Conic section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic[1] to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables which can be written in the form The geometric properties of the conic can be deduced from its equation.
In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.
History
Menaechmus and early works
It is believed that the first definition of a conic section was given by Menaechmus (died 320 BC) as part of his solution of the Delian problem (Duplicating the cube).[b][25] His work did not survive, not even the names he used for these curves, and is only known through secondary accounts.[26] The definition used at that time differs from the one commonly used today. Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone (such a line is called a generatrix). Three types of cones were determined by their vertex angles (measured by twice the angle formed by the hypotenuse and the leg being rotated about in the right triangle). The conic section was then determined by intersecting one of these cones with a plane drawn perpendicular to a generatrix. The type of the conic is determined by the type of cone, that is, by the angle formed at the vertex of the cone: If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola (but only one branch of the curve).[27]
Euclid (fl. 300 BC) is said to have written four books on conics but these were lost as well.[28] Archimedes (died c. 212 BC) is known to have studied conics, having determined the area bounded by a parabola and a chord in Quadrature of the Parabola. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and Spheroids.[29]
Given two pencils of lines at two points (all lines containing and resp.) and a but not perspective mapping of onto . Then the intersection points of corresponding lines form a non-degenerate projective conic section.[49][50][51][52]
projective
In the complex projective plane
In the complex plane C2, ellipses and hyperbolas are not distinct: one may consider a hyperbola as an ellipse with an imaginary axis length. For example, the ellipse becomes a hyperbola under the substitution geometrically a complex rotation, yielding . Thus there is a 2-way classification: ellipse/hyperbola and parabola. Extending the curves to the complex projective plane, this corresponds to intersecting the line at infinity in either 2 distinct points (corresponding to two asymptotes) or in 1 double point (corresponding to the axis of a parabola); thus the real hyperbola is a more suggestive real image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity.
Further unification occurs in the complex projective plane CP2: the non-degenerate conics cannot be distinguished from one another, since any can be taken to any other by a projective linear transformation.
It can be proven that in CP2, two conic sections have four points in common (if one accounts for multiplicity), so there are between 1 and 4 intersection points. The intersection possibilities are: four distinct points, two singular points and one double point, two double points, one singular point and one with multiplicity 3, one point with multiplicity 4. If any intersection point has multiplicity > 1, the two curves are said to be tangent. If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be superosculating.[62]
Furthermore, each straight line intersects each conic section twice. If the intersection point is double, the line is a tangent line.
Intersecting with the line at infinity, each conic section has two points at infinity. If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola. If the points at infinity are the cyclic points (1, i, 0) and (1, –i, 0), the conic section is a circle. If the coefficients of a conic section are real, the points at infinity are either real or complex conjugate.
given the two conics and , consider the pencil of conics given by their linear combination
identify the homogeneous parameters which correspond to the degenerate conic of the pencil. This can be done by imposing the condition that and solving for and . These turn out to be the solutions of a third degree equation.
given the degenerate conic , identify the two, possibly coincident, lines constituting it.
intersect each identified line with either one of the two original conics; this step can be done efficiently using the dual conic representation of
The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections.
In particular two conics may possess none, two or four possibly coincident intersection points.
An efficient method of locating these solutions exploits the homogeneous matrix representation of conic sections, i.e. a 3 × 3 symmetric matrix which depends on six parameters.
The procedure to locate the intersection points follows these steps, where the conics are represented by matrices:[69]
Generalizations
Conics may be defined over other fields (that is, in other pappian geometries). However, some care must be used when the field has characteristic 2, as some formulas can not be used. For example, the matrix representations used above require division by 2.
A generalization of a non-degenerate conic in a projective plane is an oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line.
Generalizing the focus properties of conics to the case where there are more than two foci produces sets called generalized conics.
The intersection of an elliptic cone with a sphere is a spherical conic, which shares many properties with planar conics.