Stereographic projection

From Academic Kids

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In cartography and geometry, the stereographic projection is a mapping that projects each point on a sphere onto a tangent plane along a straight line from the antipode of the point of tangency (with one exception: the center of projection, antipodal to the point of tangency, is not projected to any point in the Euclidean plane; it is thought of as corresponding to a "point at infinity"). One approaches that point at infinity by continuing in any direction at all; in that respect this situation is unlike the projective plane, which has many points at infinity.

Contents

Notable properties

Two notable properties of this projection were demonstrated by Hipparchus:

  • this mapping is conformal, i.e., it preserves the angles at which curves cross each other, and
  • this mapping transforms those circles on the surface of the sphere that do not pass through the center of projection to circles on the plane. It transforms circles on the sphere that do pass through the center of projection to straight lines on the plane (these are sometimes thought of as circles through a point at infinity).
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A stereographic projection is conformal and perspective but not equal area or equidistant.

Formula

Polar coordinates

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Stereographic_Projection_Northern_Hemisphere.png
Stereographic projection of the Northern hemisphere

On a sphere, let φ be azimuth and θ be co-latitude (angular distance from the pole). Let R be the radius of the sphere. Let the points of the sphere be projected stereographically onto a plane which is tangent to the pole. Let the points of the projection have coordinates ρP (radial distance away from origin) and θP. Then the projection is

<math> \theta_P = \phi, \qquad \qquad (1) <math>
<math> \rho_P = 2 R \tan {\theta \over 2}. \qquad \qquad (2) <math>

If θL is, instead, the latitude, then the equation for ρP changes to

<math> \rho_P = 2 R \tan {{\pi \over 2} - \theta_L \over 2 } \qquad \qquad (3) <math>

or, equivalently,

<math> \rho_P = 2 R ( \sec \theta_L - \tan \theta_L). <math>

Cartesian coordinates

There are a number of ways to perform stereographic projection onto a sphere, based on your choice of where you put the plane and the sphere. One can treat the plane as being complex numbers, and use the following pair of transformations:

<math>\begin{pmatrix} \xi \\ \eta \\ \zeta \end{pmatrix} =

\begin{pmatrix} \frac {2 a} {1 + \bar z z} \\ \\ \frac {2 b} {1 + \bar z z} \\ \\ \frac {1 - \bar z z} {1 + \bar z z} \end{pmatrix} <math>

<math>z = a + b i = \frac {\xi + \eta i} {1 - \zeta}. <math>

Loxodromes on a stereographic projection

It is possible to find the equations of loxodromes on the stereographic projection. A loxodrome on a sphere is described by

<math> \phi = a \ln \left| \tan \left( {\theta_L \over 2} + {\pi \over 4} \right) \right|. <math>

Substituting equation (1) we obtain

<math> \theta_P = a \ln \left| \tan \left( {\theta_L \over 2} + {\pi \over 4} \right) \right|. \qquad \qquad (4) <math>

Equation (3) can be solved for θL:

<math> \theta_L = {\pi \over 2} - 2 \arctan \rho_P. \qquad \qquad (5)<math>

Substitute equation (5) into equation (4), then simplify,

<math> \theta_P = a \ln \left| \tan \left( {\pi \over 2} - \arctan \rho_P \right) \right|. \qquad \qquad (6)<math>

Apply the following trigonometric identity

<math> \tan \left({\pi \over 2} - \theta\right) = { 1 \over \tan \theta } <math>

to equation (6), yielding

<math> \theta_P = a \ln \left| {1 \over \tan \left( - \arctan \rho_P \right)} \right|<math>
<math> \theta_P = a \ln \left| {1 \over - \rho_P} \right| = a \ln \left| {1 \over \rho_P} \right| = -a \ln \rho_P.<math>

Let b = −1/a; then

<math> \rho_P = e^{b \theta_P}, <math>

therefore a loxodrome on a stereographic projection is a equiangular spiral.

Loxodromes may also found by transforming any point with a Mbius transformation, in particular one with a "characteristic constant" that has an nonzero argument and a modulus not equal to one, and which has fixed points that map to diametrically opposite points on the sphere. Continuous iteration may be done by scaling the log of the characteristic constant. These loxodromes are a family of S-shaped Spirals with varying degrees of "tightness".

See also

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