9P-s-Cu1 9P-Cubic

A cubic in general is defined by 9 random points.
A cubic with 7 random points and the 2 imaginary circular points at infinity is called a circular cubic and can be found at 7P-s-Cu1.

See picture and Ref-63, page 207. 
See also remarks at 7P-s-Cu1.

9P s Cu1 Cubic 02


• A regular cubic is determined by 9 points. Through 8 random points infinitely many cubics can be drawn. However according to the Cayley-Bacharach theorem they will all pass through a common point, named the Cayley-Bacharach point (8P-s-P1). Therefore by varying one defining point P9 on the cubic many new cubics will occur with one common point 8P-s-P1 (apart from the other 8 defining points). And therefore each point on a cubic has a corresponding Cayley-Bacharach point. See also 7P-s-Tf1.
• When the cubic is circular (see 7P-s-Cu1) then there is the corresponding Cayley-Bacharach point 6P-s-P2. See also 5P-s-Tf1.