| (From Text Book of Topographical Surveying, by permission of the Controller of H. M. Stationery Office.) |
| Fig. 14.—Part of the Atlantic Ocean on a Meridian Central Projection. The shortest path between any two points is shown on this projection by a straight line. |
Fig. 14 is an example of a meridian central projection of part of the Atlantic Ocean. The term “gnomonic” was applied to this projection because the projection of the meridians is a similar problem to that of the graduation of a sun-dial. It is, however, better to use the term “central,” which explains itself. The central projection is useful for the study of direct routes by sea and land. The United States Hydrographic Department has published some charts on this projection. False notions of the direction of shortest lines, which are engendered by a study of maps on Mercator’s projection, may be corrected by an inspection of maps drawn on the central projection.
There is no projection which accurately possesses the property of showing shortest paths by straight lines when applied to the spheroid; one which very nearly does so is that which results from the intersection of terrestrial normals with a plane.
We have briefly reviewed the most important projections which are derived from the sphere by direct geometrical construction, and we pass to that more important branch of the subject which deals with projections which are not subject to this limitation.
Conical Projections.
Conical projections are those in which the parallels are represented by concentric circles and the meridians by equally spaced radii. There is no necessary connexion between a conical projection and any touching or secant cone. Projections for instance which are derived by geometrical construction from secant cones are very poor projections, exhibiting large errors, and they will not be discussed. The name conical is given to the group embraced by the above definition, because, as is obvious, a projection so drawn can be bent round to form a cone. The simplest and, at the same time, one of the most useful forms of conical projection is the following:
| Fig. 15. |
Conical Projection with Rectified Meridians and Two Standard Parallels.—In some books this has been, most unfortunately, termed the “secant conical,” on account of the fact that there are two parallels of the correct length. The use of this term in the past has caused much confusion. Two selected parallels are represented by concentric circular arcs of their true lengths; the meridians are their radii. The degrees along the meridians are represented by their true lengths; and the other parallels are circular arcs through points so determined and are concentric with the chosen parallels.
Thus in fig. 15 two parallels Gn and G′n′ are represented by their true lengths on the sphere; all the distances along the meridian PGG′, pnn′ are the true spherical lengths rectified.
Let γ be the co-latitude of Gn; γ′ that of Gn′; ω be the true difference of longitude of PGG′ and pnn′; hω be the angle at O; and OP = z, where Pp is the representation of the pole. Then the true length of parallel Gn on the sphere is ω sin γ, and this is equal to the length on the projection, i.e. ω sin γ = hω (z + γ); similarly ω sin γ′ = hω (z + γ′).