Barycentric coordinates
At the end of the discussion on Ceva's Theorem, we arrived at the conclusion that, for any point K inside ΔABC, there exist three masses w_{A}, w_{B}, and w_{C} such that, if placed at the corresponding vertices of the triangle, their center of gravity (barycenter) coincides with the point K. August Ferdinand Moebius (1790-1868) defined (1827) w_{A}, w_{B}, and w_{C} as the barycentric coordinates of K. (It's possible to generalize and consider also negative masses for points outside the triangle. This is not necessary for my purposes.) As defined, the barycentric coordinates are not unique. Masses kw_{A}, kw_{B}, and kw_{C} have exactly same barycenter for any
(*) | w_{A} + w_{B} + w_{C} = 1 |
the barycentric coordinates are defined uniquely for every point inside the triangle. (Barycentric coordinates that satisfy (*) are known as areal coordinates because, assuming the area of ΔABC is 1, the weights w are equal to the areas of triangles KBC, KAC, and KAB.) Since the center of gravity of any two points lies on the connecting segment, w_{A} = 0 for points on BC, w_{B} = 0 for points on AC, and w_{C} = 0 on AB. Vertices A,B,C have coordinates (1,0,0), (0,1,0), and (0,0,1), respectively.
The usual way to define the barycenter of three points with given masses (let's call such points material) is first to place the sum of masses of any two points at their barycenter. Now repeat the same (1-dimensional) operation pairing the new and the third remaining point. (The argument is formalized within affine geometry. I prefer to keep the discussion intuitive.) Assuming (*), if w_{A} is kept fixed, so will be the sum w_{B} + w_{C}. It then follows that for two possible positions D and D' of the barycenter of B and C, we have
If M_{b} and M_{c} are midpoints of AC and AB, respectively, then the equation of M_{b}M_{c} is
Let's have a look at the diagram on the right. In the blue triangle, all three coordinates are less than 1/2. Therefore, the first digit of their binary representation is zero. In the red triangles, two coordinates are less than 1/4 while the third is between 1/2 and 3/4. Therefore, in the red triangles, all three coordinates have their second binary digit zero. This leads to the trema removal procedure for constructing the Sierpinski gasket and provides a clue to its Description in the barycentric coordinates.
Barycentric coordinates arise naturally whenever variable quantities have a constant sum. Three glass problem, where we are pouring water from one glass to another under the unrealistic assumption that in the process no drop of water is going to be spilled, is a salient example. The problem illustrates beautifully the concept of barycentric coordinates.
|Contact| |Front page| |Contents| |Geometry|
Copyright © 1996-2018 Alexander Bogomolny