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Conics, also known as conic sections to emphasize their three-dimensional geometry, arise as the intersection of a plane with a cone. Degeneracy occurs when the plane contains the apex of the cone or when the cone degenerates to a cylinder and the plane is parallel to the axis of the cylinder. See Conic section#Degenerate cases for details.
A degenerate case thus has special features which makes it non-generic, or a special case. However, not all non-generic or special cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate cases often correspond to singularities, either in the object ...
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).
All the conic sections share a reflection property that can be stated as: All mirrors in the shape of a non-degenerate conic section reflect light coming from or going toward one focus toward or away from the other focus. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going ...
Degenerate conics follow by continuity (the theorem is true for non-degenerate conics, and thus holds in the limit of degenerate conic). A short elementary proof of Pascal's theorem in the case of a circle was found by van Yzeren (1993), based on the proof in (Guggenheimer 1967). This proof proves the theorem for circle and then generalizes it ...
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 ...
The space of (possibly degenerate) conics in the complex projective plane CP 2 can be identified with the complex projective space CP 5 (since each conic is defined by a homogeneous degree-2 polynomial in three variables, with 6 complex coefficients, and multiplying such a polynomial by a non-zero complex number does not change the conic).
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form A x 2 + B x y + C y 2 + D x + E y + F = 0 with A , B , C not all zero. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with ...