Coordinate system that is defined by points instead of vectors
Not to be confused with Barycentric coordinates (astronomy).
See also: Affine space § Barycentric coordinates
Barycentric coordinates on an equilateral triangle and on a right triangle.A 3-simplex, with barycentric subdivisions of 1-faces (edges) 2-faces (triangles) and 3-faces (body).
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.
Every point has barycentric coordinates, and their sum is never zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity.
Barycentric coordinates were introduced by August Möbius in 1827.[1][2][3] They are special homogenous coordinates. Barycentric coordinates are strongly related with Cartesian coordinates and, more generally, to affine coordinates (see Affine space § Relationship between barycentric and affine coordinates).
Barycentric coordinates are particularly useful in triangle geometry for studying properties that do not depend on the angles of the triangle, such as Ceva's theorem, Routh's theorem, and Menelaus's theorem. In computer-aided design, they are useful for defining some kinds of Bézier surfaces.[4][5]
^Möbius, August Ferdinand (1827). Der barycentrische Calcul. Leipzig: J.A. Barth.
Reprinted in Baltzer, Richard, ed. (1885). "Der barycentrische Calcul". August Ferdinand Möbius Gesammelte Werke. Vol. 1. Leipzig: S. Hirzel. pp. 1–388.
^Max Koecher, Aloys Krieg: Ebene Geometrie. Springer-Verlag, Berlin 2007, ISBN 978-3-540-49328-0, S. 76.
^Hille, Einar. "Analytic Function Theory, Volume I", Second edition, fifth printing. Chelsea Publishing Company, New York, 1982, ISBN 0-8284-0269-8, page 33, footnote 1
^Josef Hoschek, Dieter Lasser: Grundlagen der geometrischen Datenverarbeitung. Teubner-Verlag, 1989, ISBN 3-519-02962-6, S. 243.
^Gerald Farin: Curves and Surfaces for Computer Aided Geometric Design. Academic Press, 1990, ISBN 9780122490514, S. 20.
and 27 Related for: Barycentric coordinate system information
In geometry, a barycentriccoordinatesystem is a coordinatesystem in which the location of a point is specified by reference to a simplex (a triangle...
coordinates are used in the Hamiltonian treatment of mechanics. Barycentriccoordinatesystem as used for ternary plots and more generally in the analysis...
Barycenter) BarycentricCoordinate Time, a coordinate time standard in the Solar systemBarycentric Dynamical Time, a former time standard in the Solar System In...
BarycentricCoordinate Time (TCB, from the French Temps-coordonnée barycentrique) is a coordinate time standard intended to be used as the independent...
1, resulting in another point. These coefficients define a barycentriccoordinatesystem for the flat through the points. Any vector space may be viewed...
Barycentric Dynamical Time (TDB, from the French Temps Dynamique Barycentrique) is a relativistic coordinate time scale, intended for astronomical use...
The barycentric celestial reference system (BCRS) is a coordinatesystem used in astrometry to specify the location and motions of astronomical objects...
frame, e.g. a clock located at the solar system barycenter would not measure the coordinate time of the barycentric reference frame, and a clock located at...
other reference cardinal direction Barycentric and geocentric celestial reference systems – Celestial coordinatesystem Celestial sphere – Imaginary sphere...
triangles. The triangle case can be solved easily by use of a barycentriccoordinatesystem, parametric equation or dot product. The dot product method...
Cartesian coordinatesystem is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat...
linear function) n-simplex (e.g. tetrahedron) interpolation (see barycentriccoordinatesystem) Inverse distance weighting ABOS - approximation based on smoothing...
1 {\textstyle w_{j}=\prod _{m\neq j}(x_{j}-x_{m})^{-1}} (called the barycentric weight), and a part representing the displacement from x j {\textstyle...
spelled out by the ICRS. More specifically, the ICRF is an inertial barycentric reference frame whose axes are defined by the measured positions of extragalactic...
different ways of coordinatizing the plane in hyperbolic geometry are used. This article tries to give an overview of several coordinatesystems in use for the...
and how TT is realised. BarycentricCoordinate Time (TCB) is the analog of TCG, used for calculations relating to the Solar System beyond Earth orbit. TCG...
vector spaces based on different kinds of scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which...
provides an approximation for the exact solution of a partial differential equation which is parametrized barycentriccoordinatesystem (mathematics) v t e...
general, the incenter is not the same as the centroid; the centroid has barycentric coordinates 1 : 1 : 1 (these being proportional to actual signed areas...
visualized as billiard moves in the (clipped) coordinatesystem on a triangular lattice. The barycentric plot on the right gives two solutions to the 8...
trilinear or barycentric coordinates given above for the circumcenter: all three coordinates are positive for any interior point, at least one coordinate is negative...
In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free...
for generating lightmaps, normal maps, or low level of detail models. Barycentric coordinates Three-element coordinates of a point inside a triangle. Beam...
as projective coordinates of the point R on this line, and are termed barycentric coordinates. Another way of interpreting the process here is the mechanical...
BCRS may refer to: Barycentric celestial reference system, a coordinatesystem used in astrometry to specify the location of astronomical objects Bureau...
Global Information
