Editing Barycentric coordinate system (section)

== Definition ==
Let <math>A_0, \ldots, A_n</math> be {{math|''n'' + 1}} points in a [[Euclidean space]], a [[flat (geometry)|flat]] or an [[affine space]] <math>\mathbf A</math> of dimension {{mvar|n}} that are [[affinely independent]]; this means that there is no [[affine subspace]] of dimension {{math|''n'' − 1}} that contains all the points,<ref name="Reventós Tarrida">Reventós Tarrida, Agustí. "Affine Maps, Euclidean Motions and Quadrics". Springer, 2011, {{ISBN|978-0-85729-709-9}}, page 11</ref> or, equivalently that the points define a [[simplex]]. Given any point <math>P\in \mathbf A,</math> there are [[scalar (mathematics)|scalars]] <math>a_0, \ldots, a_n</math> that are not all zero, such that 
<math display=block> ( a_0 + \cdots + a_n ) \overset{}\overrightarrow{OP} = a_0 \overset{}\overrightarrow {OA_0} + \cdots + a_n \overset{}\overrightarrow {OA_n}, </math>
for any point {{mvar|O}}. (As usual, the notation <math>\overset{}\overrightarrow {AB}</math> represents the [[translation (geometry)|translation vector]] or [[free vector]] that maps the point {{mvar|A}} to the point {{mvar|B}}.)

The elements of a {{math|(''n'' + 1)}} tuple <math>(a_0: \dotsc: a_n)</math> that satisfies this equation are called ''barycentric coordinates'' of {{mvar|P}} with respect to <math>A_0, \ldots, A_n.</math> The use of colons in the notation of the tuple means that barycentric coordinates are a sort of [[homogeneous coordinates]], that is, the point is not changed if all coordinates are multiplied by the same nonzero constant. Moreover, the barycentric coordinates are also not changed if the auxiliary point {{mvar|O}}, the [[origin (mathematics)|origin]], is changed.

The barycentric coordinates of a point are unique [[up to]] a [[scaling (geometry)|scaling]]. That is, two tuples <math>(a_0: \dotsc: a_n)</math> and <math>(b_0: \dotsc: b_n)</math> are barycentric coordinates of the same point [[if and only if]] there is a nonzero scalar <math>\lambda</math> such that <math>b_i=\lambda a_i</math> for every {{mvar|i}}.

In some contexts, it is useful to constrain the barycentric coordinates of a point so that they are unique. This is usually achieved by imposing the condition
<math display=block>\sum a_i = 1,</math> 
or equivalently by dividing every <math>a_i</math> by the sum of all <math>a_i.</math> These specific barycentric  coordinates are called '''normalized''' or '''absolute barycentric coordinates'''.<ref name=Deaux>Deaux, Roland. "Introduction to The Geometry of Complex Numbers". Dover Publications, Inc., Mineola, 2008, {{ISBN|978-0-486-46629-3}}, page 61</ref> Sometimes, they are also called [[affine coordinates]], although this term refers commonly to a slightly different concept.

Sometimes, it is the normalized barycentric coordinates that are called ''barycentric coordinates''. In this case the above defined coordinates are called ''homogeneous barycentric coordinates''.

With above notation, the homogeneous barycentric coordinates of {{mvar|A{{sub|i}}}} are all zero, except the one of index {{mvar|i}}. When working over the [[real number]]s (the above definition is also used for affine spaces over an arbitrary [[field (mathematics)|field]]), the points whose all normalized barycentric coordinates are nonnegative form the [[convex hull]] of <math>\{A_0, \ldots, A_n\},</math> which is the [[simplex]] that has these points as its vertices.

With above notation, a tuple <math>(a_1, \ldots, a_n)</math> such that
<math display=block>\sum_{i=0}^n a_i=0</math>
does not define any point, but the vector 
<math display=block> a_0 \overset{}\overrightarrow {OA_0} + \cdots + a_n \overset{}\overrightarrow {OA_n}</math>
is independent from the origin {{mvar|O}}. As the direction of this vector is not changed if all <math>a_i</math> are multiplied by the same scalar, the homogeneous tuple <math>(a_0: \dotsc: a_n)</math> defines a direction of lines, that is a [[point at infinity]]. See below for more details.