Families of Conics

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Families of Conics

1 Background Information

1.1 The Configuration of Pappus

The configuration of Pappus is obtained by taking two distinct lines, any three distinct points on one line and any three distinct points on the other (subject to the condition that none of the six points lies at the point of intersection of the two lines). Label the three points on the first line A, C, and E, and the three points on the second line by B, D, and F. We use the notation X=AB.DE to mean that X is the point of intersection of the lines AB and DE. Similarly, let Y=BC.EF and Z=CD.FA. Then the three points X, Y, and Z are collinear, and the line containing X, Y, and Z is called the Pappus Line of the configuration.

1.2 The configuration of Pascal

The configuration of Pascal is obtained by taking any six distinct points lying on a non-degenerate conic. Label these points A, B, C, D, E, and F. Then the three points X=AB.DE, Y=BC.EF, and Z=CD.FA are collinear, and the line containing X, Y, and Z is called the Pascal Line of the configuration

1.3 Steiner points

Let A, B, C, D, E, and F be the six points of either a Pappus or a Pascal configuration. Denote the Pappus Line or Pascal Line defined above by PL(A, B, C, D, E, F). Now consider two more lines, similarly defined by the same six points but taken in different orders, namely PL(A, D, C, F, E, B), and PL(A, F, C, B, E, D). These three lines meet in a single point, called a Steiner Point. There is a second Steiner Point, given by the common intersection of the three lines PL(A, B, C, F, E, D), PL(A, F, C, D, E, B) and PL(A, D, C, B, E, F).

At first glance it might seem that there could be many Pappus Lines associated with one fixed set of six points on two lines, but in fact there are only six, and they are all listed here.

1.4 Twelve Steiner conics

The locus of either of the above two Steiner Points, as a function of any one of the original six points, is a conic. Since we have six points which may be use to generate the locus, and since we have two Steiner Points to work with, we get 12 conics.

1.5 A characterization of the twelve conics

Consider one of the twelve conics. We take S to be one of the two Steiner points, say the one on the line PL(A, B, C, D, E, F), and any one of the original six points used to define the configuration. Without loss of generality, let A be the chosen point. Now consider the conic defined to be the locus of S as a function of A. This conic passes through the five points C, E, BC.FE, DC.BE, and FC.DE. These five points are distinct and no three are collinear, hence they characterize the conic. If we look at the other Steiner point, its locus, as a function of A, is the conic through the five points C, E, BE.FC, DE.BC, and FE.DC.

1.6 Another six conics

In the case determined by a Pappus Configuration, the six points BC.FE, FE.DC, DC.BE, BE.FC, FC.DE, and DE.BC form a hexagon whose opposite sides meet by pairs (BC and BE), (DC and DE), and (FC and FE) in three collinear points, namely B, D, and F. Since these points are collinear they lie on a conic. This one happens to be determined by A and it does not meet the line CE, but it meets the line BD in two points.

Just as A was used to define this conic, B, C, D, E, and F can also be used, giving six in all. The three determined this way, by A, C, and E all meet the line BD in the same two points. Similarly the three conics determined this way all meet the line AC in two points.

2 Families of conics

2.1 Families of 12 conics

The accompanying figures illustrate families of conics associated with the configurations of Pappus and Pascal. In each figure, there are 12 conics each of which is the locus of a Steiner point associated with the configuration.

2.2 Families of 18 conics

In the case of a Pappus configuration, there are an additional six conics which, interestingly, meet by two pairs of three in points on the original base lines of the Pappus configuration.

2.3 A Special Case

In the case of the Pascal configuration, when the original conic happens to be a circle, there is a fascinating configuration that everyone has seen before. It arises when the six points of the Pascal configuration are equally spaced alternately around the circle. The resulting 12 conics become six circles and six degenerate conics (straight lines).

2.4 Summary

For each of the subject conics, whether it comes from a family of 12 or 18, there is a certain set of five points that characterizes that conic. Each of the these 5 points is either one of the original six points that defined the configuration, or is a cross point that comes directly from the original six points.

The author:
Prof. Leroy J. Dickey
Department of Pure Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1

 

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