무게중심좌표 Barycentric Coordinates

http://2000clicks.com/mathhelp/GeometryTriangleBarycentricCoordinates.aspx

http://mathworld.wolfram.com/BarycentricCoordinates.html

Barycentric coordinates are triples of numbers  corresponding to masses placed at the vertices of a reference triangle . These masses then determine a point , which is the geometric centroid of the three masses and is identified with coordinates . The vertices of the triangle are given by , , and .

Barycentric coordinates normalized so that they become the actual areas of the subtriangles are called homogeneous barycentric coordinates. Barycentric coordinates normalized so that

so that the coordinates give the areas of the subtriangles normalized by the area of the original triangle are called areal coordinates. Barycentric and areal coordinates can provide particularly elegant proofs of geometric theorems such as Routh's theorem, Ceva's theorem, and Menelaus' theorem.

1. Areal coordinates of point P are normalized barycentric coordinates.  They are normalized so that they become the areas (hence the name, areal) of the subtriangles formed by connecting P with each of the triangle's vertices, and where the area is measured in terms of the area of triangle ABC.  This way of normalizing the coordinates is equivalent to specifying t₁+t₂+t₃=1.

2. Homogeneous barycentric coordinates of point P are normalized so that they represent the actual areas of the subtriangles formed by connecting P with each of the triangle's vertices.