Information About

Stereographic Projection
  CATEGORIES ABOUT STEREOGRAPHIC PROJECTION
   
cartographic projections
   
projective geometry
   
conformal mapping
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



In , which has many points at infinity.


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).




FORMULA


Polar Coordinates


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

: heta_P = \phi, \qquad \qquad (1)
: ho_P = 2 R an { heta \over 2}. \qquad \qquad (2)

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

  :<math> \phi a \ln \left an \left( { heta_L \over 2} + {\pi \over 4} ight) ight </math>
  :<math> Heta P a \ln \left an \left( { heta_L \over 2} + {\pi \over 4} ight) ight \qquad \qquad (4) </math>
  :<math> Heta P a \ln \left an \left( {\pi \over 2} - \arctan ho_P ight) ight \qquad \qquad (6)</math>
  :<math> Heta P a \ln \left {1 \over an \left( - \arctan ho_P ight)} ight</math>
  :<math> Heta P a \ln \left {1 \over - ho_P} ight = a \ln \left {1 \over ho_P} ight = -a \ln ho_P</math>