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This case always occurs as a degenerate conic in a pencil of circles. However, in other contexts it is not considered as a degenerate conic, as its equation is not of degree 2. The case of coincident lines occurs if and only if the rank of the 3×3 matrix is 1; in all other degenerate cases its rank is 2. [3]: p.108
In such a degenerate case, the solution set is said to be degenerate. For some classes of composite objects, the degenerate cases depend on the properties that are specifically studied. In particular, the class of objects may often be defined or characterized by systems of equations. In most scenarios, a given class of objects may be defined by ...
If the conic is non-degenerate, the conjugates of a point always form a line and the polarity defined by the conic is a bijection between the points and lines of the extended plane containing the conic (that is, the plane together with the points and line at infinity). If the point p lies on the conic Q, the polar line of p is the tangent line ...
The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying these three axioms either have at most one line, or are projective spaces of some dimension over a division ring , or are non-Desarguesian planes .
Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 2 5 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative ...
It includes real circles, imaginary circles, and two degenerate point circles called the Poncelet points of the pencil. Each point in the plane belongs to exactly one circle of the pencil. A parabolic pencil (as a limiting case) is defined where two generating circles are tangent to each other at a single point. It consists of a family of real ...
A special case is Pascal's theorem, in which case the two cubics in question are all degenerate: given six points on a conic (a hexagon), consider the lines obtained by extending opposite sides – this yields two cubics of three lines each, which intersect in 9 points – the 6 points on the conic, and 3 others. These 3 additional points lie ...
If the conic is a circle, then another degenerate case says that for a triangle, the three points that appear as the intersection of a side line with the corresponding side line of the Gergonne triangle, are collinear. Six is the minimum number of points on a conic about which special statements can be made, as five points determine a conic.