Map projections.

Map - conventional image of the entire Earth's surface or a part of the plane, made in a reduced form by a specific law. Such an image can not be obtained accurately because, in principle, can not deploy the spherical surface of the Earth on a plane without gaps or wrinkles in certain places, that is, without distortion. Depending on the purpose of the map up to a certain scale.

Have to start with the banal: the Earth, in general, the round, the card is completely flat. Objective: To display the round Earth on a flat map is not redrawn, such as "New Earth" by hand, and to project it so that it was established mathematical correspondence between the geographic coordinates on Earth and the grid coordinates on the paper. To ensure this compliance and apply mathematical projection. For most types of projections is fairly simple mathematical tool that takes the geographical coordinates on the ellipsoid in Cartesian coordinates on the paper.

Globe certainly gives the truest representation of the relative positions of the continents and oceans , rivers , towns, mountains. But this model of our planet is not very convenient to operate. Globes with all its advantages are very small-scale and unwieldy. Thus, if the globe was made in a 1:1 000 000 , it had a diameter 12.7 m In addition, it is difficult to produce linear measurements , to determine horizontal coordinates of points applied to the images of geographic features.

As well go to the map of the globe , how to move the spherical surface of the Earth on a plane ? But try to make a plane from orange peel and you will understand what is the core problem of cartography : the surface of a sphere or ellipsoid can not be transferred to a plane without gaps or creases. Without deforming the world can not do.

Although there are thousands of ways Earth image on the plane , none of them provide an accurate playback. Always have to sacrifice something . On some maps correctly depict the outlines of continents and oceans, but it distorts their size . On the other - save space, but it distorts the shape of the continents .

Map projection - mathematically certain way to render the surface of the earth ellipsoid on the plane. Map projection sets the analytical relationship between the geographical coordinates of the points of the earth ellipsoid and rectangular coordinates of points on the same plane.

The surface of the ellipsoid ( sphere) can not be deployed on the plane , like the surface of a cone or a cylinder , so the continuity and uniqueness of the image is achieved by deforming ( non-uniform variations in the expansion or contraction) of the ellipsoid when carrying it on the plane.

Mapping the design process is carried out in two stages:

Reducing the size of the earth ellipsoid to ellipsoid model ( Globe ), respectively, to the selected scale images .

Mathematical design of the ellipsoid to the sphere , and then the image of a sphere onto a plane.

In the process of mapping a mismatch scale of the map in its different parts . Published on the main map scale, which corresponds to only some parts of the map .

If you compare the infinitesimal segment on the projection in this direction ( ie, a segment with the strain after projecting a sphere on the plane) , respectively, with an infinitesimal segment of the surface of the earth ellipsoid, we get the particular scale.