Curvilinear Coordinates - Orthogonal Curvilinear Coordinates in 3D - Coordinates, Basis, and Vectors

Coordinates, Basis, and Vectors

For now, consider 3d space. A point in 3d space can be defined using Cartesian coordinates (x, y, z), or in another system (q₁, q₂, q₃), as shown in Fig. 1. The latter is a curvilinear coordinate system. The quantities (q₁, q₂, q₃ ) are the curvilinear coordinates of a point P.

The surfaces q₁ = constant, q₂ = constant, q₃ = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates.

The relation between the coordinates is given by the invertible transformations:

Any point can be written as a position vector r in either coordinate system. For Cartesian coordinates:

where x, y, z are the coordinates of the position vector with respect to the basis vectors e_x, e_y, e_z. The Cartesian basis vectors are the standard basis set of vectors.

In terms of the curvilinear system, the same r can be written:

where h₁, h₂, h₃ are scale factors (also called Lamé coefficients after Gabriel Lamé) that account for the deformation from the rectangular Cartesian coordinates to the curvilinear system (see below), h₁q₁, h₂q₂, h₃q₃ are the coordinates of this position vector, and b₁, b₂, b₃ are the curvilinear basis.

In a curvilinear system, the basis vectors b_i depend on the coordinates q_i (i = 1, 2, 3), and are not necessarily orthogonal. If they are, the basis is an orthogonal basis and the coordinates are orthogonal coordinates. Curvilinear coordinates allow the generality of basis vectors not all mutually perpendicular to each other, and are not required to be of unit length: they can be of arbitrary magnitude and direction. The use of an orthogonal basis makes vector manipulations simpler than for non-orthogonal. However, but in some areas of physics and engineering, particularly fluid mechanics and continuum mechanics, require non-orthogonal bases to describe deformations and fluid transport to account for complicated directional dependences of physical quantities.

A basis whose vectors change their direction and/or magnitude from point to point is called local basis. All bases associated with curvilinear coordinates are necessarily local. Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.

Note: usually all basis vectors are denoted by e, for this article e is for the standard basis (Cartesian) and b is for the curvilinear basis.