Barycentric coordinates: Difference between revisions
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In [[geometry]], '''barycentric coordinates''' form a homogeneous coordinate system based on a reference [[simplex]]. The location of a point with respect to this system is given by the masses which would need to be placed at the reference points in order to have the given point as [[barycentre]]. | In [[geometry]], '''barycentric coordinates''' form a homogeneous coordinate system based on a reference [[simplex]]. The location of a point with respect to this system is given by the masses (by definition positive) which would need to be placed at the reference points in order to have the given point as [[barycentre]]. However, for a point outside the simplex some coordinates must be negative. | ||
In an [[affine space]] or [[vector space]] of [[dimension (vector space)|dimension]] ''n'' we take ''n''+1 points <math>s_0, s_1, \ldots, s_n</math> in general position (no ''k''+1 of them lie in an affine subspace of dimension less than ''k'') as a simplex of reference. The barycentric coordinates of a point ''x'' are an (''n''+1)-tuple <math>x_0, x_1, \ldots, x_n</math> such that | In an [[affine space]] or [[vector space]] of [[dimension (vector space)|dimension]] ''n'' we take ''n''+1 points <math>s_0, s_1, \ldots, s_n</math> in general position (no ''k''+1 of them lie in an affine subspace of dimension less than ''k'') as a simplex of reference. The barycentric coordinates of a point ''x'' are an (''n''+1)-tuple <math>x_0, x_1, \ldots, x_n</math> such that | ||
:<math>(x_0+\cdots+x_n) x = x_0 s_0 + \cdots + x_n s_n | :<math>(x_0+\cdots+x_n) x = x_0 s_0 + \cdots + x_n s_n \,</math> | ||
The coordinates are not affected by scaling, and it may be convenient to take <math>x_0+\cdots+x_n = 1</math>. | and at least one of <math>x_0, x_1, \ldots, x_n</math> does not vanish. | ||
The coordinates are not affected by scaling, and it may be convenient to take <math>x_0+\cdots+x_n = 1</math>, in which case they are called ''areal coordinates''.[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:00, 16 July 2024
In geometry, barycentric coordinates form a homogeneous coordinate system based on a reference simplex. The location of a point with respect to this system is given by the masses (by definition positive) which would need to be placed at the reference points in order to have the given point as barycentre. However, for a point outside the simplex some coordinates must be negative.
In an affine space or vector space of dimension n we take n+1 points in general position (no k+1 of them lie in an affine subspace of dimension less than k) as a simplex of reference. The barycentric coordinates of a point x are an (n+1)-tuple such that
and at least one of does not vanish. The coordinates are not affected by scaling, and it may be convenient to take , in which case they are called areal coordinates.
