Infinity and below: points on lines and conics

(under revision; last revised 12-27-06)

If one knows points on the line at infinity, then one can map those points, both to conics and points on conics. we could start anywhere, but the equation of the line at infinity x+y+z = 0 is so simple and, given its affine invariance, it is a good place to start.

Peter Moses has made a list of the ETC points at infinity, so I began there. Unfortunately it was not very relevant. The ETC collection of infinite points is poor. Most are irrelevant to my investigation of this line, and as far as I can tell, irrelevant to anything triangle geometers are likely to be interested in. This is the first time ETC was not very informative. I include an appendix with an assesment of the ETC infinite points, after dropping 2/3 of them.

So I tried another track which was very productive.

I want to generate points on the line at infinity as I generated those on G—P; i.e., find the basic forms (the "primes") from which the others can be generated. I find his an enlightening way to think about points on lines. Although being a center is not restrictive enough to be interesting, nor is being on the line at infinity, the combination is.

We get lots of help because lines and conics hide infinite points which can easily be recovered.

From a point on a line to a point at infinity
If P0 = (x0:y0:z0) is on line lx+my+nz = 0, then l x0+m y0+n z0 = 0 so that the point ( l x0 : m y0, n z0 ) is on the line at infinity. Very neat and very simple.

Using the line G—Io = : c - a :, whose points are listed in the first table; we have the infinite points in the next table. And come to think of it, I lied. We are not beginning with the points at infinity, but rather those on G—Io, and using those to find the points at infinity. By doing this for the incenter Io, we are really doing it for all points, since Io contains all the affine structure of the plane (a very important statement!). For general affine points just replace (a,b,c) with (l,m,n).

Points on G—Io
name middle coordinate comments m d
P0
:1:
 centroid itself
itself
P1
Io = :b:
 incenter So
No
P2
∞•(G—Io) =
: c+a–2b :
 infinite point itself
itself
P3
~So•(G—Io) =
: b2–ca :
 Steiner inverse of Io ~Io•(G—Io) =
:a2–bc+c2–ab:
a2–ab–b2+ac–bc+c2
P4
: b2(c+a) :
 P0,P1; P4, P5 harmonic conjugates;
the pro-Steiner point;
center of homothety*
a2b–ab2+a2c–b2c+ac2+bc2
P5
:b(ab + bc – ca):
 P0,P1; P4, P5 harmonic conjugates;
a2b–ab2+a2c–abc–b2c+ac2+bc2
P6
b(ab + bc – 2ca)
 P0,P1; P5, P6 harmonic conjugates;
where tripolar of Io meets G—Io
a2b–ab2+a2c–2abc–b2c+ac2+bc2
b sb2
 = P4 of (No=dIo)
a3–3a2b+3ab2–b3–a2c+2abc +3b2c–ac2–3bc2+c3
P7
(c+a)SB
–ab2–b3+a2c–b2c+ac2
(a–2b+c)3
 = P4 of P2
= p7 of p2
(a+c)2(a+2b+c) = p4 of mP1
(a+c)(a2–2b2+c2) = p6 of mP1
(c+a)(SB±const) harmonically conjugate with P8 and So = mIo;
related to Fermat points if constant = Sπ/3

Notation: I use Conway notation where sa = s – a, sab = sa sb, and SA = (b2+c2 – a2)/2, and SBC = SB SC.

The line at infinity is an affine invarient line, and there are basic at affine invarient forms for points on this line. For example if (l:m:n) is a central point,  m-n is the x-coordinate of an infinite point, as is the formula m+n-2l. There are many more of these forms which are listed on in the chart.

But the nice thing about these affine forms is that, once defined, any point will then fit into them. This means that the most important points in the triangle plane each lead to many infinite points. I usually create four groups of infinite points. One set is from the incenter, then the Gergonne, while another is from the symmedian point, and another from the orthocentre. This means that we can both generate as many infinite points as we need, and we can customize those points to fit certain circumstances. Now why would we want to know lots of infinite points? Because infinite points can be mapped easily onto most other figures. And why would we want to customize infinite points? Because we would like to find the best possible points on that figure. For example if I wanted to find points on the Kiepert hyperbola, I would use the infinite points that came from the symmedian point or the orthocentre. There is always one very special point, being the incenter. The incenter carries the affine information embedded in the plane of the triangle, and so the infinite points from the incenter are abnormally effective in producing interesting points on other objects.

When I first came up with a scheme, it seemed little more than an academic exercise. But when I began applying this technique I found that in most cases this technique generated exactly the same points in the ETC.  ETC represents the wisdom of geometrical investigation by our community, and my natural mapping, as I call it, reproduces this well. 

Points as primes

Our goal is to find central points. Clark has given a definition of these points which works well in this context. My preferred definition is that a point is central if it's construction is invarient under all six permutations of ABC. It turns out that centrality gives a series of forms that behave in some ways like primes. These forms can be applied to any point including to infinite points. When I form a of an infinite point is applied to another infinite point, this generates a third infinite point, giving us an operation on infinite points. Under this operation most infinite points or composite, but many are not; these behaves like the primes. These primes are at affine invarient forms, into which any point in the triangle plane may be placed. As such they are uniquely powerful.

What we have done in the next chart is to place the generating points of line G—Io into the line equation to find infinite points. Remember that Io contains all the affine structure of the plane, so that what we do for Io is perfectly general.

The points on G—Io are explained here, and are listed in the above and following charts. Blue numbers are from ETC.

From infinity to Seiner ellipse 1/x+1/y+1/z = 0. (x:y:z) -> (1/x:1/y:1/z).
From infinity to Circumcircle a2/x+b2/y+c2/z = 0. (x:y:z) -> (a2/x:b2/y:c2/z ), the isogonal conjugate.
From infinity to Kiepert hyperbola (b2–c2)/x+(c2–a2)/y+(a2–b2)/z = 0. (x:y:z) -> ((b2–c2)/x : (c2–a2)/y : (a2–b2)/z).
From infinity to Seiner inellipse √x+√y+√z = 0. (x:y:z) –> (x2:y2:z2).

Table of infinite points, primes are listed at I1, I2, I3, I4, I5, I6. All the other points in the chart can be generate by them or m, d.

point on G—Io
coordinates of infinite point
comments about infinite point
G
I1 c – a 514
direction of ~Io
 
I2 c+a–2b 519
direction of G—Io
Io
b(c–a) 513
direction of ~tIo
 
bc+ca–2ab 536
direction of G—tIo
So
c2 – a2 523
direction of ~K; note that Io naturally generates the structure for K.
No
sb (c – a) 522
direction of ~Go
pSo = :b2(c+a):
b2(c2 – a2) 512
direction of ~tK; with ~K this completes the natural generation of the anallagmatic structure of K
b(bc+ab–ca)
I3 b(bc+ab–ca)(c–a)
b sbb
I4 b sbb (c–a)
direction of ~Go2
(c+a)SB
(c2–a2)SB 512
infinite point on ~H, ~O, ~N; duals of points on the Euler line are in this direction.
(b2–ca)
I5 (b2–ca)(c–a) 812
I6 (b2–ca)(c+a)
Direction of Io—tIo
(a2–bc+c2–ab)
(c2–+a2–)(c–a)
(c+a–2b)
I7 (c–a)(c+a–2b) 900
Infinite point on G—wS (X1086)
b(c2–+a2–)/(c–a)
I8 b(c2–+a2–)
518
Tarry Form
I9 b(c-a)sbSB


This chart lists some points on the line at infinity and on some conics. Green rows are generated from the incenter, blue from the symmedian points, and purple from the orthocenter.

line at infinity

affine form
n–l
Steiner Form
n+l–2m
m(n–l)
mn+ml–2nl
(n+l–m)(n–l)
  (n+l–2m)(n–l)
m2(mn+nl–2lm)(n–l) 
m(n+l–m)2(n–l) 
(m2–nl)(m–l) 
Tarry form 
(n–l)(n+l–m + const)
Fermat Form
 
 
 
 
 
 
 
 
 
on line at infinity from Io
c – a
514
c+a–2b
519
b(c–a)
513
:bc+ab–2ca:  
536
sb (c–a)
(c-a)(c+a–2b)
900
b2(bc+ab–2ca)(c–a)
b sbb (c-a)
(b2-ca)(c-a)
b(c2-+a2-)
518
  (c–a)(sb+const)
 
 
 
 
 
 
 
 
 
from K
c2–a2
523
tS
c2+a2–2b2
524
nS
b2(c2–a2)
∞•(~tK)
b2c2+a2b2–2c2a2  
SB(c2–a2)
  (c2–a2)(c2+a2–2b2)
b2(b2c2+a2b2–2c2a2)(c2–a2)2 
b2 SBB (c2–a2)
(b4–)(c2–a2)
b2(c4–+a4–)
gT 511
  (c2–a2)(SB+const)
 
 
 
 
 
 
 
 
 
from D
c2–a2
523
c2+a2–2b2
SB(c2–a2)
 SBC+SAB–2SCA
30
(b2–SB)(c2–a2)
 (c2–a2)(c2+a2–2b2) 
b2(c2–a2)( SBC+SAB–2SCA)
b2SBB(c2–a2)
SB2–(c2–a2)
 
SB(c2–a2+const)
 
 
 
 
 
 
 
 
 
Lemoine line
b2(c – a)
649
b2(c+a–2b)
902
b3(c–a)
667
 : b2(bc+ab–2ca): 
b2 sb (c – a)
a2(c–a)(c+a–2b)
1960
b3(bc+ab–2ca)(c–a)
b3 sbb (c-a)
b2(b2–ca)(c–a)
b4(c4–+a4–)
(b2–SB+const)(c2–a2)
 
 
 
 
 
 
 
 
 
 
b2(c2–a2)
b2(c2+a2–2b2)
b4(c2–a2)
669
b2 (b2c2+a2b2–2c2a2  )
b2 SB(c2–a2)
 
b4(b2c2+a2b2–2c2a2)(c2–a)2
b4 SBB (c2–a2)
b2(b4-)(c2-a2)
 
 
 
 
 
 
 
 
 
 
 
 
b2(c2–a2)
512
b2(c2+a2–2b2)
b2SB(c2–a2)
b2(SBC+SAB–2SCA)  
1495
b2(b2–SB)(c2–a2)
 
 
b4 SBB(c2–a2)
b2 SB2–(c2-a2)
 
 
 
 
 
 
 
 
 
 
 
Euler line from Io 1/(c+a)SB
b/(c+a)SB
 b sbb /(c+a)SB
K
H
(c2+a2–2b2)/SB(c2–a2)
b2/SB
25 Gob's perspector
H
(c2+a2–2b2)/SB
SB( SBC+SAB–2SCA)
1/b2SB
tO
(b4–)/SB
(SB+const)/SB
H
H
 (c2+a2–2b2)/SB(c2–a2) SB(c2–a2)
SBC+SAB–2SCA
30
Steiner ellipse 
1/(c – a)
190
1/(c+a–2b)
903
1/(b(c–a))
:1/(bc+ab–2ca):  
1/(sb (c – a))
 
1/(b(bc+ab-ca)(c-a))
1/(b sbb (c–a))
1/(b2–ca)(c–a)
1/(b(c2–+a2–))
 
 
 
 
 
 
 
 
 
 
 
1/(c2–a2)
1/(c2+a2–2b2)
1/(b2(c2–a2))
1/(b2c2+a2b2–2c2a2)  
1/(SB(c2–a2))
 
 
1/(b2 SBB (c2–a2))
1/(b4–(c2–a2))
1/(bb(c4–+a4–))
 
 
 
 
 
 
 
 
 
 
 
1/(c2–a2)
1/(c2+a2–2b2)
1/(SB(c2–a2))
648
 1/(SBC+SAB–2SCA)
1494
1/((b2–SB)(c2–a2))
 
 
1/(b2SBB(c2–a2))
1/(SB2–(c2–a2))
 
 
 
 
 
 
 
 
 
 
 
Circumcircle
b2/(c – a)
101
b2/(c+a–2b)

b/(c–a)
100

: b2/(bc+ab–2ca):
739
b2/(sb (c – a))
109
1/((c-a)(c+a–2b))
901
b/(bc+ab–ca)(c–a))
b/(sbb (c–a))
b2/(b2–ca)(c–a)
b/(c2–+a2–)
 
 
 
 
 
 
 
 
 
 
 
b2/(c2–a2)
110
b2/(c2+a2–2b2)
1/(c2–a2)
b2/(b2c2+a2b2–2c2a2)  
b2/(SB(c2–a2))
 
 
1/( SBB (c2–a2))
b3/((b4–)(c2–a2))
b2/(c4–+a4–)
 
 
 
 
 
 
 
 
 
 
 
b2/(c2–a2)
110
b2/(c2+a2–2b2)
b2/(SB(c2–a2))
  b2/(SBC+SAB–2SCA)
-
b2/((b2–SB)(c2–a2))
 
 
1/SBB(c2–a2))
b2/(SB2-(c2-a2))
 
 
 
 
 
 
 
 
 
 
 
Kiepert
So
(c2–a2)/(c+a–2b)
:(c+a)/b:
321
: (c2–a2)/(bc+ab–2ca):  
:(c+a)/sb: = :(c+a)sca:
:(c+a)/(c+a–2b):
G
(c2–a2)/(c2+a2–2b2) 
R
(c2–a2)/(b2c2+a2b2–2c2a2)  
H
 
 
1/(b2SBB)
1/b4-
 
 
 
 
 
 
 
 
 
 
 
G
(c2–a2)/(c2+a2–2b2)
H
 (c2–a2)/(SBC+SAB–2SCA)
2394
tddK
1/(b2SBB)
1/ SB2–
 1/(SB+const)
includes Fermat
 Steiner in-ellipse  (c–a)2
1086
 (c+a-2b)2  
 b2(c–a)2
:(bc+ab–2ca)2:  
  sb2(c–a)2
(c–a)2(c+a–2b)2
  b2(bc+ab–ca)2(c–a)2
  b2 sbbbb (c–a)2
(b2–ca)2(c–a)2
  b2(c2–+a2–)2
 
 
 
 
 
 
 
 
 
 
   (c2–a2)2   (c2+a2–2b2)2   b4(c2–a2)2
(b2c2+a2b2–2c2a2)2  
 SB2(c2–a2)2
  (c2–a2)2(c2+a2–2b2)2  b4(b2c2+a2b2–c2a2)2(c2–a2)2  b4SBBBB (c2–a2)2                        
 
 (c2–a2)2
115
  (c2+a2–2b2)2
SB2(c2–a2)2
 (SBC+SAB–2SCA)2
(b2–SB)2(c2–a2)2
    
b4SBBBB(c2–a2)2
(SB2–)2(c2–a2)2
                      
                                         
                                         
                             
 
          

 Appendix: ETC POINTS AT INFINITY

These are the portions of the ETC list of infinite points the I consider relevant. Post points follow a few affine forms in the table of infinite above. Most are of the form I1= :n–l: or I2 = :n+l–2m: for some choce of (l:m:n). The particular forms follow in blue; I2 represents the second form (the second row).

X(30) Euler ∞ = SBC + SAB – 2SCA I2 for P = H
X(511) g98 = t290 = bb(c4- + a4-) Tarry form
X(512) g99 = t670 = bb(cc–aa) I1 for P = tK
X(513) g100 = t 668 = bc – ab = b(c–a) I1 for P = tIo
X(514) g101 = t190 = c – a I1 for P = Io
X(515) g102
X(516) g 103 =
X(517) b (abc - c SC-b SB)
X(518) b/(a2- + c2-) Tarry form for P = Io
X(519) c+a-2b = g106 I2 for P = Io
X(520) g107 = sin2B(sin 2C - Sin 2A) = bbSB( ccSC – aaSA) I1 for P = tO
X(521) g108 = sec C - sec A = abSAB – bcSBC = bSB(cSC – aSA) I1 for P = tWo, a Clawson point
X(522) g109 = cos C - cos A = c SC – a SA I1 for P = Wo
X(523) g(110) = cc - aa I1 for P = K
X(524) g111 = t671 = cc + aa – 2bb I2 for P = K
X(525) g112 = t648 = cc SC – aa SA I1 for P = O
X(526) (sin A) (1 + 2 cos 2A)sin(B – C) = aaSAA(bb–cc)
X(536)
X(674) g 6750 = aa[b3 + c3 – a(b2 + c2)]
X(688) g689 = b4(c4–a4) I1 for P = tpK
X(838)
888,891 ----
X(900) g901 = (c – a)(c + a – 2b) I1 for 1/(c-a)
912 = aSA(b SB + c SC – aaa)
916 ----
x918 g919 = t666 = (b – c)(b2 + c2 – ab – ac)
924,
926, a sa(b sb – c sc) I1 for P = tMo
928,
952, [2a4 – 2a3(b + c) – a2(b2 – 4bc + c2) + (2a – b – c)(b – c)(b2 – c2)] = 2 S2 – 8 a sabc – a2SA
971 a5(b + c) - 2a4(b2 + c2 – bc) + 2a2(b – c)2(b2 + c2 + bc) – a(b – c)2(b + c)3
X(972) g972 = a5(b + c) – 2a4(b2 + c2 – bc) + 2a2(b – c)2(b2 + c2 + bc) – a(b - c)2(b + c)3, X(3)X(9). X(4)X(7) interesting
X(1154) ∞.e_H = a2( S2-3SCC)(S2 - 3SAA) a new prime I bet
1499, (b2 - c2)(b2 + c2 - 5a2) note: (b2 - c2)(b2 + c2 - n a2) is infinite
1503, g1297 = (b6 + c6 - 2a6 )+ (a4b2 + a4c2 - b4c2 - c4b2) sum of two infinite pts
1510,
1912,
X(1938) a( (cos A cos B - cos2C) sin B - (cos A cos C - cos2B) sin C)) I1? for P = ?
X(2771) g2687 = ugly