64 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
In a third class of projections, the perspective projections, the eye of the 
observer is considered at some definite point in space from which the points 
on the surface of the sphere are viewed. The points of intersection of the 
lines of sight with the fixed plane of projection are then the desired projec- 
tion points. The projection thus obtained is dependent both on the posi- 
tions of the eye-point and on that of the plane of projection. 
In Fig. 39 let BMC be a plane tangent at M to the sphere MPS; let GP 
be a direction in space which includes an angle p with the vertical axis. 
Then if the eye be at E, the intersection, e', of the line EP with the plane 
BC is the projection point of GP. From the figure it is evident that 
DP : Me' = (EG+GD) : (EG+GM) 
On substituting in this equation DP = sin p, DG = cos p, GE = c, and Me' = x, 
we find 
sin p :*=(c+cosp) : (c+i) 
or 
x= (i+c)sinp ^y 
C + COS p 
If the plane of projection be the central equatorial plane KGL, then 
c sin p 
x=- (10) 
c + cos p 
Equations (i) and (ia) represent the general form of zenithal perspective 
projections on a horizontal plane and from them the different special types 
can be readily derived. 
If the eye-point be at G (c = o;x = = tan p) and the projection plane 
cosp 
the horizontal tangent plane (Fig. 40), the projection is the gnomonic pro- 
jection. In this projection all great circles are represented by straight lines 
and the small vertical circles by hyperbolas. Plate 10 is a. gnomonic meridian 
projection (radius of projection sphere = 5 cm.), the interval between the suc- 
cessive great circles (straight lines) and also the small circles (hyperbolas) 
being 2. The gnomonic projection is best suited to crystallographic work, 
since by its use all crystal faces are reduced to points and all zones to straight 
lines. 
If the eye be located at 5 (c= i) (Fig. 39), and the plane of projection 
is the equator, the projection is the stereographic* 
( sin p p\ 
(x= - -=tan-). 
V I+COSp 2/ 
(Fig. 41.) The stereographic projection is unique in that all circles, whether 
great or small, appear in the projection as circles instead of ellipses, as might 
be supposed at first thought. Moreover, the angle which two great circles 
make with each other is preserved unaltered in the projection. The pro- 
jection is thus angle-true. In Plate 3! the portions of great circles of the 
S L. Penfield. this Journal (4) II, i. 115; E Pedorow. Zcitschr. Kryst . 26, 27, 29, and G. Wutff. 
Zfitschr. Kryst., 21,. 249. Q89J: **, 14-1*. 1907, have given complete descriptions of the stereographic 
projection; also H. K. Boeke. Die Anwcndunc der stereographischen Projektion bei krystallographischen 
Untenuchungen. 1911. 
t Photolithographic reproduction of the meridian stereographic projection plat by Prof. G. Wulff, Zeitschr. 
Kryttall M, 14. 1002. 
