Map projections . are equal in length to thelength of the corresponding parallels on the Earth multipliedby the representative fraction of the map. Having given the radius and the length of the arc of parallelwe obtain at once the angle which it subtends at the pole of theprojection. If this angle is ft. 2tt, then ;/ is what we have calledthe constant of the cone. We shall find it convenient to examine projections syste-matically in the following order of properties : (a) radius and length of the standard parallel, or parallels. {b) constant of the cone n. (c) radii of the other parallels. 78
Map projections . are equal in length to thelength of the corresponding parallels on the Earth multipliedby the representative fraction of the map. Having given the radius and the length of the arc of parallelwe obtain at once the angle which it subtends at the pole of theprojection. If this angle is ft. 2tt, then ;/ is what we have calledthe constant of the cone. We shall find it convenient to examine projections syste-matically in the following order of properties : (a) radius and length of the standard parallel, or parallels. {b) constant of the cone n. (c) radii of the other parallels. 78 THE SIMPLE MATHEMATICS (d) linear scale along the meridians and parallels. (e) scale of areas. (/) alteration of angles. (g) construction by rectangular coordinates. Simple conical projection with one standard parallel. The radius of the standard parallel is the length of thetangent drawn from a point in the polar axis produced totouch the sphere at the standard parallel ; that is r0 = j? cot ^)0 or =i?tan^0 (i).. Fig. 13. Simple Conic. The length of an element of the standard parallel is R cos cf)0. AX. Equating this to the alternative expression for its length,viz. r0. 0 we have R cos 0. 0,whence we have for the constant of the cone h = — = sin 00 .(2). OF PROJECTIONS 79 The lengths along the meridians are true. Hence thegeneral expression for the radius is r=r0-R((J3- 4>n)= AMtcot0„-((/>-(/,,)} (3). In practice we draw the standard parallel, and lay off alongthe central meridian the true distances to the other parallelswhich can then be described. We divide the standard paralleltruly; and join the points of division to the pole of the projectionto obtain the other meridians. We can easily allow for the ellipti-city of the Earth by taking the true distances from geodetic tables. The only difficulty in constructing the projection graphicallylies in the awkwardness of describing circles of very large this reason it is often preferable to calculate
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