Page:Encyclopædia Britannica, Ninth Edition, v. 10.djvu/217

Rh MATH}-31IATICAL.] GEOGRAPHY 203 is to be drawn. When pin is greater than pa, the ellipse P0P’, and 221’, vP' cutting lr in pp’: these are the projections touches the circle in two points; these points divide the ellipse into two parts, one of which, being on the other side of the meridian plane aqr, is invisible. SlereograpIu'c .Pro_ject£on.—Ii1 this case the point of vision is on the surface, and the projection ,_, P is made on the plane of the great 0 circle whose pole is V. Let /l1)l’V (fig. 1:3) be a great circle through the point I of vision, and ms the trace of the ‘ plane of projection. Let c be the 0 centre of a small circle whose radius is cp=cl ; the straight line pl repre- sents this small circle iii orthographic ‘_v projection. 1'13“ 12- We have ﬁrst to show that the stereographic projection of the small circle pl is itself a circle ; that is to say, a straight line through V, moving along the circumference of 2;], traces a circle on the plane of projection 01's. This line generates an oblique cone standing on a circular base, its axis being cV (since the angle pVc= angle cVl) ; this cone is divided symmetrically by the plane of the great circle Ir];/, and also by the plane which passes through the axis V4‘, perpendicular to the plane kpl. Now V7‘-V22, being = V0 sec /cVp -V/5 cos /t-V1"; = V0 -V/.', is equal to Vs-Vl ; there- fore the triangles Vrs, V71) are similar, and it follows that the section of the cone by the plane 7's is similar to the section by the plane pl. But the latter is a circle, hence also the projection is a circle; and since the repre- sentation of every inﬁnitely small circle on the surface is itself a circle, it follows that in this projection the represen- tation of small parts is (as we have before shown) strictly similar. Another inference is that the angle in which two lines on the sphere intersect is represented by the same angle in the projection. This may otherwise be proved by means of fig. 13, where "ole is the diameter ofl the sphere passing through the point of vision, jg//e the plane of projection, /at a great circle, passing of course through V, and mm the line of intersection of these two planes. A tangent plane to the surface at t cuts the plane of projection in the line rvs perpendicular to on; tv is a tangent to the circle lit at 1, tr and is are any two tangents to the surface at 1‘. Now the angle via (a being the projection of t) is 90° — otV = 90° — oVt = oaV = tzw, therefore tv is equal to av; and since ms and zws are right angles, it follows that the angles arts and «ms are equal. Hence the angle 2-ls also is equal to its projection rats; that is, any angle formed by two intersecting lines on the surface is truly represented in the stereographic projection. Ve have seen that the projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere ; otherwise, a line anywhere. It follows that meridians and parallels are represented in a projection on the horizon of any place by two systems of orthogonally cutting circles, one Bystem passing through two ﬁxed points, namely, the poles; and the projected meridians as they pass through the poles show the proper differences of longitude. To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle vl/l-r (ﬁg. 14) with the diameters hr, lr at right angles ; the latter is to represent the central meridian. Take lcoP equal to the co-latitude of the given place, say a; draw the diameter Fig. 1:3. of the poles, through which all the circles representing meridians have to pass. All their centres then will be in a line smn which crosses pp’ at right angles through its middle point m. lTow to describe the meridian whose west longitude is co, draw pn making the angle opn = 90° — to, then or is the centre of the required circle, whose direction as it passes through p will make an angle op_r/=1» with pp’. The lengths of the several lines are e P d 3 m n 34/ r P’ Fig. 14. op =tan§u ; op’=cot.1_.u; om = eotu ; 771% = coscc u cot (0. Again, for the parallels, take Pb = Pc equal to the co-latitude, say c, of the parallel to be projected ; join vb, vc cutting I r in e, (l. Then ed is the diameter of the circle which is the required projection ; its centre is of course the middle point of ad, and the lengths of the lines are od = tan §(u — c); oc = tan ,1,(u + c). The line sn itself is the projection of a parallel, namely, that of which the co-latitude c = 180° — a, a parallel which passes through the point of vision. A very interesting connexion, noted by Professor Cayley, exists between the stereographic projection of the sphere on a meridian plane (i.e., when a point on the equator occupies the centre of the drawing) and the projection on the horizon of any place whatever. The very same circles that represent parallels and meri- dians in the one case represent them in the other case also. In g. 15, abs being a projection ,, in which an equatorial point is in the centre, draw any chord ab per- pendicular to the centre meridian ° cos, and on ab as diameter describe a circle, when the property referred to will be observed. This smaller circle is now the stereographic pro- jection of the sphere on the horizon F, ' of some place whose co-latitude we lg‘ may call a. The radius of the first circle being unity, let ac=sin.r, then by what has been proved above co=sin.-1; cotu = cos.-r ; therefore a = a‘, and ac = sin 21-. Although the meridian circles dividing the 360° at the pole into equal angles must be actually the same in both systems, yet a parallel circle whose co-latitude is c in the direct projection abs belongs in the oblique system to some other co-latitude as c’. To determine the connexion between c and c’, con- sider the point t (not marked), in which one of the parallel circles crosses the line soc. In the direct system, 112 being the pole,

C 3 15., = __ _ o_. = 2 pt 1 tan §(9O c; -F +cot éc and in the oblique, pt = ac (tan -32¢ — tan §(u —- c')), which, replacing ac by its value sin a, becomes 2 sin ﬁlo sin §c’_ 2 cos -.§(u — c’) -1 +cot 51¢ cot 23¢: I D thererore can J_,~c= tan §c' tan 213415 is the required relation. Notwithstanding the facility of construction, the stereo- graphic projection is not much used in map~making. _ But it may be made very useful as a means of graphical inter- polation for drawing other projections in which points are represented in their true azimuths, but with an arbitrary