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 point of tangency, with one exception: the center of projection, antipodal to the point of tangency, is not projected to any point (sometimes it is said to be projected to the point at infinity).
Two remarkable properties of this projection were demonstrated mathematically by Hipparchus:
- This mapping is conformal, i.e. it preserves angles at which curves cross each other, and
- This mapping transforms circles on the surface of the sphere that do not pass through the center of projection, to circles in the plane. It transforms circles on the sphere that do pass through the center of projection to straight lines in the plane, which are sometimes thought of as circles through the 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 ρP (radial distance away from origin) and θP. Then the projection is
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If θL is, instead, the latitude, then the equation for ρP changes to
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or, equivalently,
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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
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Substituting equation (1) we obtain
- .
Equation (3) can be solved for θL:
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Substitute equation (5) into equation (4), then simplify,
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Apply the following trigonometric identity
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to equation (6), yielding
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Divide both numerator and denominator by tangent of π/2:
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Let b=-1/a, then
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therefore a loxodrome on a stereographic projection is a equiangular spiral.
See also
