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Coordinate Systems

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A coordinate system is simply a method of assigning numbers to points in space in order to provide a mathematical description of the universe. In order to be useful, the method of assignment must provide a unique set of coordinate values for each point in space. In other words, there must be a one-to-one correspondence between points in space and coordinate values. Other than that, any method of assignment is valid.

However, the number of coordinates must always be the same as the dimensionality of the space. In one-dimensional space (a line, curved or straight), one coordinate is sufficient. In two dimensions (a surface), two coordinates are necessary. In three-dimensional space (a volume), three coordinates are needed. And in four-dimensional space (spacetime), four coordinates are needed.

The three most commonly used coordinate systems in three-dimensional space are (1) the familiar, right-handed, rectangular Cartesian coordinate system (x, y, z); (2) the cylindrical polar coordinate system (r, f, z); and (3) the spherical polar coordinate system (r, q, f). These coordinate systems are illustrated below for an arbitrary point P in three-dimensional space.

Generalized coordinates q_i or xⁱ are sometimes used to represent any of these coordinates. For example, if q_i represents spherical polar coordinates, q₁ = r, q₂ = q, and q₃ = f. Or if xⁱ represents rectangular coordinates, x¹ = x, x² = y, and x³ = z.

Because of the one-to-one correspondence between points in space and coordinate values, it is always possible to transform one set of coordinates into another set. The equations that accomplish this are called coordinate transformation equations. And the equations that reverse the transformation are called the inverse coordinate transformation equations. Examples of coordinate transformation equations and inverse transformation equations are given below:

Transformation Equations	Inverse Transformation Equations
x = r cos f	r = sqrt(x^2 + y^2)
y = r sin f	f = arctan(y/x)
z = z	z = z

Transformation Equations	Inverse Transformation Equations
x = r sin q cos f	r = sqrt(x^2 + y^2+ z^2)
y = r sin q sin f	q = arctan((sqrt(x^2 + y^2))/z)
z = r cos q	f = arctan(y/x)

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