
to the course it would have ta.ken if merely reflected a t the
point of inc.idence.
The geometrical relntions on a sphere of indefiilitely great ra.dius, figure 7, readily lead to approprin te forinulne
for computation, Formulae 11. It is a ~orollary of the
Law of Reflection, that the source and the image lie a t
equal distances from the geometric pole of every great circle normal to the reflecting plane (any such-pole is on
the great circle PP). Furtherinore, the projectioiis of the
incident and the reflected rays on any plane through the normal to the reflecting surface make equal angles with
the normal, just as if they were nc.tua1 rays. These principles constitute Bravais' Laws of Reflection. The
deviation (angular displacement of the image from the
true position of an indefinitely distant source) is away
from the normal N. I n the case of prismatic refraction accompanied by an internal reflection from a principal plane, figure 8, the
projection of the ray on the principal plane is not altered
by the reflection; nnd hence the positions of the images
produced with and without suc,h a reflection are syni- metric with respect to the princ,ipal plane. The calcu-
1a.tion of the ima.ge, figure 9, is the,refore acconiplished
with Formulae I except t1in.t the altitude relnt,ive to the principal pla.ne is --h and
(6*) COS D=COS* h(l+cos D')-l (I*)
cos h sin D' (7*) sin A= sin
by the Law of Cosines and t,he Law of Sines, respect,ively.
The deviation is toward V and away froin P'. Prismatic refraction accompanied by an internal reflec-
tion from m y arbitrarily given plane, figure 10, requires
the more complicated Formulae 111; see figure 11. When
I The figure3 am numbered consecutively with those In paper 11.
55
the reflecting plane coincides with a principal plane, p=O" and the formulae reduce to the previous ones;
when the reflecting plane coincides with a lateral face,
p=90.
Multiple internal reflection will, of course, take place
if the once-reflected ray meets the next interface a t too
great an angle with the normal. The positions of the images formed by light from the
sun or the nioon incident on a particular face of an ice
crystal of specified form in a given orientttion may now
be calculated by superposing the appropriate figures on the celestial sphere, with S in the position of the luminary:
The preceding formulae give the. position of an imnge relative to the source; and from t h s , the position relative
to the horizon is readily found for any given altitucle of the sun or moon, by solving the spherical triangles involved.
The formulae required will be given in the next section.
I f I \- \
.
FIGUhE 'I.-Calculalion of the Image Produced by Simple Refldion: See Formulae 11,
PPP reflecting plane CN, normal at point of incidence; N. pole of reflecting plane. on sahm side of plane as source 6; S', image; I), deviation; C , observer. The deviation, D, is away from N.
Formulae I1
CALCULATION OF IMAGE PRODUCED BY SIMPLE REFLECTION
Given To Find
Angle of incidence, i Coordinates of S relative to Relative to reflecting plane: reflecting plane: Altitude 90'- i Relative Azimuth 0' Relative Azimuth Oo
Coorainates of S':
Altitude
Relative to S Deviation, D Positiou Angle 180'
a
Computation
(1) D=18Oo-2i oo<ii900
-90<a<00 D (2) a =-- 2
[See figure 7.1
56 MONTHLY WEATHER REVIEW FEBRUARY 1937
FIGURE 8.-Intcrnal Refleelion iron& a Principal Plnne: SC.OCS, path of ray; PI, principal plane. CN, CN, normalc. 0, point of internal reflection; SS, plane of re- fraction a t incidence; xs, plane oirefraction at emergence.
FIGURE 9.-Cnleci/alion of the Imagr Prodicecd by Inlernd Re/kcIion from a Principal Plunet See Fmnulae I *. PPV, principal plane; P pole or principal plane on same side of plane as 6; V vertex of refracting angle; S, source; 8 image; D, deviation; A position angle of irn&e; h. inclination of incident ray to principal plsne; R, T, pro: jections of source and of image on principal plane; li, deviation nf projection of ray on principal plane; C. observer. Deviation is always toward V and away from P.
FIGURE lO.-Rismalic Refraction aecompanitd by Infernal Refleelion: SCOCS path of rav. PP principal plane of refracting angle. O F normal to reflecting plane; XS plane of refraction at incidence; FCOC, plane of incident and reflected rays; ktS8, plane bf r&afraetion a t emergence; CN, CN, Lormals to faces of incidence and emirgence, rcupectively.
FEBRUARY 1937 MONTHLY WEATHER REVIEW 57
1 I
FIGURE 11.-Calculation of Image Produced by Refraction and Infernal Rcprlretion: See Formulae 111. PP, principal plane. with pole PI; LL reflecting plane, with pole F; 8 source; E, image; R. T, ,projections of source and image on principal plane; D deviation of projection of ra>--un principal plane; D, diviation; A, position angle. IC, observer at center of sphere IS omitted from the diagram; SIC, YIC would correspond to CO. CO, respectlvely, in h u r e 10.1
Formulae I11 Computatio I I
}Table 3 CALCULATION OF IMAGE PRODUCED BY PRISMATIC REFRAC- (1) sit1 =p sin k
TION COMBINED WITH INTERNAL REFLECTION (2 ) A= X+z1--r,
To Find (3) cos w=sin k cos O+cos k sin COS X Given . sin p sin X 11. a; h; 8, x; i 1 Coordinates of S (4) sin y= Coordinates of S: Relative to Principal Plane sin w
Relative to Principal Plane Altitude -h (5) sin k=sin k cos 2 wfcos k sin 2 w cos y
Altitude h Relative Azimuth f D (6) sin h=p sin k Relative Azimuth 0
Deviation u Position Angle f A
Relative to S sin 2 w sin y
cos k (7) sin z=
(8) T~=LT+z--T~ (Internal reflection occurs if
r:>$ for h) (9) D=il+il--a Table 3 with h=h
(10) cos D=cos h cos h cos D-sin h sin h \I
cos h sin D
sin L) (11) sin A=-
[See figure 11. The t.rigonometrica1 relations on the sphere follow from t.he Law of Cosiiies nnd the Law of Sines.]