The representation, on the plane, of all or part of the terrestrial ellipsoid. Usually this representation is determined having in mind the drawing of a map.
Cartographic projections are drawn in a specified scale. By reducing the ideal terrestrial ellipsoid times, one obtains a geometric model of it, a globe, the mapping of which, in natural size, onto the plane gives a map of the surface of this ellipsoid. The ratio represents the representative fraction called the principal scale of the map. The fundamental characteristic of a map projection at any point of it is, however, the actual scale . This is the inverse of the fraction obtained by dividing an infinitely small segment on the terrestrial ellipsoid by its image on the plane: . The number depends on the position of the point on the ellipsoid and on the direction of the chosen segment. The ratio is called the relative scale or linear distortion, the difference is called the linear deformation. The numerical value of the principal scale is taken into account only in the calculation of the coordinates of points of a map projection and in the use of maps; in the study of map projections one sets .
In cartography, in most cases the flattening of the Earth can be ignored or taken into account in one way or another. Therefore, below mappings onto the -plane of a spherical surface with respect to the geographical coordinates (latitude) and (longitude) are considered.
The equation of a map projection has the form
where and are functions satisfying some general conditions. (A map projection can also be defined by equations in which some planar coordinates other than the rectangular ones are used.) The representations of the meridians (lines of constant longitude) and the parallels (lines of constant latitude) under a given map projection form a cartographic network (or graticule).
Network representing lines of the globe (normal and rotated)
The globe and its orthographic projections A. Normal B. Transverse C. Oblique
Cylindrical projections A. Mercator (conformal) B. Rectangular C. Orthographic (equal area)
Conical projections A. Conformal B. Equidistant C. Equal area
Azimuthal projections A. Conformal (stereographic) Transverse Oblique B. Equidistant Transverse Oblique C. Equal area Transverse Oblique
I liked UCSB
I transferred in too, from University of Nevada, Las Vegas.
UCSB is definitely what you make of it, moreso than many of the other schools I've seen. In other words, there are fantastic opportunities. Many of the faculty in science and engineering are well respected (or even near the top of their field). There are plenty of students, which is a good thing. That means there are plenty of courses that can be offered. So if you decided you wanted to become a serious mathematician, you could get up to speed quickly.
There's also a lot of interdisciplinary stuff. For example, there was a center for quantum computation that was set up after I graduated, but I understand that it draws all sorts folks from physics, engineering, and math