For a pdf version of these notes look
here.
Points in an affine space are utilized to position ourselves within the space. The operations on the vectors of an affine space are numerous - addition, scalar multiplication, dot products, cross products - but the operations on the points are limited. In this section we discuss the basic operations on points - affine combinations.
Let
and
be points in an affine space. Consider the
expression
We note that if
then
is somewhere on the line segment
joining
and
.
This expression allows us to define a basic operation on points. We utilize the following notation
We can generalize this to define an affine combination of an arbitrary
number of points. If
are points and
are scalars such that
, then
To construct
an excellent example of an affine combination consider three
points
,
and
. A point
defined by
In fact, it can be easily shown that if
then the point
will be
within (or on the boundary) of the triangle. If any
is
less than zero or greater than one, then the point will lie outside
the triangle. If any
is zero, then the point will lie on
the boundary of the triangle.
![]() |
In this form, the values
are called the
barycentric coordinates of
relative to the points
Vectors can also be expressed in barycentric form by letting
To give a simple example of barycentric coordinates, consider two
points
and
in the plane. If
and
are scalars such that
, then the point
defined by
To give a slightly more complex
example of barycentric coordinates, consider three points
,
,
in the plane. If
,
,
are scalars such that
, then
the point
defined by
Thus barycentric coordinates are another method of introducing coordinates into an affine space. If the coordinates sum to one, they represent a point ; if the coordinates sum to zero, they represent a vector.
Given a set of points
, we can form affine
combinations of these points by selecting
, with
and form
the point
If each
is such that
, then the
points
is called a convex combination of the points
.
To give a simple example of this, consider two points
and
.
Any point
on the line passing through these two points can be
written as
which is an affine
combination of the two points. The points
and
in the
following figure are affine combinations of
and
.
However, the point
is a convex combination, as
, and any point on the line segment
joining
and
can be written in this way.
Given any set of points, we say that the set is a convex set, if given any two points of the set, any convex combination of these two points is also in the set. The following figure illustrates both a convex set (on the left) and a non-convex set (on the right).
This concept is actually quite intuitive, in that if one can draw a straight line between two points of the set that is not completely contained within the set, the the set is non-convex.
The set of all points
that can be written as convex combinations
of
is called the convex hull of
the points
. This convex hull is the
smallest convex set that contains the set of points
. The following figure illustrates the convex hull of a
set of six points:
One of the six points does not contribute to the boundary of the convex hull. If one looked closely at the coordinates of the point, one would find that this point could be written as a convex combination of the other five.
Most of this material was adapted from Tony DeRose's wonderful treatment of affine spaces given in [1].