coordinate surfaces

Given these functions, coordinate surfaces are defined by the relations:

In three-dimensional space the intersection of two coordinate surfaces is a coordinate curve.

Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae.

One immediate consequence is that the constant time coordinate surfaces form a family of (isometric) spatial hyperslices.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.

Hence, the coordinate surfaces are confocal parabolic cylinders.

Focaloid (shell given by two coordinate surfaces)

More general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories.

For example, the coordinate surfaces obtained by holding ρ constant in the spherical coordinate system are the spheres with center at the origin.

Equations with boundary conditions that follow coordinate surfaces for a particular curvilinear coordinate system may be easier to solve in that system.

In general, a focaloid could be understood as a shell consisting out of two closed coordinate surfaces of a confocal ellipsoidal coordinate system.

The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved.

A system of skew coordinates is a curvilinear coordinate system where the coordinate surfaces are not orthogonal, in contrast to orthogonal coordinates.

Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

The sloping nature of the coordinate surfaces does require additional interpolation of the pressure gradient force, and the smoothing of terrain can often cause it to extend beyond the true boundaries of land.