```Date: Jun 30, 1998 9:22 PM
Author: steve sigur
Subject: Conway on Trilinear vs Barycentric coordinates

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.Steve Sigur  ----------------   On 12/18/97 john Conway wrote    I'm sending this to several people interested in triangles, andhope they'll confirm receipt (except for rkg, who might be out ofemail touch), and send any further substantial messages about trianglesto everyone else on the list.   There are three standard systems that use three coordinates torepresent 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 longername since in fact all three systems are trilinear.  Each may or maynot 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 allthese into account the barycentric system emerges as the clear winner.   The letters B,A,BA,OT before each numbered point below show how thisdecision was reached.  A letter N indicates that normalized coordinatesare 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 ] OT   6.  The isogonal conjugate (or "conjugal") is        co-P =  ( a^2/X : b^2/Y : c^2/Z )      [ 1/X : 1/Y : 1/Z ] BA   7.  The isotomic conjugate (or "isotome") is       iso-P = ( 1/X : 1/Y : 1/Z )    [ 1/(x.a^2) : 1/(y.b^2) : 1/(z.c^2) ] 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 conceptthat its expression (see #7) in barycentrics cannot involve the edgelengths 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:     sub-P = ( Y+Z : Z+X : X+Y )   [ (by+cz)/a : (cz+ax)/b : (ax+by)/c ]   super-P = (Y+Z-X:Z+X-Y:X+Y-Z)   [(by+cz-ax)/a:(cz+ax-by)/b:(ax+by-cz)/c] (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 veryuseful 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:       [ b/a + c/a - 1 : c/b + a/b - 1 : a/c + b/c - 1]   and       [ b/a + c/a + 1 : c/b - a/b - 1 : -a/c + b/c - 1].   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 casesbut one (the definition of A!)  B is at least a joint winner.  I have deliberately preferred conceptual reasons for preferringone system to another, rather than mere comparisons of the simplicityof the coordinates for particular points.  Some points are simplerunder one system rather than another, and often it's OT that would givethe simpler ones.  But this difference can never be great, since[x,y,z] translates to [ax,by,cz]; and in barycentrics the simplicityoften 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, morecomplicated 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 thesub-incenter (Spieker point)          (b+c:c+a:a+b)  =  ( (b+c)/a : (c+a)/b : (a+b)/c ).   The "Morley perspectors" look simpler in OT: [ cos(A/3) : cos(B/3) : cos(C/3) ] and [ sec(A/3) : sec(B/3) : sec(C/3) ]but again this apparent simplicity disappears when we want a bit more:the Barycentric versions       ( a.cos(A/3) : ... )  and  ( a.sec(A/3) : ... )can be conjugated to get all the "companion Morley perspectors"just as easily.     In summary, the simplicity of coordinates for particular points can goeither way, and in any case is not a strong argument.  It's their theoreticalproperties that make barycentrics the clear winner.             John Conway
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