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Stereographic projection

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

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).

Formula

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 ρ_P (radial distance away from origin) and θ_P. 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 θ_L is, instead, the latitude, then the equation for ρ_P 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>

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 θ_L:

<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 ({\pi \over 2} - \theta) = { 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.

See also

• map projection
• Riemann sphere
• projective geometry