Cubic curves reflect geometrical structure

The probability that three or more points placed randomly in the plane will be colinear is zero. Hence in triangle geometry we take three colinear points (or three concurrent lines) as a sign of intentional struncture.

A straight line intersects a cubic curve three times, creating an instant colinearity. Other properties of cubics, explained below, organize points and lines into groups of four. Collectively these properties are known as the group structure of points on a cubic curve, making them as powerful an organizing structure in triangle geometry as anything we know.

Cubics are of third degree but their class varies with the type of cubic. The class of a curve is the number of tangent lines that can be drawn to it from any point in the plane. The cubics we will discuss are third degree, but for many points on the cubic there are 4 tangent lines than can be drawn from other points on the cubic. This gives points on the cubic a natural four-fold structure.

Desmic structure, the basis for the group law

The 16 lines-12 points projective configuration (the projective cube, see here for an example) with 4 lines meeting each point and 3 points on each line is the basis for the group structure on a cubic. John Conway has named it the Desmic (linking) structure. The 12 points may be thought of as 3 quadrangles. Two of these quadrangles can be thought to be projected tetrahedra, which can be considered opposite vertices in a projective cube. In this way, as shown in the link picture, 16-12 configureation may be thought of as a projected cube.

This structure arises so often in tringle geometry because two triangles in perspective can usually be extended to three triangles mutually in perspective. In this case we count the 12 points as 9 for the three triangles and 3 perspectors. In the chart below, we may consider that points D, A, B, C form a quadrangle or that ABC is a triangle with D a perspector. Here is a picture of this arrangement.

A cubic curve of the type we study in geometry naturally creates this stucture. Choosing a point P on the cubic, there will be 4 points on the cubic whose tangents meet at P. These 4 points form a natural grouping, a quadrangle in the above sense. The tangent from P meets the cubic at another point Q. Three other points will have tangents through Q. These group with P to form another quadrangle. Since a line intersects a cubic three times, each line contains three points of the cubic. To create a desmic 16-12 structure, find three colinear points and for each point find the quadrangle of points whose tangents meet it. These will be a desmic triple.

Pictures of cubics

These pictures emphasize three things: the relation of the cubic to the triangle; points that occur on the cubic, And the organization of points and and lines in the plane, as represented by the group table and the lines drawn on the picture.

The group table

We have explained the horizontal structure of the group table. This horizontal structure is the Klein 4 group. The vertical structure is the group Z, represented by the integers on the left. The points in three rows form a desmic triple if their numbers add to a particular constant, indicated on the group table. For the Neuberg cubic this constant is 2.

The Neuberg cubic

The greatest example of organizational stucture, other than the triangle itself, is the Neuberg cubic, also known as the 32 point cubic (although in the modern age we know a lot more than 32 points on it). Since its pivot is at infinity in the direction of the Euler line, it organizes many points on lines parallel to Euler's line. This cubic can be defined as consisting of all points P such that the line P—gP is parallel to the Euler line. gP is the isogonal conjugate of P.

The colored regions in this picture represent self-isogonal regions. The boundary lines between colors are the 6 angle bisectors. If a point is in a region of a given color, its conjugate is in another region of the same color. A conjugal cubic will always inhabit two colored regions. Since this cubic is self conjugate, each part of the cubic must stay in regions of the same color. Conjugates must therefore be on the same branch of the cubic.

This version of the picture includes points compiled by the Hyacinthos group of geometers and compiled on Bernard Gibert's webpage on cubics, a great resource.

The group table in this picture represents a few more than the original 32 known points. Here is a larger group table with 112 known points on this cubic.

Reading the group table

The numbers of the rows must add to the constant, which is 2 for this cubic. The columns must be as shown in the chart above. If we use a row twice, we are getting the tangent to the cubic from that point.

0 + 0 + 2. These rows add to two, the constant, so tells us that the tangents from the points in row 0 all meet at the conjugate of the Euler infinity point. We infer that the asymptote of the cubic is parallel to the Euler line and goes through to conjugate of the Euler infinity point.

1 + 1 + 0 adds to the constant. Hence tangents from Io, Ia, Ib, Ic all meet at the Euler infinity point, meaning that the tangents to the cubic at these points are parallel to the Euler line.

(1+n) + (1-n) + 0 add to the constant, so that the line through corresponding Fermat and Isodynamic points is parallel to the Euler line.

The Lucas cubic

My favorite because this cubic deals with the Nagel-Gergonne structure, which is the model for so much triangle geometry. The GerGer and NagNag points come from an unusual structure. Each point in the plane is a part of a family of points, often called its harmonic associates. The Gergonne and Nagel points are part of a second family which John Conway has called its extraversions. The first family is a purely projective structure. The second family is about as unprojective as you can get. The miracle is that these two sets are related as two parts of a desmic structure, meaning that when they are connected by lines, the lines concur at new points. The the Gergonne points, we call these the GG_x points, where x = o, a, b, c, one of the four extraversion subscripts.

The four incenters are included in the picture, although they are not part of the cubic, because the lines Ix—Gx (for x = 0, a, b, c) concur at L and are tangent to the cubic at Gx. The tangents at Nx also go through G and Ix.

The Darboux antipodal cubic

This is a very simple cubic with two operations, isogonal conjugacy, and reflection in the circumcenter. The colored regions are bounded by the 6 angle bisectors as for the Neuberg cubic. I am swithcing to this form of picture because it seems to show more and looks nice.

In this table g represents the isogonal conjugate operation and o represents reflection in the circumcenter. These are the two operations that generate the points on the cubic. A_L is the cevian trace of L, the deLongchamps point.

The group table organizes the points of the cubic, as group tables always do, but a nice bonus is to give many lines that are parallel to the asymptotes which are the perpendicular bisectors of the sides. For rows that are equally above and below O on the table, whose indices have the same absolute value, the defined lines are parallel to these asymptotes. For example, the line I_o—oI_a is parallel to the a-perpendicular bisector, as is H—A, the altitude, and L—oA, and gL—oA_L.

The Thomson cubic

The medial of the Lucas cubic. More about this when I get a chance.

updated 8-29-05