7P-s-Cu1   7P-Circular Cubic

A circular cubic is defined here as a cubic through 7 random points and 2 imaginary circular points at infinity.
These imaginary circular points at infinity are the same points that lie on each circle. The defining points of a circle aren’t actually 3 points, but 3 points + the 2 imaginary circular points at infinity, making a total of 5 points, the number of points defining any conic. And a circle is a conic as it appears now.
In the same way like a cubic is determined by 9 points (see 9P-s-Cu1), the circular cubic (7P-s-Cu1) is determined by 7 points + 2 imaginary circular points at infinity.
And just like a circle is defined by 3 points without mentioning the circular points at infinity, so the circular cubic is defined by 7 points again without mentioning the circular points at infinity. The circular property is just mentioned in the name.

Focus of the Circular Cubic

A circular cubic has a focus F (also called singular focus), being the point where the tangents at the circular points at infinity intersect. This focus is 7P-s-P1.
If F lies on the curve, it is called a Van Rees circular focal cubic (like QL-Cu1 with focus QL-P1 and QA-Cu7 with focus QA-P41). If not, it is just a circular cubic, like QA-Cu1 with focus QA-P9.
See Ref-66, QPG-messages #665, #666.

Asymptote of the Circular Cubic

A circular cubic has a real asymptote and is cutting its asymptote in a point Q.
This point Q is the tangential of the focus of the circular cubic.
Any line through Q cuts the circular cubic in 2 points equidistant from the focus.
See Ref-66, QPG-messages #152-154, #678, #679. Anallagmatic Cubics

All circular cubics are anallagmatic, the pole of inversion being the point of the cubic where the tangent is parallel to the real asymptote. See Ref-62.
Explanation:
A curve is said to be anallagmatic if it is globally invariant by inversion. See Ref-13, keyword “anallagmatic” and Ref-63, page 276.
In other words, when there is a pole M on a curve with the property that taking another point X on the curve with 3rd intersection point Y of line MX with that curve, then there is a fixed number p=|MX|.|MY|, independent of the location of point X.
Consequently point Y can be mapped, when given points M and X on the cubic and the power sqrt[p]. See Ref-63, page 276.

Construction
See picture with description from Ref-63, p.212/213 in the French language from Roger Cuppens. Clarification
Here’s an clarification of Bernard Keizer, mentioned in Ref-66, QPG-messages #658, #659.

1) Homography is a projective bijective transformation keeping the cross-ratio.
2) A set of 4 lines define a cross-ratio, which is the same for the 4 points intersection of any line with this set.
3) 2 sets of 3 lines through 2 summits define a homography : the intersections of a line of the 1rst set with the corresponding line of the 2nd give 3 points, the se 3 points determine with the 2 summits a conic and the lines of the 2 sets partners in the homography intersect on this conic.
4) A set of 4 curves of degree n through a basis of n(n+3)/2 points (1 for a line, 5 for a conic, 9 for a cubic, 14 for a quartic ...) has the cross-ratio of the 4 tangents in a point of the basis
5) An homography of 2 sets of curves of degrees n and n' determine as intersection of 2 corresponding curves a curve of degree n + n' (n = n' = 1 gives a conic, n = 1 and n' = 2 gives a cubic)
Hence the construction of a cubic through 9 points (for a circular cubic through 7 points, the construction is the same providing you replace the conics through 4 points by circles through 2 points)
1) 9 points A to I
2) set of 5 conics through A, B, C, D and E (resp F, G, H, I)
3) set of 5 tangents e, f, g, h and i through A to these conics
4) The center of homography S between e, f, g, h and i and E, F, G, H and I is the unique point S for which the lines e, f, g, h and i and the lines SE, SF, SG, SH and SI intersect in the same conic through A and S. This point S is the point named by Cotterill focus of the QA ABCD wrt the cubic.
a) Choose the conic defining the homography between g, h and i and EG, EH and EI (conic though the 3 intersections g', h' and i' of the corresponding lines and the summits A and E.
b) e cuts this conic in A and a 2nd point e' ; the transformed of e in the homography in Ee'
c) Draw the conic through E,G,H and I and tangent in E to Ee'
d) Do the same construction by replacing E by F and e by f ; the 4th intersection of the 2 last conics (other than G, H and I) is S
5) Having S, draw the conic through the 5 intersections of the lines e, f, g, h and i and the lines SE, SF, SG, SH and SI.
6) For any point P on this conic, AP and SP are transformed in the homography and SP cuts the conic though A, B, C and D tangent in A to AP in 2 points Q and R describing the cubic when P describes the conic. (My construction is here slightly different from Cuppens, but that doesn't change the result ...)

Correspondence with ETC/EQF

Application in Triangle Geometry
• The most remarkable circular cubic in Triangle Geometry is the Neuberg cubic. Literally hundreds of triangle centers are known to lie on this circular cubic. See Cubics in the Triangle Plane from Bernard Gibert Ref-17c.
QA-Cu1 is the Circular Cubic through the 4 vertices of a Reference Quadrangle and the 3 vertices of its Diagonal Triangle QA-Tr1. Its focus is QA-P9.
QA-Cu7 is also a Circular Cubic with focus QA-P41.
QL-Cu1 is a Circular Cubic through the 6 intersection points of the 4 defining lines of a Quadrilateral and the Miquel Point QL-P1, which is its focus.

Eckart Schmidt and Bernard Keizer figured out next scheme:

• bipartite focal circular cubics can be taken as
... QA-Cu1 (YES), QL-Cu1 (YES), QA-Cu7 (YES).
• bipartite nonfocal circular cubics can be taken as
... QA-Cu1 (YES), QL-Cu1 (NO), QA-Cu7 (NO).
• monopartite focal circular cubics can be taken as
... QA-Cu1 (NO), QL-Cu1 (YES), QA-Cu7 (NO).
• monopartite nonfocal circular cubics can be taken as
... QA-Cu1 (NO), QL-Cu1 (NO), QA-Cu7 (NO).
A cubic is called focal, when its singular focus (see 7P-s-P1) lies on the curve.
See Ref-66, QPG-message #882.

Properties

• A circular cubic is determined by 7 points and the circular points at infinity. Through 6 random points infinitely many circular cubics can be drawn. According to the Cayley-Bacharach theorem for circular cubics they all pass through a common point, called the Cayley-Bacharach point for circular cubics (6P-s-P2).
• When we consider a circular cubic Cux = 7P-s-Cu1(P1,P2,P3,P4,P5,P6,Px), where Px is some point on 6P-s-Ci1, then 6P-s-Tf1(X) = 6P-s-Tf1(Px) for all points X on Cux. Hence we have a pencil of circular cubics, all passing through 6P-s-P2 and each corresponding with its own unique point on circle 6P-s-Ci1. See Ref-66, QPG-messages #653 and #657.
7P-s-Cu1 is 7P-s-Tf1 invariant with a pivot 7P-s-P2 in a common intersection of X.7P-s-Tf1(X) on the curve, whose tangential 7P-s-P3 is its 7P-s-Tf1 partner. See Ref-66, Eckart Schmidt, QPG-message#652.