Map projections . onical orthomorphic projections. Consider the general properties of a conical projection de-fined by the equation r= ;//(tan£x)1 (1), where, as usual, ^ is the co-latitude and 11 is the constant ofthe cone. It is easy to show that any conical projection of this familyis orthomorphic, whatever the value of m and n. For the scalealong the meridian at any point is dr _ mn (tan ^%)n_1 i sec2 \-% (2). (3), R sin ^ And the scale along the parallel at any point is rdd nr Rsm-^.dX R sin ^ the same as the scale alontr the meridian. 94 THE SIMPLE MATHEMATICS Also, since the projection
Map projections . onical orthomorphic projections. Consider the general properties of a conical projection de-fined by the equation r= ;//(tan£x)1 (1), where, as usual, ^ is the co-latitude and 11 is the constant ofthe cone. It is easy to show that any conical projection of this familyis orthomorphic, whatever the value of m and n. For the scalealong the meridian at any point is dr _ mn (tan ^%)n_1 i sec2 \-% (2). (3), R sin ^ And the scale along the parallel at any point is rdd nr Rsm-^.dX R sin ^ the same as the scale alontr the meridian. 94 THE SIMPLE MATHEMATICS Also, since the projection is conical, the meridians andparallels cut at right angles. This, combined with the pro-perty we have just proved, that at any point the scale alongthe meridians and parallels is the same, shows that the familyof projections defined by (1) are all orthomorphic, whatever thevalues of ;// and n. We have still these constants at our disposal. The constantm is evidently a simple scale constant. If we want to make the. Fig. 17. Conical Orthomorphic. parallel of co-latitude %0 the standard parallel, of true length,we must have 2irn . m (tan ^%0)n = 2irR sin %0 (4). whence R sin Xs_»(taniX.)B and our scale constant cannot be determined numerically,though we have selected our standard parallel, until we haveselected further the value which we shall give to n, the con-stant of the cone. But suppose, for the moment, that we have decided to giveto n the same value that it has in the simple conical projection, OF PROJECTIONS 95 namely cos ^0. We shall then find the corresponding value ofm from (5), and we shall have an orthomorphic projection whichwe may consider as constructed upon the tangent cone at theselected standard parallel. Now the scale at any point not on this standard parallel istoo large. If then we consider an) scale value, larger than the true,we shall be able to find a pair of parallels, one on each sideof the standard, possessing this scale value. And it is evidentthat b
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