Coordinates and parameterizations

For any open set $U \subseteq M$ and function $\phi: U \rightarrow {\mathbb{R}}^n$ such that $\phi$ is a homeomorphism onto a subset of ${\mathbb{R}}^n$, the pair $(U,\phi)$ is called a coordinate neighborhood (or chart in some literature). The values $\phi(p)$ for some $p \in U$ are called the coordinates of $p$.

Example 8..11 (Coordinate Neighborhoods on ${\mathbb{S}}^1$)   A simple example can be obtained for the circle $M = {\mathbb{S}}^1$. Suppose $M$ is expressed as the unit circle embedded in ${\mathbb{R}}^2$ (the set of solutions to $x^2 + y^2 = 1$). Let $(x,y)$ denote a point in ${\mathbb{R}}^2$. Let $U$ be the subset of ${\mathbb{S}}^1$ for which $x > 0$. A coordinate function $\phi: U \rightarrow (-\pi/2,\pi/2)$, can be defined as $\phi(x,y) = \tan^{-1}(y/x)$.

Let $W = \phi(U)$ (the range of $\phi$) for some coordinate neighborhood $(U,\phi)$. Since $U$ and $W$ are homeomorphic via $\phi$, the inverse function $\phi^{-1}$ can also be defined. It turns out that the inverse is the familiar idea of a parameterization. Continuing Example 8.11, $\phi^{-1}$ yields the mapping $\theta \mapsto (\cos\theta,\sin\theta)$, which is the familiar parameterization of the circle but restricted to $\theta \in (-\pi/2,\pi/2)$. $\blacksquare$

To make differentiation work at a point $p \in M$, it will be important to have a coordinate neighborhood defined over an open subset of $M$ that contains $p$. This is mainly because defining derivatives of a function at a point requires that an open set exists around the point. If the coordinates appear to have no boundary, then this will be possible. It is unfortunately not possible to cover all of $M$ with a single coordinate neighborhood, unless $M = {\mathbb{R}}^n$ (or $M$ is at least homeomorphic to ${\mathbb{R}}^n$). We must therefore define multiple neighborhoods for which the domains cover all of $M$. Since every domain is an open set, some of these domains must overlap. What happens in this case? We may have two or more alternative coordinates for the same point. Moving from one set of coordinates to another is the familiar operation used in calculus called a change of coordinates. This will now be formalized.

Figure 8.9: An illustration of a change of coordinates.

Suppose that $(U,\phi)$ and $(V,\psi)$ are coordinate neighborhoods on some manifold $M$, and $U \cap V \not = \emptyset$. Figure 8.9 indicates how to change coordinates from $\phi$ to $\psi$. This change of coordinates is expressed using function composition as $\psi \circ \phi^{-1}: {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$ ($\phi^{-1}$ maps from ${\mathbb{R}}^n$ into $M$, and $\psi$ maps from a subset of $M$ to ${\mathbb{R}}^n$).

Example 8..12 (Change of Coordinates)   Consider changing from Euler angles to quaternions for $M = SO(3)$. Since $SO(3)$ is a 3D manifold, $n=3$. This means that any coordinate neighborhood must map a point in $SO(3)$ to a point in ${\mathbb{R}}^3$. We can construct a coordinate function $\phi: SO(3) \rightarrow {\mathbb{R}}^3$ by computing Euler angles from a given rotation matrix. The functions are actually defined in (3.47), (3.48), and (3.49). To make this a coordinate neighborhood, an open subset $U$ of $M$ must be specified.

We can construct another coordinate function $\psi: SO(3) \rightarrow {\mathbb{R}}^3$ by using quaternions. This may appear to be a problem because quaternions have four components; however, the fourth component can be determined from the other three. Using (4.24) to (4.26), the $a$, $b$, and $c$ coordinates can be determined.

Now suppose that we would like to change from Euler angles to quaternions in the overlap region $U \cap V$, in which $V$ is an open set on which the coordinate neighborhood for quaternions is defined. The task is to construct a change of coordinates, $\psi \circ \phi^{-1}$. We first have to invert $\phi$ over $U$. This means that we instead need a parameterization of $M$ in terms of Euler angles. This is given by (3.42), which yields a rotation matrix, $\phi^{-1}(\alpha,\beta,\gamma) \in SO(3)$ for $\alpha$, $\beta$, and $\gamma$. Once this matrix is determined, then $\psi$ can be applied to it to determine the quaternion parameters, $a$, $b$, and $c$. This means that we have constructed three real-valued functions, $f_1$, $f_2$, and $f_3$, which yield $a = f_1(\alpha,\beta,\gamma)$, $b = f_2(\alpha,\beta,\gamma)$, and $c = f_3(\alpha,\beta,\gamma)$. Together, these define $\psi \circ \phi^{-1}$. $\blacksquare$

There are several reasons for performing coordinate changes in various contexts. Example 8.12 is motivated by a change that frequently occurs in motion planning. Imagine, for example, that a graphics package displays objects using quaternions, but a collision-detection algorithm uses Euler angles. It may be necessary in such cases to frequently change coordinates. From studies of calculus, you may recall changing coordinates to simplify an integral. In the definition of a smooth manifold, another motivation arises. Since coordinate neighborhoods are based on homeomorphisms of open sets, several may be required just to cover all of $M$. For example, even if we decide to use quaternions for $SO(3)$, several coordinate neighborhoods that map to quaternions may be needed. On the intersections of their domains, a change of coordinates is necessary.

Now we are ready to define a smooth manifold. Changes of coordinates will appear in the manifold definition, and they must satisfy a smoothness condition. A smooth structure^8.5 on a (topological) manifold $M$ is a family^8.6 ${\cal U}= \{U_\alpha,\phi_\alpha\}$ of coordinate neighborhoods such that:

1. The union of all $U_\alpha$ contains $M$. Thus, it is possible to obtain coordinates in ${\mathbb{R}}^n$ for any point in $M$.
2. For any $(U,\phi)$ and $(V,\psi)$ in ${\cal U}$, if $U \cap V \not = \emptyset$, then the changes of coordinates, $\psi \circ \phi^{-1}$ and $\phi \circ \psi^{-1}$, are smooth functions on $U \cap V$. The changes of coordinates must produce diffeomorphisms on the intersections. In this case, the coordinate neighborhoods are called compatible.
3. The family ${\cal U}$ is maximal in the sense that if some $(U,\phi)$ is compatible with every coordinate neighborhood in ${\cal U}$, then $(U,\phi)$ must be included in ${\cal U}$.

A well-known theorem (see [133], p. 54) states that if a set of compatible neighborhoods covers all of $M$, then a unique smooth structure exists that contains them.^8.7 This means that a differential structure can often be specified by a small number of neighborhoods, and the remaining ones are implied.

A manifold, as defined in Section 4.1.2, together with a smooth structure is called a smooth manifold.^8.8

Example 8..13 ( ${\mathbb{R}}^n$ as a Smooth Manifold)   We should expect that the concepts presented so far apply to ${\mathbb{R}}^n$, which is the most straightforward family of manifolds. A single coordinate neighborhood ${\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$ can be used, which is the identity map. For all integers $n \in \{1,2,3\}$ and $n > 4$, this is the only possible smooth structure on ${\mathbb{R}}^n$. It is truly amazing that for ${\mathbb{R}}^4$, there are uncountably many incompatible smooth structures, called exotic ${\mathbb{R}}^4$ [291]. There is no need to worry, however; just use the one given by the identity map for ${\mathbb{R}}^4$. $\blacksquare$

Example 8..14 ( ${\mathbb{S}}^n$ as a Smooth Manifold)   One way to define ${\mathbb{S}}^n$ as a smooth manifold uses $2(n+1)$ coordinate neighborhoods and results in simple expressions. Let ${\mathbb{S}}^n$ be defined as

$${\mathbb{S}}^n = \{(x_1,\ldots,x_{n+1}) \in {\mathbb{R}}^{n+1} \vert \;x_1^2 + \cdots + x_{n+1}^2 = 1\}.$$ (8.22)

The domain of each coordinate neighborhood is defined as follows. For each $i$ from $1$ to $n+1$, there are two neighborhoods:

$$U^+_i = \{(x_1,\ldots,x_{n+1}) \in {\mathbb{R}}^{n+1} \vert \;x_i > 0\}$$ (8.23)

and

$$U^-_i = \{(x_1,\ldots,x_{n+1}) \in {\mathbb{R}}^{n+1} \vert \;x_i < 0\} .$$ (8.24)

Each neighborhood is an open set that covers half of ${\mathbb{S}}^n$ but misses the great circle at $x_i = 0$. The coordinate functions can be defined by projection down to the $(n-1)$-dimensional hyperplane that contains the great circle. For each $i$,

$$\phi^+_i(x_1,\ldots,x_{n+1}) = (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)$$ (8.25)

over $U^+_i$. Each $\phi^-_i$ is defined the same way, but over $U^-_i$. Each coordinate function is a homeomorphism from an open subset of ${\mathbb{S}}^n$ to an open subset of ${\mathbb{R}}^n$, as required. On the subsets in which the neighborhoods overlap, the changes of coordinate functions are smooth. For example, consider changing from $\phi^+_i$ to $\phi^-_j$ for some $i \not = j$. The change of coordinates is a function $\phi^-_j \circ (\phi^+_i)^{-1}$. The inverse of $\phi^+_i$ is expressed as

$$\begin{split}& (\phi^+_i)^{-1}(x_1,\ldots,x_{i-1},x_{i+1},\ld... ...- x_{i+1}^2-\cdots-x_n^2},x_{i+1},\ldots,x_{n+1}) . \end{split}$$ (8.26)

When composed with $\phi^-_j$, the $j$th coordinate is dropped. This yields

$$\begin{split}\phi^-_k \circ (\phi^+_i)^{-1}&(x_1,\ldots,x_{i-... ... & \;\;x_{i+1},\ldots,x_{j-1},x_{j+1},\ldots,x_n) , \end{split}$$ (8.27)

which is a smooth function over the domain $U^+_i$. Try visualizing the changes of coordinates for the circle ${\mathbb{S}}^1$ and sphere ${\mathbb{S}}^2$.

The smooth structure can alternatively be defined using only two coordinate neighborhoods by using stereographic projection. For ${\mathbb{S}}^2$, one coordinate function maps almost every point $x \in {\mathbb{S}}^2$ to ${\mathbb{R}}^2$ by drawing a ray from the north pole to $x$ and mapping to the point in the $x_3 = 0$ plane that is crossed by the ray. The only excluded point is the north pole itself. A similar mapping can be constructed from the south pole. $\blacksquare$

Example 8..15 ( ${\mathbb{RP}}^n$ as a Smooth Manifold)   This example is particularly important because ${\mathbb{RP}}^3$ is the same manifold as $SO(3)$, as established in Section 4.2.2. Recall from Section 4.1.2 that ${\mathbb{RP}}^n$ is defined as the set of all lines in ${\mathbb{R}}^{n+1}$ that pass through the origin. This means that for any $\alpha \in {\mathbb{R}}$ such that $\alpha \not = 0$, and any $x \in {\mathbb{R}}^{n+1}$, both $x$ and $\alpha x$ are identified. In projective space, scale does not matter.

A smooth structure can be specified by only $n+1$ coordinate neighborhoods. For each $i$ from $1$ to $n+1$, let

$$\phi_i(x_1,\ldots,x_{n+1}) = (x_1/x_i,\ldots,x_{i-1}/x_i,x_{i+1}/x_i,\ldots,x_n/x_i) ,$$ (8.28)

over the open set of all points in ${\mathbb{R}}^{n+1}$ for which $x_i \not = 0$. The inverse coordinate function is given by

$$\phi^{-1}_i(z_1,\ldots,z_n) = (z_1,\ldots,z_{i-1},1,z_i,\ldots,z_{n+1}) .$$ (8.29)

It is not hard to verify that these simple transformations are smooth on overlapping neighborhoods.

A smooth structure over $SO(3)$ can be derived as a special case because $SO(3)$ is topologically equivalent to ${\mathbb{RP}}^3$. Suppose elements of $SO(3)$ are expressed using unit quaternions. Each $(a,b,c,d)$ is considered as a point on ${\mathbb{S}}^3$. There are four coordinate neighborhoods. For example, one of them is

$$\phi_b(a,b,c,d) = (a/b,\;c/b,\;d/b) ,$$ (8.30)

which is defined over the subset of ${\mathbb{R}}^4$ for which $b \not = 0$. The inverse of $\phi_b(a,b,c,d)$ needs to be defined so that a point on $SO(3)$ maps to a point in ${\mathbb{R}}^4$ that has unit magnitude. $\blacksquare$