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 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 a point at infinity).
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 ρP (radial distance away from origin) and θP. Then the projection is \n:\n:\nIf θL is, instead, the latitude, then the equation for ρP changes to\n:\nor, equivalently,\n:
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
- .
Substituting equation (1) we obtain
-
Equation (3) can be solved for
θL:
-
Substitute equation (5) into equation (4), then simplify,
-
Apply the following
trigonometric identity
-
to equation (6), yielding
- \n:
Let
b=-1/a, then
-
therefore a loxodrome on a stereographic projection is a
equiangular spiral.
See also
\n* map projection\n*
Riemann sphere\n*
projective geometry
Category:Cartography\nCategory:Geometry
