How are Stereographic Projections Derived in Differential Geometry?

In summary, the derivation for the equations for stereographic projection is straightforward, but the other direction is more difficult.
  • #1
niall14
5
0
In cartesian coordinates (x.y,z) on the and (X,Y) on the plane, the projection and its inverse are given by the following formulae:

(X,Y)=(x/1-z,y/1-z)

(x,y,z)=(2X/1+X^2+Y^2, 2Y/1+X^2+Y^2, -1+X^2+Y^2/1+X^2 +Y^2)

This relates to the field of differntial geo.Anybody have a proof to where thes equations are derived?

Thanks very much
 
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  • #2
The derivation is straightforward if not somewhat cumbersome. Do you know how stereographic projection is defined? You take the straightline through a pole and a point on the sphere and project it onto the plane. Consider a fixed point on the sphere (x,y,z) and find the line through the North pole and (x,y,z). Then find where it intersects the plane z=0.
 
  • #3
thanks for your reply.not really to sure what u mean by 'project'.if you could give me the first few lines of the proof I am sure i could figure it out.
 
  • #4
I'll give you one direction, the other follows similarly but is definitely the harder direction. Consider the north pole [itex] (0,0,1) [/itex] and let [itex] (x_0,y_0,z_0) [/itex] be a point on the sphere. Then the line between the north pole and this point is given by
[tex] x = x_0 t, \qquad y = y_0 t, \qquad z = 1 + (z_0 - 1)t [/tex]
which is easily calculated via basic geometric results. Notice that when [itex] t=0 [/itex] we get the North pole, and when [itex] t = 1 [/itex] we get the point on the sphere. Now we want to calculate where this line hits the z=0 plane, so in particular
[tex] z= 0 = 1+(z_0 - 1)t \quad \Rightarrow t = \frac1{1-z_0} [/tex]
Substituting this value into the equation of the line we get the intersection of the line with the plane
[tex] \left( \frac{x_0}{1-z_0}, \frac{y_0}{1-z_0}, 0 \right) [/tex]
and we drop the third coordinate since we want to identify the sphere with the plane.

Again, the other direction is a bit harder and more technical, but follows precisely the same idea. Give it a try yourself and see if you can get the right answer.
 
  • #5
thanks ill give it a go
 

What is a stereographic projection?

A stereographic projection is a method of mapping points on a sphere onto a flat surface, typically a plane. It preserves angles and shapes, but distorts areas and distances.

What is the purpose of a stereographic projection?

Stereographic projections are commonly used in cartography to create maps of the Earth's surface. They also have applications in mathematics, crystallography, and astronomy.

How is a stereographic projection created?

A stereographic projection is created by projecting points on a sphere onto a tangent plane that touches the sphere at a single point, known as the "point of projection". This is typically done using mathematical equations or geometric constructions.

What are the advantages of using a stereographic projection?

One advantage of a stereographic projection is that it preserves angles, making it useful for navigational purposes. It also shows the entire sphere on a single flat surface, making it more visually appealing and easier to interpret.

What are the limitations of a stereographic projection?

One limitation of a stereographic projection is that it distorts distances and areas, particularly near the edges of the map. It also cannot accurately represent the entire surface of a sphere, as points near the "point of projection" are disproportionately enlarged.

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