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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 algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4.
In inversive geometry, a line is a degenerate case of a circle, with infinite radius. Two parallel lines also form a degenerate parabola. A line segment can be viewed as a degenerate case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints, and the eccentricity goes to one.
If a set of points is not in general linear position, it is called a degenerate case or degenerate configuration, which implies that they satisfy a linear relation that need not always hold. A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).
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.
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 ...
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 ...
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 .