高维重心坐标的几何意义

GEOMETRIC MEANING OF HIGER-DIMENSIONAL BARYCENTRIC COORDINATES

设Ｓ是以Ｅｎ中点Ａｉ（ａｉ１，ａｉ２，…，ａｉｎ），（ｉ＝０，１，…，ｋ，１≤ｋ≤ｎ）为顶点的ｋ维定向非退化单形，Ａｉ（ｉ＝０，１，…，ｋ）确定的ｋ维平面上任意一点Ａ（ｘ１，ｘ２，…，ｘｎ）将Ｓ分成ｋ＋１个ｋ维定向单形，若它们的ｋ维带号体积比Ｖ［ＡＡ１…Ａｋ］∶Ｖ［Ａ０ＡＡ２…Ａｋ］∶…∶Ｖ［Ａ０Ａ１…Ａｋ－１Ａ］＝ｕ０∶ｕ１∶…∶ｕｋ，则有①点Ａ的重心坐标为（ｕ０ｕ，ｕ１ｕ，…，ｕｋｕ），②ｘｉ＝∑ｋｊ＝０ｕｊａｊｉｕ（ｉ＝１，…，ｎ），其中ｕ＝∑ｋｊ＝０ｕｊ。

Let S be a directed nondegenerate k dimensional simplex,with A i(a i1 ,a i2 ,…,a in ),(i=0,1,…,k,1≤k≤n) as its vertexes.Any point A(x 1,x 2,…,x n) on the k dimensional plane determined by A i(i=0,1,…,n), divides S into k+1 directed k dimensional simplexes in the signed volume ratio V[AA 1…A k]∶V[A 0AA 2…A k]∶…∶V[A 0A 1…A k-1 A]=u 0∶u 1∶…∶u k. Then barycentric coordinates of point A are (u 0/u 1,u 1/u,…,u k/u) and x i=∑kj=0u ja ji /u(i=1,2,…,n), where u denotes ∑kj=0u j.