Distance coordinates with respect to a triangle of reference

Abstract

With respect to a triangle of reference A IA 2A3 • each point P in the plane of the triangle . has unique area coordinates: P = (b l • b, . b3 ) with 6 1 + 'b2 + b3 = 1. Distance coordinates are introduced such that P= (d l • d 2 • d3 ], with d,. the di stance from P to A k • It is shown that there is an' explicit function [(XI. X,. X3) such thatf(d;. d~. d5) = 0 is necessary and sufficient for P = [d l • d2 • d3 ], each d k nonnegative. The partial derivatives fdxI. X2. X3) = a[(x l . X2. X 3 ) laxk are such that b,. = fdd;. d;' d~) for each k. Other results relating the bk and the dj,' are given. The use of f(x" x •• X3) in solving geometric problems is shown.