Affine space/Affine bases/Introduction/Section


A family of points , , in an affine space over a -vector space is called an affine basis of , if there exists an such that the family of vectors

is a basis

of .

Because of

the basis vectors with respect to the origin can be expressed as linear combinations of the corresponding vectors with respect to any other origin point of the family. Therefore, the property of being an affine basis is independent from the chosen .

The barycentric coordinates in the plane, where the affine basis points are the corners of a triangle.
The barycentric coordinates in the plane, where the affine basis points are the corners of a triangle.


For a family , , of points in an affine space and a tuple , , in satisfying

(for infinite, only finitely many of the are allowed to be different from ), the sum is called a barycentric combination of the . The corresponding point in is given by

where is an arbitrary point in .


For a family , , of points in an affine space , a barycentric combination

defines a unique point in .

Proof



Let , , denote an affine basis in an affine space over the -vector space . Then, for every point , there exists a unique barycentric representation

Let be fixed. In , we have a unique representation

We set

Then , and

Therefore, there exists such a representation with as origin. Uniqueness follows from the facts that the , , are uniquely determined as the coefficients of the vector space basis, and that is determined by the baryzentric condition.


The colors for additive mixing with primary colors red, blue, green (this corresponds to the three cone cells of the human eye). The eye is only interested in the mixing of the three colors, therefore, only linear combinations '"`UNIQ--postMath-00000037-QINU`"' with '"`UNIQ--postMath-00000038-QINU`"' (and non-negative coefficients) are relevant. Thus, colors are described by barycentric coordinates, saving one dimension.
The colors for additive mixing with primary colors red, blue, green (this corresponds to the three cone cells of the human eye). The eye is only interested in the mixing of the three colors, therefore, only linear combinations with (and non-negative coefficients) are relevant. Thus, colors are described by barycentric coordinates, saving one dimension.


Let , , denote an affine basis in an affine space over the -vector space . For a point , the uniquely determined numbers

such that

holds, are called the barycentric coordinates

of .


Let , , be an affine basis in an affine space over the -vector space . Then the point () has the barycentric coordinates , where the is at the -th place ( being finite and ordered).