Parabolic coordinatesParabolic 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:
then a cross-section is obtained; the coordinates become confined to the x-z
(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:
factor out the x
cancel out common factors from both sides,
take the square root,
is the geometric mean
. 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:
zc = zb
, 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 half-plane with x>0, because x<0 corresponds to φ=π.
Thus a pair of coordinates η and ξ specify a unique point on the half-plane. Then letting φ range from 0 to 2π the half-plane revolves with the point (around the z-axis as its hinge): the parabolas form paraboloids. A pair of opposing paraboloids specifies a circle, and a value of φ specifies a half-plane 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.