204 G E O G law of distance, as p =_/'(u). We may thus avoid the calcu- lation of all the distances aml azimuths (with reference to the selected centre point) of tlie intersections of meridians and parallels. Construct a stereographie projection of the globe on the horizon of the given place; then on this pro- jection draw concentric circles (according to the stereo- graphic law) representing the loci of points whose distances from the centre are consecutively 5°, 10", 15‘, 20", &c., up to the required limit, and a system of radial lines at inter- vals of 5°. Then to construct any other projection.—com- mence by drawing concentric circles, of which the radii are q_,i/V F1G.l6. —Stereographic Projection. previously calculated by the law p =_f(u), for the successive values of 1:, 5°, 10°, 15°, 20°, the, up to the limits as before, and a system of radial lines at intervals of 5°. This being completed, it remains to transfer the points of intersection from the stereographic to the new projection by graphic interpolation. We now come to the general case in which the point of vision has any position outside 0 the sphere. Let abccl (fig. 17) be 5, m the great circle section of the ‘ sphere by a plane passing through 1, e j c, the central point of the por- (L 0 tion of surface to be represented, and V the point of vision. Let )3)’ perpendicular to Vc be the plane of representation, join mV cut- *1 ting pi inf, thenfis the projec- tion of any point an in the circle F" l;!. abc, and qf is the representation of cm. Let the angle com =21, then, since ef : cl’ = my : _r/V, _ I: sin n -1: +cos u ’ 'e=l-, Yo=lz, ljf=p; which gives the law connecting a spherical distance u with its rectilinear representation p. The relative scale at any point in this system of projection is given (keeping to our previously adopted notation) by _ 1+hcosu _ , I: — (h+cos u)° ’ :IL+Cos u ’ the former applying to measurements made in a direction which passes through the centre of the map, the latter to the transverse direction. The product 00-’ gives the exaggera- tion of areas. With respect to the alteration of angles we have _ it + cos n LL. 1+hcosu' and the greatest alteration of angle is —-1/L—1 . ., w
- .~i:i tan- .
‘ll. 7-]. 2 I A 1’ 11 Y ' This varnishes when It = 1, that is, if the projection be stereo- graphic; or for 1(=0, that is, at the centre of the map. At a distance of 90’ from the centre, the greatest alteration is 90° — 2 C0t_l //1. (‘See 1’/tlilusop/I. .1/all/,, April 1h'G'_?_) The constants It and k can be determined, so that the total misrepresentation, viz., M:/B -IU [.r.s.'rm:.r.s.'r1c.I.. {(cr- 1)'3 + to" — 1Y3} sin lulu, shall be a minimmn, /3 being the f'I‘c:1te.~‘t value of 21, or the spherical radius of the map. On substituting the expres- sions for 0- and o-' the integration is elfectel without diffi- culty. Put A21-COS B; v—».(/z— 1)A , h+cos B II:y—-(h+1)l0g,()+l), 11'_—h_:_](2— y+ 5.-'=). Then the value of .I is . 117.4 sin‘: §,B+'2/.l{ + 15:11’. When this is a minimum, (lrI_
- 7, _
, (l.l_ o, (75.0. .-.1.-11'+1I:o; 2'”‘+A (1/1. H Therefore M = 4 sin'3 — and I: must be determined so as to make H3: H’ a maximum. In any particular case this maximum can only be ascertained by trial, that is to say, log H3 — log H’ must be calculated for certain equidistant values of It, and then the particular value of I: which cor- responds to the required maximum can be obtained by interpolation. Thus we find that if it be required to make the best. possible perspective representation of a hemisphere, the values of It and I; are /L = 1'47 and 1:: 22034 ; so that in this case _:lII' _ 0. (UL- _‘_"034 sin 11 P ‘ 1”-T: ll. ' For a map of Africa or South America, the limiting radius /3 we may take as -10’; then in this case _ 2'5-13 sin 21. ‘ 1 -025 + cos_zo ' For Asia, /3=54 , and the distance It of the point of sigh Fig. 1%. Fig. 1.9 is a map of Asia having the meridians and parallels laid down on this system.
in this C150 is 1131.