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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 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 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 ...
To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let β be the determinant of the 3 × 3 matrix of the conic section—that is, β = (AC − B 2 / 4 )F + BED − CD 2 − AE 2 / 4 ; and let α = B 2 − 4AC be the discriminant.
The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2.This is because the n-th degree terms are ,, …,, numbering n + 1 in total; the (n − 1) degree terms are ,, …,, numbering n in total; and so on through the first degree terms and , numbering 2 in total, and the single zero degree term (the constant).
Steiner observed that the conics tangent to a given conic form a degree 6 hypersurface in CP 5. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 6 5 = 7776, which was Steiner's incorrect ...
The set of such intersections is called a projective conic, and in acknowledgement of the work of Jakob Steiner, it is referred to as a Steiner conic. Suppose a projectivity is formed by two perspectivities centered on points A and B , relating x to X by an intermediary p :