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, 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 ]

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 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:

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 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:

[ 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 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)

(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 go

either way, and in any case is not a strong argument. It's their

theoretical

properties that make barycentrics the clear winner.

John Conway