Table of conics, 1728 In, a conic section (or simply conic) is a obtained as the intersection of the of a with a. The three types of conic section are the, the, and the. The is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when undertook a systematic study of their properties. The conic sections of the have various distinguishing properties.
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Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a, and some particular line, called a directrix, are in a fixed ratio, called the. The type of conic is determined by the value of the eccentricity.
In, a conic may be defined as a of degree 2; that is, as the set of points whose coordinates satisfy a in two variables. This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions.
In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane (adding a line at infinity) this apparent difference vanishes, and the commonality becomes evident. Further extension, by expanding the coordinates to admit coordinates, provides the means to see this unification algebraically.
The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. A conic is the curve obtained as the intersection of a, called the cutting plane, with the surface of a double (a cone with two nappes). It shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called and some authors do not consider them to be conics at all.
Unless otherwise stated, 'conic' in this article will refer to a non-degenerate conic. There are three types of conics, the,. The is a special kind of ellipse, although historically it had been considered as a fourth type (as it was by Apollonius). The and the arise when the intersection of the cone and plane is a. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola.
In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. Eccentricity, focus and directrix. Main article: defined a conic as the point set given by all the absolute points of a that has absolute points. Von Staudt introduced this definition in Geometrie der Lage (1847) as part of his attempt to remove all metrical concepts from projective geometry. A polarity, π, of a projective plane, P, is an involutory (i.e., of order two) between the points and the lines of P that preserves the. Thus, a polarity relates a point Q with a line q and, following, q is called the polar of Q and Q the pole of q.
An absolute point ( line) of a polarity is one which is incident with its polar (pole). A von Staudt conic in the real projective plane is equivalent to a. Constructions A conic can not be constructed as a continuous curve (or two) with straightedge and compass.
However, there are several methods that are used to construct as many individual points on a conic, with straightedge and compass, as desired. One of them is based on the converse of Pascal's theorem, namely, if the points of intersection of opposite sides of a hexagon are collinear, then the six vertices lie on a conic. Specifically, given five points, A, B, C, D, E and a line passing through E, say EG, a point F that lies on this line and is on the conic determined by the five points can be constructed.
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Let AB meet DE in L, BC meet EG in M and let CD meet LM at N. Then AN meets EG at the required point F. By varying the line through E,as many additional points on the conic as desired can be constructed.