ABSTRACT
This chapter is devoted to the mathematical fundamentals of map projection. The object of the practice of map projection is to transform from the terrestial angular coordinates, that is, the geographic coordinates, to a Cartesian map system in the direct transformation. The reverse is also introduced: transformation from Cartesian to geographic coordinates by the inverse transformation. The basic transformation matrices are derived in forms useful for both the equal area and the conformal projections. The chapter presents the derivation of the first fundamental form of a surface. The first fundamental form is useful in dealing with arc length, area, angular measure on a surface, and the normal to the surface. The chapter then presents the derivation of the second fundamental form that provides the means for evaluating the curvature of a surface and determining the principal directions on the surface. The second fundamental form, in conjunction with the first fundamental form, defines the curvature.
