Scribd
Upload
262 views4 pages

Stereographic Projection

1) The document describes stereographic projection, which maps points on a sphere to the extended complex plane. 2) It establishes a one-to-one correspondence between points on the unit sphere (excluding the north pole) and the complex plane by projecting points onto the complex plane along lines through the north pole. 3) The mapping establishes that stereographic projection transforms the sphere onto the entire extended complex plane, providing a coordinate system to represent spherical geometry on a flat plane.

Uploaded by

BERNARDITA E. GUTIB
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
262 views4 pages

Stereographic Projection

1) The document describes stereographic projection, which maps points on a sphere to the extended complex plane. 2) It establishes a one-to-one correspondence between points on the unit sphere (excluding the north pole) and the complex plane by projecting points onto the complex plane along lines through the north pole. 3) The mapping establishes that stereographic projection transforms the sphere onto the entire extended complex plane, providing a coordinate system to represent spherical geometry on a flat plane.

Uploaded by

BERNARDITA E. GUTIB
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
You are on page 1/ 4

Stereographic Projection

Consider the unit sphere S in R3 given by


S = {(x1, x2, x3) : x12, x22, x32 = 1}.

Let N = (0, 0, 1) which is called north pole on S.

We identify ℂ with

{(x1, x2, 0) : x1, x2 ∈ ℝ}

So that ℂ cuts along the equator.

For each z ∈ ℂ, consider the straight line


in ℝ3 through z and N. The line intersects
S at exactly one point P ≠ N.

If |z| > 1, then P is in the northern hemisphere of S.


If |z| < 1, then P is in the southern hemisphere of S.

If z=0 , then P is actually the South Pole(0,0 ,−1)

If|z|=1, then z=P , ie, z=P lies along the equator

As|z|→ ∞, P approaches to N .

Recall first that the symmetric equation of line L in R3 (use x , y , z coordinate) passing through the
points P0=( x 0 , y 0 , z 0) and P1=( x 1 , y 1 , z 1) is given by (using P0)

x−x 0 y − y 0 z−z 0
= = ,
x1− x0 y 1− y 0 z 1−z 0

that is, for some t ∈ R , the parametric equation of the line passing through P0 and P1 is as follows:

x=x 0 + ( x 1−x 0 ) t , y= y 0 + ( y 1 − y 0 ) t , z=z 0+ ( z 1−z 0 ) t

More precisely,
L=¿

Suppose that z=x +iy and P=(x 1 , x 2 , x 3) be the corresponding point on S at which the line through
z and N intersects S. The line through P0=(x 0 , y 0 , z 0) and N=(0,0,1) in R3 (use x 1 , x 2 , x 3
coordinate) is given by

{ x 0−x 0 t , y 0− y 0 t , z 0−z 0 t :t ∈ R } .
Taking( x 0 , y 0 , z 0 )=( x , y , 0), we have

{( (1−t ) x , ( 1−t ) y ,t ) : t ∈ R } .
Hence, for some t ∈ R

x 1=( 1−t ) x , x 2=( 1−t ) y ∧x3 =t .

But P=x 1 , x 2 , x 3 is on S and so ( 1−t )2 x 2 + ( 1−t )2 y 2 +t 2=1

Now,

( 1−t )2 x 2 + ( 1−t )2 y 2 +t 2=1

( 1−t )2 ( x 2+ y 2)+t 2=1

( 1−2t +t2 )|z|2 +t 2−1=0


(1+| z|2) t 2−2|z|2 t+|z|2−1=0

By quadratic formula,
2
−B ± √ ❑
t=
❑
2
−(−2|z| )± √❑
t=
❑
2
2|z| ± √ ❑
t=
❑
2
2|z| ± √ ❑
t=
❑
2
2|z| ±2
t=
2 ( 1+|z| )
2
2
|z| ± 1
t= 2
1+|z|

2
|z| ± 1
Hence, t= 2 since Z ≠ N and t ≠ 1.
1+|z|

Solving for x 1 , x 2 , x 3, we have

( )
2
|z| ± 1 2x 2 ℜ( x) z + z
x 1=( 1−t ) x = 1− 2 x= 2 = 2 = 2 ,
|z| +1 |z| +1 |z| +1 |z| +1

( )
2
|z| ± 1 2y 2∈(z) −i(z−z )
x 2=( i−t ) y = i− 2 y= 2 = 2 = 2
,
|z| +1 |z| +1 |z| + 1 |z| +1
2
|z| ± 1
x 3=t= 2
1+|z|

( )
2
z + z −i(z− z) |z| ± 1
So, P= 2
, 2
, 2
|z| +1 |z| + 1 1+|z|
Thus,
2
x1 +i x 2 2z 1+|z|
= × =z
1−x3 1+|z|2 2

x 1+i x 2
Therefore, if we start with P=x 1 , x 2 , x 3 ∈ S, we have z= ∈ C∞.
1−x 3

Define the mapping φ : S →C ∞ by

x1 x2
φ ( x ¿ ¿ 1 , x 2 , x 3)={ +i ∞ ( x ¿ ¿ 1 , x 2 , x 3 )≠ N ( x ¿ ¿ 1 , x 2 , x 3 )=N ¿ ¿¿
1−x3 1−x 3

This mapping φ is a one-to-one correspondence between S and C ∞ .

The sphere S is called the Riemann sphere and this 1-1 correspondence is called the stereographic
projection.

You might also like