Map Projections - projection, plane, equator, equal, parallels and lines

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MAP PROJECTIONS In the construction of maps, one finds the sphere cannot be opened out into a plane like the cone or cylinder; consequently in a plane representation of configurations on a sphere it is impos sible to retain throughout the proportions of lines or areas or equality of angles, though we may have in the representation sim ilarity to all very small portions of the original, but at the expense of the areas, which will be quite misrepresented; or we may retain equality of areas if we give up the idea of similarity. A globe of ordinary dimensions serves scarcely any other purpose than to convey a clear conception of the earth's surface as a whole. For this purpose it is absolutely essential. The construction of a map resolves itself into the drawing of two sets of lines, one set to represent meridians, the other to represent parallels.

Cylindrical Equal Area Projec tion.—Let us suppose a model of the earth to be enveloped by a cylinder in such a way that the cylinder touches the equator, and let the plane of each parallel such as PR be prolonged to intersect the cylinder in the circle pr (fig. 12). Now unroll the cylinder and the projection will appear as in fig. 13. The whole world is now represented as a rectangle, each parallel is a straight line, and its total length is the same as that of the equator, the distance of each parallel from the equator is sin 1 (where l is the latitude and the radius of the model earth is taken as unity). The meridians are parallel straight lines spaced at equal distances.

This projection possesses an important property : each strip of the projection is equal in area to the zone on the model which it represents, and that each portion of a strip is equal in area to the corresponding portion of a zone. Any figure, of any shape on the model, is correctly represented as regards area by its corre sponding figure on the projection. This is "the cylindrical equal area projection." It is clear that in this case all meridian lengths are too small and all lengths along the parallels, except the equator, are too large ; thus al though the areas are preserved the shapes are, especially away from the equator, much distorted.

The property of preserving areas is, however, a valuable one when the purpose of the map is to ex hibit areas. Mercator's projection commonly used in atlases, preserves local shape at the expense of area.

In perspective drawings of the sphere, the plane on which the representation is actually made may generally be any plane per pendicular to the line joining the centre of the sphere and the point of vision. If V be the point of vision, P any point on the spherical sur face, then p, the point in which the straight line VP intersects the plane of the representation, is the projection of P.

Orthographic Projection.

In this projection the point of vision is at an in finite distance and the rays consequently parallel; in this case the plane of the drawing may be supposed to pass through the centre of the sphere. Let the circle (fig. 14) represent the plane of the equator on which we propose to make an orthographic representation of meridians and parallels. The centre of this circle is clearly the projection of the pole, and the parallels are projected into circles having the pole for a com mon centre. The diameters aa', bb' being at right angles, let the semicircle bab', be divided into the re quired number of equal parts ; the diam eters drawn through these points are the projections of meridians. It is clear that, when the points of division are very close, the parallels will be very much crowded towards the outside of the map; so much so, that this projection is not much used.

For an orthographic projection of the globe on a meridian plane let qnrs (fig. Is) be the meridian, ns the axis of rotation, then qr is the projection of the equator. The parallels will be represented by straight lines passing through the points of equal division ; these lines are, like the equator, perpendicular to ns. The meridians will in this case be ellipses described on ns as a common major axis, the distances of c, of d and of e from ns being the minor semiaxes.