In response to frequent mention of Trilinear coordinates on this newsgroup, John Conway admonished us that Barycentric coordinates were to be preferred. He sent some of us the following comparison of the two. Since my next post will comment on this, I thought I would post it first in its unadulterated form. I have learned much Geometry studying this.
On 12/18/97 john Conway wrote
I'm sending this to several people interested in triangles, and hope they'll confirm receipt (except for rkg, who might be out of email touch), and send any further substantial messages about triangles to everyone else on the list.
There are three standard systems that use three coordinates to represent a point in the plane of a give triangle, namely
BARYCENTRICS (B), AREALS (A), and ORTHOGONAL TRILINEARS (OT).
Often the last is shortened to "TRILINEARS", but I prefer the longer name since in fact all three systems are trilinear. Each may or may not be normalized. For the not-necessarily normalized versions I'll use:
(X:Y:Z) for B and A, and [x:y:z] for OT
and for the normalized ones
(X,Y,Z) for B, and [x,y,z] for OT.
All three (unnormalized) systems are very similar - indeed they coincide for Barycentrics and Areals, so I'll usually call these jointly BA, and the conversion from TO to these is very simple:
[x:y:z] becomes (ax:by:cz).
So for many purposes it hardly matters which system one uses. However, there ARE ways in which one or other of the systems is better than another, and it is the purpose of this note to point out that when one takes all these into account the barycentric system emerges as the clear winner.
The letters B,A,BA,OT before each numbered point below show how this decision was reached. A letter N indicates that normalized coordinates are involved.
OT 0N. The distances of P from the sides are
2X.Delta/a, 2Y.Delta/b, 2Z.Delta/c [ x, y, z ].
A 1N. The areas of PBC, PCA, PAB are the normalised areal coordinates
X.Delta, Y.Delta, Z.Delta [ ax/2, by/2, cz/2 ]
B 2N. If VA, VB, VC are vectors to A, B, C , then
P = X.VA + Y.VB + Z.VC
[Of course these are just the definitions of the three systems.]
BA 3. The Cevian ratios are
Y:Z, Z:X, X:Y [ by:cz, cz:ax, ax:by ].
BA 4. The Menelean ratios of the lines
PX + QY + RZ = 0 and px + qy + rz = 0 are -Q:R -R:P -P:Q and -cq:br -ar:cp -bp:aq
B 5. The normalizing condition is
X + Y + Z = 1 (B) or Delta (A), [ ax + by + cz = 2Delta ]
B 8. AFFINE INVARIANCE. If an affine transformation takes
A, B, C, and P = (X,Y,Z) to A1, B1, C1 and P1,
then the barycentric coordinates of P1 with respect to the new triangle A1 B1 C1 are still (X,Y,Z).
(It is because isotomic conjugation is an affinely invariant concept that its expression (see #7) in barycentrics cannot involve the edge lengths of the triangle.)
BA 9. The concepts of subordinate and superior points sub-P and super-P are particularly important in the theory. These points are the images of P in the subordinate (or "medial") triangle, whose vertices are the midpoints of the edges of ABC, and the superior (or "anticomplementary") triangle, the midpoints of whose edges are A,B,C. We have:
(Again the simplicity of the BA coordinates is due to affine invariance.)
B 10. RATIONALITY. X,Y,Z are rational functions of the Euclidean coordinates of the points A,B,C,P. If P is a point that's rationally defined from A,B,C, then its barycentric coordinates are rational functions of a^2, b^2, c^2. This is true, for instance, of the centroid, orthocenter, circumcenter, symmedian point, Brocard points, and so on.
(This is an extremely important point, and extends to give the very useful property below.)
BA 11. ALGEBRAIC CONJUGATES. Many points (such as the incenter) can be obtained by solving algebraic equations that have other (algebraically conjugate) solutions. Passing to these other solutions then yields further points that have essentially the same geometric properties (in this way, we get from the incenter to the excenters). We can get the barycentric coordinates of such "companions" as the appropriate algebraic conjugates of those of the original.
The simplest case of this is when the coordinates are rational functions of a,b,c but not of a^2, b^2, c^2. So for example if a point that's rationally constructed from the incenter has barycentric coordinates
( X(a,b,c), Y(a,b,c), Z(a,b,c) )
then the corresponding point obtained from the a-excenter is simply obtained by "changing the sign of a", thus:
( X(-a,b,c), Y(a,-b,c), Z(a,b,-c) ).
For example the Nagel point is the
"super-incenter" (b+c-a : c+a-b : a+b-c),
and so its a-companion is (b+c+a : c-a-b : -a+b-c).
The OT coordinates of these points are much harder to understand:
Well, that will do for now. I'll just survey the "winners"
0 1 2 3 4 5 6 7 8 9 10 11 OT A B BA BA B OT BA B BA B BA
In only two cases is OT the winner, and in all other cases but one (the definition of A!) B is at least a joint winner.
I have deliberately preferred conceptual reasons for preferring one system to another, rather than mere comparisons of the simplicity of the coordinates for particular points. Some points are simpler under one system rather than another, and often it's OT that would give the simpler ones. But this difference can never be great, since [x,y,z] translates to [ax,by,cz]; and in barycentrics the simplicity often has a useful conceptual meaning.
For example (1:1:1) = [ 1/a : 1/b : 1/c ] is the centroid, and its simplicity in barycentrics comes from its affine invariance.
On the other hand (a:b:c) = [1:1:1] is the incenter, more complicated in barycentrics since it has algebraic conjugates (-a:b:c) (a:-b:c) (a:b:-c). The apparent simplicity in trilinears disappears when we pass to the sub-incenter (Spieker point)