Parabolic coordinates
Parabolic coordinates are an alternative system of coordinates for three dimensions. They are
orthogonal. Conversion from Cartesian to parabolic coordinates is effected by means of the following equations:
If
φ=0 then a crosssection is obtained; the coordinates become confined to the
xz plane:

If
η=c (a constant), then

This is a
parabola whose focus is at the origin for any value of
c. The parabola's axis of symmetry is vertical and the concavity faces upwards.
If ξ=c then

This is a parabola whose focus is at the origin for any value of
c. Its axis of symmetry is vertical and the concavity faces downwards.
Now consider any upward parabola η=c and any downward parabola ξ=b. It is desired to find their intersection:
 ,
regroup,
 ,
factor out the
x,

cancel out common factors from both sides,
 ,
take the square root,

x is the
geometric mean of
b and
c. The
abscissa of the intersection has been found. Find the
ordinate. Plug in the value of
x into the equation of the upward parabola:

then plug in the value of
x into the equation of the downward parabola:

z_{c} = z_{b}, as should be. Therefore the point of intersection is
Draw a pair of tangents through point
P, each one tangent to each parabola. The tangential line through point
P to the upward parabola has slope:

The tangent through point
P to the downward parabola has slope:
The products of the two slopes is

The product of the slopes is
negative one, therefore the slopes are perpendicular. This is true for any pair of parabolas with concavities in opposite directions.
Such a pair of parabolas intersect at two points, but when φ is restricted to zero, it actually confines the other coordinates η and ξ to move in a halfplane with x>0, because x<0 corresponds to φ=π.
Thus a pair of coordinates η and ξ specify a unique point on the halfplane. Then letting φ range from 0 to 2π the halfplane revolves with the point (around the zaxis as its hinge): the parabolas form paraboloids. A pair of opposing paraboloids specifies a circle, and a value of φ specifies a halfplane which cuts the circle of intersection at a unique point. The point's Cartesian coordinates are [Menzel, p. 139]:
See also: spherical coordinates, cylindrical coordinates, Cartesian coordinates.
 Menzel, Donald H., Mathematical Physics, Dover Publications, 1961.