Affine space/Affine bases/Introduction/Section

Definition

A family of points ${}P_{i}\in E$, ${}i\in I$, in an affine space ${}E$ over a ${}K$-vector space ${}V$ is called an affine basis of ${}E$, if there exists an ${}i_{0}\in I$ such that the family of vectors

${\overrightarrow {P_{i_{0}}P_{i}}},i\in I\setminus \{i_{0}\},$

is a basis

of ${}V$.

Because of

${}{\overrightarrow {P_{i_{0}}P_{i}}}={\overrightarrow {P_{i_{0}}P_{i_{1}}}}+{\overrightarrow {P_{i_{1}}P_{i}}}=-{\overrightarrow {P_{i_{1}}P_{i_{0}}}}+{\overrightarrow {P_{i_{1}}P_{i}}}\,,$

the basis vectors ${}{\overrightarrow {P_{i_{0}}P_{i}}}$ with respect to the origin ${}P_{i_{0}}$ can be expressed as linear combinations of the corresponding vectors with respect to any other origin point ${}P_{i_{1}}$ of the family. Therefore, the property of being an affine basis is independent from the chosen ${}P_{i_{0}}$.

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

Definition

For a family ${}P_{i}$, ${}i\in I$, of points in an affine space ${}E$ and a tuple ${}a_{i}$, ${}i\in I$, in ${}K$ satisfying

${}\sum _{i\in I}a_{i}=1\,$

(for ${}I$ infinite, only finitely many of the ${}a_{i}$ are allowed to be different from ${}0$), the sum ${}\sum _{i\in I}a_{i}P_{i}$ is called a barycentric combination of the ${}P_{i}$. The corresponding point in ${}E$ is given by

${}\sum _{i\in I}a_{i}P_{i}=Q+\sum _{i\in I}a_{i}{\overrightarrow {QP_{i}}}\,,$

where ${}Q$ is an arbitrary point in ${}E$.

Lemma

For a family ${}P_{i}$, ${}i\in I$, of points in an affine space ${}E$, a barycentric combination

$\sum _{i\in I}a_{i}P_{i}$

defines a unique point in ${}E$.

Proof

See exercise.

$\Box$

Theorem

Let ${}P_{i}$, ${}i\in I$, denote an affine basis in an affine space ${}E$ over the ${}K$-vector space ${}V$. Then, for every point ${}P\in E$, there exists a unique barycentric representation

${}P=\sum _{i\in I}a_{i}P_{i}\,.$

Proof

Let ${}i_{0}\in I$ be fixed. In ${}V$, we have a unique representation

${}{\overrightarrow {P_{i_{0}}P}}=\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\,.$

We set

${}a_{i_{0}}:=1-\sum _{i\in I\setminus \{i_{0}\}}a_{i}\,.$

Then ${}\sum _{i\in I}a_{i}=1$, and

${}{\begin{aligned}P&=P_{i_{0}}+{\overrightarrow {P_{i_{0}}P}}\\&=P_{i_{0}}+\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\\&=P_{i_{0}}+a_{i_{0}}{\overrightarrow {P_{i_{0}}P_{i_{0}}}}+\sum _{i\in I\setminus \{i_{0}\}}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}\\&=P_{i_{0}}+\sum _{i\in I}a_{i}{\overrightarrow {P_{i_{0}}P_{i}}}.\end{aligned}}$

Therefore, there exists such a representation with ${}P_{i_{0}}$ as origin. Uniqueness follows from the facts that the ${}a_{i}$, ${}i\neq i_{0}$, are uniquely determined as the coefficients of the vector space basis, and that ${}a_{i_{0}}$ is determined by the baryzentric condition.

$\Box$

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 ${}(r,g,b)$ with ${}r+g+b=1$ (and non-negative coefficients) are relevant. Thus, colors are described by barycentric coordinates, saving one dimension.

Definition

Let ${}P_{i}$, ${}i\in I$, denote an affine basis in an affine space ${}E$ over the ${}K$-vector space ${}V$. For a point ${}P\in E$, the uniquely determined numbers

$(a_{i},i\in I){\text{ with }}\sum _{i\in I}a_{i}=1$

such that

${}P=\sum _{i\in I}a_{i}P_{i}\,$

holds, are called the barycentric coordinates

of ${}P$.

Example

Let ${}P_{i}$, ${}i\in I$, be an affine basis in an affine space ${}E$ over the ${}K$-vector space ${}V$. Then the point ${}P_{j}$ (${}j=I$) has the barycentric coordinates ${}(0,\ldots ,0,1,0,\ldots ,0)$, where the ${}1$ is at the ${}j$-th place (${}I$ being finite and ordered).