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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
Objects such as the digon and monogon can be viewed as degenerate cases of polygons: valid in a general abstract mathematical sense, but not part of the original Euclidean conception of polygons. A random variable which can only take one value has a degenerate distribution ; if that value is the real number 0, then its probability density is ...
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 ...
whose values for =,,,, are ,,,, – there are no curves of degree 0 (a single point is a point and is thus determined by a point, which is codimension 2), 2 points determine a line, 5 points determine a conic, 9 points determine a cubic, 14 points determine a quartic, and so forth.
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.
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 ...
A limiting case is sometimes a degenerate case in which some qualitative properties differ from the corresponding properties of the generic case. For example: A point is a degenerate circle, whose radius is zero. A parabola can degenerate into two distinct or coinciding parallel lines. An ellipse can degenerate into a single point or a line ...
2 The number of lines meeting 4 general lines in space; 8 The number of circles tangent to 3 general circles (the problem of Apollonius). 27 The number of lines on a smooth cubic surface (Salmon and Cayley) 2875 The number of lines on a general quintic threefold; 3264 The number of conics tangent to 5 plane conics in general position