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The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry.
That is, if two real non-degenerated conics are defined by quadratic polynomial equations f = 0 and g = 0, the conics of equations af + bg = 0 form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose f and g so that the given degenerate conic belongs to the pencil they determine.
Media in category "Conic sections" This category contains only the following file. Drawing an ellipse via two tacks a loop and a pen 2.jpg 480 × 640; 24 KB
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 ...
Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances ...
As a space curve, a spherical conic is a quartic, though its orthogonal projections in three principal axes are planar conics. Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors.
It is usual, when dealing with dual and common conic sections, to call the common conic section a point conic and the dual conic a line conic. In the case that the underlying field has = all the tangents of a point conic intersect in a point, called the knot (or nucleus) of the conic. Thus, the dual of a non-degenerate point conic is a subset ...
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.