On-Line Computer Graphics Notes
An affine space contains both points and vectors. Points are typically used to position ourselves in space and vectors are use to move about in space. However in the definition of an affine space, the operations on vectors are numerous, while the operations on points are sparse.
Here we define a fundamental operation on the points of an affine space, the affine combination. This operation is unique on the set of points, as it will be defined just by the points themselves.
For a postscript version of these notes look here.
What is an Affine Combination?
Given a set of points in an affine space, an affine combination is defined to be the point
where the are scalars and
We note that an affine combination of points defines a new point in space. It is not a general linear combination, but a linear combination of the points where the scalars must sum to one.
An Affine Combination of Two Points
Let and be points in an affine space. Consider the expression
By the axioms of an affine space this equation is meaningful, as is a vector, and therefore so is . Therefore is the sum of a point and a vector, which is again a point in the affine space This point represents a point on the ``line'' that passes through and .
We note that if then is somewhere on the ``line segment'' joining and .
This expression allows us to define the affine combination on two points. We define
to be the point defined by
We can then define an affine combination of two points and to be
The form is shown to be an affine transformation by setting . The form can be seen to be equivalent to the form , by setting .
An Affine Combination of an Arbitrary Number of Points
We can use the above argument this to define an affine combination of an arbitrary number of points. If are points and are scalars such that , then
is defined to be equivalent to the point
which uses the fact that .
Example -- Affine Combinations in Triangles
To construct an excellent example of an affine combination consider three points , and . A point defined by
where , gives a point in the triangle . We note that the definition of affine combination defines this point to be
The following illustration shows the point generated when and .
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.
Affine combinations define a new point from a set of points in an affine space by constructing linear combinations of points with the restrictions that the coefficients of the linear combinations must sum to one. This is the only operation on the points of an affine space. We use this concept extensively in geometric modeling, especially in the definitions of convex combinations and barycentric coordinates.
One should remember that an affine combination is a linear combination but not the other way around.
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