| 释义 |
Trilinear CoordinatesGiven a Triangle  , the trilinear coordinates of a point  with respect to are anordered Triple of numbers, each of which is Proportional to the directed distance from to one of theside lines. Trilinear coordinates are denoted  or  and also are known asBarycentric Coordinates, Homogeneous Coordinates, or ``trilinears.''
In trilinear coordinates, the three Vertices  ,  , and  are given by  ,  , and . Let the point in the above diagram have trilinear coordinates and lie at distances ,  , and  from the sides  ,  , and  , respectively. Then the distances  ,  ,and  can be found by writing  for the Area of  , and similarly for and  . We then have
so
 | (2) |
 is the Area of and  ,  , and  are the lengths of its sides. When the values ofthe coordinates are taken as the actual lengths (i.e., the trilinears are chosen so that  ), the coordinates areknown as Exact Trilinear Coordinates.
Trilinear coordinates are unchanged when each is multiplied by any constant  , so
 | (3) |
 | (4) |
 | (5) |
 | (6) |
 is the Point-Line Distance from Vertex  to the Line.
Trilinear coordinates for some common Points are summarized in the following table,where , , and are the angles at the corresponding vertices and , , and are the oppositeside lengths.
| Point | Trilinear Center Function | Centroid  |  ,  | Circumcenter  |  | | de Longchamps Point |  | | Equal Detour Point |  | Feuerbach Point  |  | Incenter  | 1 | | Isoperimetric Point |  | | Lemoine Point | | Nine-Point Center  |  | Orthocenter  |  | | Vertex | | | Vertex | | | Vertex | |
To convert trilinear coordinates to a vector position for a given triangle specified by the  - and  -coordinates ofits axes, pick two Unit Vectors along the sides. For instance, pick
where these are the Unit Vectors and . Assume the Triangle has been labeled suchthat  is the lower rightmost Vertex and  . Then theVectors obtained by traveling  and  along the sides and then inward Perpendicular tothem must meet
 | (9) |
gives
But  and  are Unit Vectors, so
And the Vector coordinates of the point are then
 | (16) |
References
Boyer, C. B. History of Analytic Geometry. New York: Yeshiva University, 1956.Casey, J. ``The General Equation--Trilinear Co-Ordinates.'' Ch. 10 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 333-348, 1893. Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, pp. 67-71, 1959. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. Coxeter, H. S. M. ``Some Applications of Trilinear Coordinates.'' Linear Algebra Appl. 226-228, 375-388, 1995. Kimberling, C. ``Triangle Centers and Central Triangles.'' Congr. Numer. 129, 1-295, 1998.
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