Coordinate system
surfaces

Given a coordinate system, if one of the coordinates of a point varies while the other coordinates are held constant, then the resulting curve is called a coordinate curve. If a coordinate curve is a straight line, it is called a coordinate line. A coordinate system for which some coordinate curves are not lines is called a curvilinear coordinate system.

A coordinate line with all constant coordinates equal to zero is called a coordinate axis.

In a Cartesian coordinate system, all coordinates curves are lines, and, therefore, there are as many coordinate axes as coordinates. Moreover, the coordinate axes are pairwise orthogonal.

A polar coordinate system is a curvilinear system where coordinate curves are lines or circles. However, one of the coordinate curves is reduced to a single point, the origin, which is often viewed as a circle of radius zero. Similarly, spherical and cylindrical coordinate systems have coordinate curves that are lines, circles or circles of radius zero.

Many curves can occur as coordinate curves. For example, the coordinate curves of parabolic coordinates are parabolas.

In three-dimensional space, if one coordinate is held constant and the other two are allowed to vary, then the resulting surface is called a coordinate surface. For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin. In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve. In the Cartesian coordinate system we may speak of coordinate planes.

Similarly, coordinate hypersurfaces are the (n − 1) -dimensional spaces resulting from fixing a single coordinate of an n -dimensional coordinate system.
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