Parabolic coordinates

From Academic Kids

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:

<math> \eta = - z + \sqrt{ x^2 + y^2 + z^2 }, <math>
<math> \xi = z + \sqrt{ x^2 + y^2 + z^2 }, <math>
<math> \phi = \arctan {y \over x}. <math>
<math>

\begin{vmatrix}d\eta\\d\xi\\d\phi\end{vmatrix} = \begin{vmatrix}

   \frac{x}{\sqrt{x^2+y^2+z^2}}

& \frac{y}{\sqrt{x^2+y^2+z^2}} &-1+\frac{z}{\sqrt{x^2+y^2+z^2}}\\

   \frac{x}{\sqrt{x^2+y^2+z^2}}

& \frac{y}{\sqrt{x^2+y^2+z^2}} &1 +\frac{z}{\sqrt{x^2+y^2+z^2}}\\ \frac{-y}{x^2+y^2}&\frac{x}{x^2+y^2}&0 \end{vmatrix} \cdot \begin{vmatrix}dx\\dy\\dz\end{vmatrix} <math>

<math>\eta\ge 0,\quad\xi\ge 0<math>

If φ=0 then a cross-section is obtained; the coordinates become confined to the x-z plane:

<math> \eta = -z + \sqrt{ x^2 + z^2}, <math>
<math> \xi = z + \sqrt{ x^2 + z^2}. <math>

If η=c (a constant), then

<math> \left. z \right|_{\eta = c} = {x^2 \over 2 c} - {c \over 2}. <math>

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

<math> \left. z \right|_{\xi = c} = {c \over 2} - {x^2 \over 2 c}. <math>

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:

<math> {x^2 \over 2 c} - {c \over 2} = {b \over 2} - {x^2 \over 2 b}, <math>

regroup,

<math> {x^2 \over 2 c} + {x^2 \over 2 b} = {b \over 2} + {c \over 2}, <math>

factor out the x,

<math> x^2 \left( {b + c \over 2 b c} \right) = {b + c \over 2}, <math>

cancel out common factors from both sides,

<math> x^2 = b c, \,<math>

take the square root,

<math> x = \sqrt{b c}. <math>

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:

<math> z_c = {b c \over 2 c} - {c \over 2} = {b - c \over 2}, <math>

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

<math> z_b = {b \over 2} - {b c \over 2 b} = {b - c \over 2}. <math>

zc = zb, as should be. Therefore the point of intersection is

<math> P : \left( \sqrt{b c}, {b - c \over 2} \right). <math>

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:

<math> {d z_c \over d x} = {x \over c} = { \sqrt{ b c} \over c} = \sqrt{ b \over c} = s_c. <math>

The tangent through point P to the downward parabola has slope:

<math> {d z_b \over d x} = - {x \over b} = { - \sqrt{ b c } \over b} = - \sqrt{ {c \over b} } = s_b. <math>

The products of the two slopes is

<math> s_c s_b = - \sqrt{ {b \over c}} \sqrt{ {c \over b}} = -1. <math>

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]:

<math> x = \sqrt{\xi \eta} \cos \phi, <math>
<math> y = \sqrt{\xi \eta} \sin \phi, <math>
<math> z = \begin{matrix}\frac{1}{2}\end{matrix} ( \xi - \eta ). <math>
<math>

\begin{vmatrix}dx\\dy\\dz\end{vmatrix} = \begin{vmatrix}

\frac{1}{2}\sqrt{\frac{\xi}{\eta}}\cos\phi

&\frac{1}{2}\sqrt{\frac{\eta}{\xi}}\cos\phi &-\sqrt{\xi\eta}\sin\phi\\

\frac{1}{2}\sqrt{\frac{\xi}{\eta}}\sin\phi

&\frac{1}{2}\sqrt{\frac{\eta}{\xi}}\sin\phi &\sqrt{\xi\eta}\cos\phi\\ -\frac{1}{2}&\frac{1}{2}&0 \end{vmatrix} \cdot \begin{vmatrix}d\eta\\d\xi\\d\phi\end{vmatrix} <math>

See also: spherical coordinates, cylindrical coordinates, Cartesian coordinates.

Reference

  • Menzel, Donald H., Mathematical Physics, Dover Publications, 1961.
Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools