Parameter Lines

Generally any point  P( x ; y ; z )  of an area A (in space) can be defined by functions fx, fy and fz in the manner

   x = fx(u,v)     y = fy(u,v)     z = fz(u,v)          (1)

with parameters  u and v  within a determined domain. We call (1) parameter representation of the area A.

The lines (curves) onto the area A, which arise from u = const. (v  variable) resp. v = const. (u  variable), in planar theory are generally called parameter lines or especially v-lines resp. u-lines. The v-linies resp. u-linies span a parameter grid onto the area. We can say that the u- and v-linies correspond to the axes x = const. and y = const. of a Cartesian system of co-ordinates.

Example:

The parameter representation of an area is given with the example of the surface of the Earth (illustration from [3] under literature).

           geographic co-ordinates

A point onto the Earth sphere can be determined by two parameters, the so called geographic longitude u and the geographic latitude v (see figure above). The equator is discribed by the u-line  v = 0, the North Pole by  v =  / 2  and the South Pole by  v = -  / 2 .

Here we have to consider a kind of handicap of the parameter method: There is no unique geographic longitude u for the poles. Thus the poles are special points in the parameter grid. This is not caused by geometry because onto the sphere all points are equitable.