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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].