For any open set
and function
such that is a homeomorphism onto a subset of
, the pair is called a *coordinate neighborhood*
(or *chart* in some literature). The values for some are called the *coordinates* of .

Let
(the range of ) for some coordinate
neighborhood . Since and are homeomorphic via
, the inverse function can also be defined. It
turns out that the inverse is the familiar idea of a
*parameterization*. Continuing Example 8.11,
yields the mapping
, which is the familiar parameterization of
the circle but restricted to
.

To make differentiation work at a point , it will be
important to have a coordinate neighborhood defined over an open
subset of that contains . 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 with a single coordinate neighborhood, unless
(or
is at least homeomorphic to
). We must therefore define
multiple neighborhoods for which the domains cover all of .
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.

Suppose that and are coordinate neighborhoods on some manifold , and . Figure 8.9 indicates how to change coordinates from to . This change of coordinates is expressed using function composition as ( maps from into , and maps from a subset of to ).

We can construct another coordinate function 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 , , and coordinates can be determined.

Now suppose that we would like to change from Euler angles to
quaternions in the overlap region , in which is an open
set on which the coordinate neighborhood for quaternions is defined.
The task is to construct a change of coordinates,
. We first have to invert over . This means that
we instead need a parameterization of in terms of Euler angles.
This is given by (3.42), which yields a rotation matrix,
for , , and
. Once this matrix is determined, then can be applied
to it to determine the quaternion parameters, , , and . This
means that we have constructed three real-valued functions, ,
, and , which yield
,
, and
.
Together, these define
.

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 . For example, even if we decide to use quaternions for , 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 is a
family^{8.6}
of coordinate neighborhoods such that:

- The union of all contains . Thus, it is possible to obtain coordinates in for any point in .
- For any and in , if
, then the changes of coordinates,
and
, are smooth functions on . The
changes of coordinates must produce diffeomorphisms on the
intersections. In this case, the coordinate neighborhoods are called
*compatible*. - The family is maximal in the sense that if some is compatible with every coordinate neighborhood in , then must be included in .

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

(8.22) |

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

(8.23) |

and

(8.24) |

Each neighborhood is an open set that covers half of but misses the great circle at . The coordinate functions can be defined by projection down to the -dimensional hyperplane that contains the great circle. For each ,

(8.25) |

over . Each is defined the same way, but over . Each coordinate function is a homeomorphism from an open subset of to an open subset of , as required. On the subsets in which the neighborhoods overlap, the changes of coordinate functions are smooth. For example, consider changing from to for some . The change of coordinates is a function . The inverse of is expressed as

(8.26) |

When composed with , the th coordinate is dropped. This yields

(8.27) |

which is a smooth function over the domain . Try visualizing the changes of coordinates for the circle and sphere .

The smooth structure can alternatively be defined using only two
coordinate neighborhoods by using *stereographic projection*. For
, one coordinate function maps almost every point
to
by drawing a ray from the north pole to and mapping to
the point in the 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.

A smooth structure can be specified by only coordinate neighborhoods. For each from to , let

(8.28) |

over the open set of all points in for which . The inverse coordinate function is given by

(8.29) |

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

A smooth structure over can be derived as a special case because is topologically equivalent to . Suppose elements of are expressed using unit quaternions. Each is considered as a point on . There are four coordinate neighborhoods. For example, one of them is

(8.30) |

which is defined over the subset of for which . The inverse of needs to be defined so that a point on maps to a point in that has unit magnitude.

Steven M LaValle 2020-08-14