91Ó°ÊÓ

One way to define a system of coordinates for the sphere \(S^{2}\), given by \(x^{2}+y^{2}+(z-1)^{2}=1\), is to consider the so-called stereographic projection \(\pi: S^{2} \sim\\{N\\} \rightarrow R^{2}\) which carries a point \(p=(x, y, z)\) of the sphere \(S^{2}\) minus the north pole \(N=(0,0,2)\) onto the intersection of the \(x y\) plane with the straight line which connects \(N\) to \(p\) (Fig. 2-12). Let \((u, v)=\pi(x, y, z)\), where \((x, y, z) \in S^{2} \sim\\{N\\}\) and \((u, v) \in x y\) plane. a. Show that \(\pi^{-1}: R^{2} \rightarrow S^{2}\) is given by $$ \pi^{-1}\left\\{\begin{array}{l} x=\frac{4 u}{u^{2}+v^{2}+4} \\ y=\frac{4 v}{u^{2}+v^{2}+4} \\ z=\frac{2\left(u^{2}+v^{2}\right)}{u^{2}+v^{2}+4} \end{array}\right. $$ b. Show that it is possible, using stereographic projection, to cover the sphere with two coordinate neighborhoods.

Step by step solution

01

Understanding Stereographic Projection

Stereographic projection maps a point on the sphere to the plane. The given sphere is defined by the equation \(x^{2}+y^{2}+(z-1)^{2}=1\), which is a sphere of radius 1 centered at (0, 0, 1). We need to find the inverse stereographic projection that maps coordinates \(u, v\) on the plane back to \(x, y, z\) on the sphere.

02

Parametrize the Projection

Given a point \(p=(x, y, z)\) on the sphere \(S^{2}\) minus the north pole \(N=(0, 0, 2)\), we need to find the intersection of the line passing through \(N\) and \(p\) with the \(xy\)-plane.

03

Derive the Formulas

Start by expressing the line passing through points \(N\) and \(p=(x, y, z)\) in parametric form: \((0,0,2) + t(x, y, z - 2) = (tx, ty, 2 + t(z-2))\). The intersection with the \(xy\)-plane occurs when \(2 + t(z-2) = 0\).

04

Solve for t

Set \(2 + t(z-2) = 0\) which simplifies to \(t = \frac{-2}{z-2}\). Substitute this value back into the parametric equation to get the coordinate projections: \(u = \frac{-2x}{z-2}\) and \(v = \frac{-2y}{z-2}\).

05

Find Inverse Formulas

Express \(x, y, z\) in terms of \(u, v\). First note that \(u = \frac{-2x}{z-2}\) and \(v = \frac{-2y}{z-2}\) implies \(x = \frac{-u(z-2)}{2}\) and \(y = \frac{-v(z-2)}{2}\). Use the equation of the sphere \(x^2 + y^2 + (z-1)^2 = 1\) to substitute \(x, y\). After calculation, this results in \(x = \frac{4u}{u^2 + v^2 + 4}\), \(y = \frac{4v}{u^2 + v^2 + 4}\), and \(z = \frac{2(u^2 + v^2)}{u^2 + v^2 + 4}\).

06

Conclude Part a

We have shown that \(\pi^{-1}(u, v) = \left(\frac{4u}{u^2 + v^2 + 4}, \frac{4v}{u^2 + v^2 + 4}, \frac{2(u^2 + v^2)}{u^2 + v^2 + 4}\right)\), which completes part a.

07

Set up Part b

To cover the sphere with two coordinate neighborhoods using stereographic projection, consider projecting from both the north pole \(N = (0, 0, 2)\) and the south pole \(S = (0, 0, 0)\).

08

South Pole Projection

Every point on the sphere, excluding the south pole, can also be projected onto the plane using a similar stereographic projection method centered at the south pole. By symmetry, this projection would cover the hemisphere including the north pole.

09

Combine Projections

By combining these two stereographic projections (one from the north pole and one from the south pole), we obtain two coordinate neighborhoods that cover the entire sphere without singularities at the poles. Thus, the entire sphere can be covered by using two stereographic projections.

10

Conclude Part b

We've shown that it is possible to cover the sphere \(S^2\) with two coordinate neighborhoods using stereographic projection from both poles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Mapping

Inverse mapping is a process wherein we compute the coordinates of points on a sphere from their projections on a plane. This is essential in stereographic projection where we map points from a sphere to a plane and then find their original location on the sphere.

In the exercise, we consider a sphere centered at (0, 0, 1) and exclude the north pole (0, 0, 2). We start with the plane coordinates \((u, v)\) and find the sphere coordinates \((x, y, z)\). Mathematically, the inverse mapping is expressed as:

• \(x = \frac{4u}{u^2 + v^2 + 4} \)
• \(y = \frac{4v}{u^2 + v^2 + 4} \)
• \(z = \frac{2(u^2 + v^2)}{u^2 + v^2 + 4} \)

Understanding this inverse relationship helps us transition smoothly between different coordinate systems and handle projections more effectively.

Coordinate Systems

Coordinate systems are frameworks that help us locate points in a given space. For the given sphere, we're primarily working with spherical and Cartesian coordinates.

In Cartesian coordinates, we describe a point in 3D space as \((x, y, z)\). For the sphere given by \(x^2 + y^2 + (z-1)^2 = 1\), this system helps in defining the shape and position of the sphere.

However, translating these coordinates into a 2D plane requires a different system. Through stereographic projection, we map these 3D coordinates onto a 2D plane with coordinates \((u, v)\). Utilizing both these systems efficiently allows us to switch between 3D and 2D representations and solve problems related to spherical geometry with clarity.

Sphere Covering

Sphere covering involves creating a way to map every point on a sphere to a coordinate system without any gaps or overlaps. In this exercise, we utilize stereographic projection to achieve this by projecting from both the north and south poles.

• North Pole Projection: Starting at the north pole (0, 0, 2), points on the sphere excluding the north pole are mapped onto the plane. This covers most of the sphere but leaves out a small region around the south pole.


• South Pole Projection: Similarly, by projecting from the south pole (0, 0, 0) onto the plane, we cover a different region of the sphere, including the north pole but excluding the south pole.

By combining these two projections, we effectively cover the entire sphere without missing any points. This creates two coordinate neighborhoods, ensuring that every point on the sphere has a corresponding coordinate on the plane.