Anallagmatic Cubics

The Princeton library has a good collection of geometry books.

I have found a nice one on cubics with such a beautiful portrayal of cubics that I will excerpt some of it here. It is Cubiques Anallagmatiques by Ad. Mineur, published by Bruxelles, Librairie J. van Dijl, 38-38a Rue des Etudiants. Mineur was a geometer and an editior of Mathesis.

The book has 59 pages reproduced as handwritten text in French with another 10 pages of notes. I am not sure if it was ever published in any but a limited edition. There is no date but some articles in 1922 are referenced. Before me, the only time it was ever checked out from the Princeton library was 1924, so I guess that it was published in 1923.

It has the best geometric description of cubics in triangle geometry I have seen. Briefly the description is this: think of a point together with its isogonic/isotomic conjugate. The intersection of the tripolars of these two points forms a third point. Now think back the other way, if this intersection moves in a straight line, then the orgininal two points describe a curve, a cubic whose pivot is the pole of the path of motion.

Here are the details. Note that this exposition assumes a good deal of knowledge about triangle geometry using barycentric and trilinear coordinates. [My comments are in brackets].

Mineur says terminology is from de Longchamps. The book begins by explaining that for each curve f(x,y,z) in barycentrics there is a corresponding curve f(x,y,z) where the coordinates are now trilinears. The book is done in barycentrics, but many theorems are understood to have a trilinear version.

Section 1

These two points are reciprocal: P1 = (1/A:1/B:1/C) and P2 = (A:B:C).
The pair is called a reciprocal couple. [All examples are worked out for the isotomic case. The isogonic conjugate is called the "inverse," and a corresponding set of statements and equations is always implied. Note: I take the meaning of anallagmatic to be invariance with respect to replacement of a point with its reciprocal (or inverse).]

They are harmonically associated with the lines

D1: Ax +By+Cz = 0 and D2: x/A + y/B + z/C = 0.

A reciprocal transformation of D1 and D2 produce two circumconics

C1: A/x +B/y+C/z = 0 and C2: 1/Ax + 1/By + 1/Cz = 0.

The points of intersection of D1 and D2 with a side of ABC are isotomic. The lines joining these points to the opposite vertex are tangent to the conics C2 and C1 so that P1 and D1 are the pole and axis of homology of C2 and vice-versa.

D1 and D2 intersect at ( : (CC-AA)/CA : ) = p1, called the first transverse point. I call this the Mineiur point.
C1 and C2 have 4th intersection at the reciprocal of p1, called the second transverse point p2.

Section 2

When the traverse point p1 moves along the line delta: lx + my + nz = 0,

then the second traverse point describes a circumscribed conic.

gamma: l/x + m/y + n/z = 0.

Since lines D1, D2, delta meet at a point we have the determinant

| A B C |
| 1/A 1/B 1/C | = 0
| m n p |

to find all the points on the curve we replace A,B,C with x,y,z to get

| x y z |
| 1/x 1/y 1/z| = 0
| m n p |

which is called an anallagmatic cubic. This cubic goes through ABC and the centroid G and its harmonic associates [which he calls "algebraic associates"].

The equation of thie cubic is ... + m y (z²-x²) + ... = 0

The line delta is the "transverse" of the cubic, while the cubic is the anti-transverse of the line.

This leads to the first theorem:

When the first transverse point describes a straight line, the second describes a circumconic of ABC, and the reciprocal couple describes a cubic that goes through G, circumscribing both ABC and the anitmedial triangle.

This theorem is understood to have an equivalent trilinear/isogonic equivalent which goes through the incenter rather than the centroid.

Section 3

More generally :y: -> : (z²-x²)/zx : replaces a point with its transverse point and a line with a cubic. This is a mapping from lines to cubics. [Note that this point is anallagmatic.]

Section 4
The point W1 = (l:m:n) is on the cubic as its reciprocal, W2 = (1/l:1/m:1/n).


on to part two.