Stereographic Projection is mathematical procedure to map non-eucledian space onto a flat or eucledian surface

1. Presentation On :
STEREOGRAPHIC PROJECTION
Presented By : RAJAT SHARMA MSC-1 (4006)
GHG KHALSA COLLEGE, GURUSAR SADHAR , LUDHIANA.

2. INTRODUCTION
Thestereographic projection is a particularmapping
(
function ) that projects a sphere onto a plane.
In layman's terms, it is a way of picturing a sphere as
a plane.

3. L
N(0,0,1)
O X
X'
Z
Z'
The equation of sphere (orange) with centre at origin (0,0,0)and unit radius in three dimensional space is :x^2
+ y^2 + z^2 = 1. Let N = (0,0,1) be the projection point and let S be the rest of the sphere.The plane Z = 0
(green) runs through the centre of the sphere.The equator (white) is the intersection of this sphere with the
plane Z = 0.
For any point P on S, there is a unique line through N and P, and this line intersects the plane Z = 0 in exactly
one point P′.
We define the stereographic projection of P to be this point P′ in the plane.
S
.P(x,y,z)
.P'(X,Y)
DEFINITION

4. L
N(0,0,1)
O X
X'
Z
Z'
In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse
are given as follows:
The equation of line joining N(0,0,1) , P(x,y,z) and P'(X,Y,0) in cartesian co-ordinate system can be given by :
(x - 0) ÷ (X - x) = ( y - 0) ÷ (Y - y) = (z - 1) ÷ (-z)
From 1st and 3rd equality after little simplification we get: X = [ x ÷ (1 - z)].......... 1
Similarly from 2nd and 3rd inequality we get Y = [ y ÷ ( 1 - z)].................2.
Therefore : Also solving (1) & (2) for x & y we get
S
.P(x,y,z)
.P'(X,Y)
FORMULATION
(X,Y) = (x÷(1-z) , y÷(1-z) ).
x = 2X ÷ ( 1 + X^2 + Y^2)
And other similar relations
for y and z.

5. L
N(0,0,1)
O X
X'
Z
Z'
S
.P(x,y,z)
.P'(X,Y)
PROPERTIES OF STEROGRAPHIC PROJECTION
•
•
•
•
The projection is defined on the entire sphere, except at one point: the projection
point N (0,0,1).
The mapping is smooth (continous) and bijective (one-one & onto).
The mapping is conformal , meaning that it preserves angles at which curves meet.
This bijective mapping is neither isometric nor area preserving.

6. HISTORY




The technique of stereographic projection was known to
Ptolemy and Hipparchus they called it planisphere projection.
The method is used by early Egyptians to create a map of
stars in the night sky popularly known as celestial charts.
In 1695, Edmund Halley ,who had much interest in star charts,
gave first mathematical proof for conformal nature of
stereographic projection.
In 16th and 17th century the angle preserving property of this
clever method is used to make well and accurate maps of
Eastern and Western hemispheres of spheroid earth on a
plane (paper).

7. APPLICATIONS





Stereographic construction have wide applications in various disciplines of
ma ematics like
Complex Analysis and
Ari emetic Geometry.
F reliable navigation ,
Cartography owes its direction preserving navigational
maps to e conf mal nature of stereographic projection.
InPlanetary Science , impact craters on a planetary body are mapped using is
me od because stereographic is e only projection capable of mapping circles on a
sphere to circles on a plane.
InCrysta ography , e ientation of crystal axes and faces can be determined
from electron diffration pattern by using stereographic projection as e diffration
pattern is often a series of curves m ting at e projection point.
Inphotography , camera software uses is technique to fix e warped image from
a fish-eye(wide angle) lens to create a pan amic image.