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Encyclopedia > Conic section
Wikibooks has more on the topic of
Conic section
Types of conic sections
Table of conics, Cyclopaedia, 1728

In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties. Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... Download high resolution version (1080x600, 168 KB)conic sections source:me File links There are no pages that link to this file. ... Download high resolution version (1080x600, 168 KB)conic sections source:me File links There are no pages that link to this file. ... Image File history File linksMetadata Download high resolution version (1689x2991, 1122 KB) This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain. ... Image File history File linksMetadata Download high resolution version (1689x2991, 1122 KB) This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain. ... 1913 advertisement for Encyclopædia Britannica. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... This article is about the geometric object, for other uses see Cone. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... This article is about the mathematical construct. ... Apollonius of Perga [Pergaeus] (ca. ...

Contents

Types of conics

Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to generator lines of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.) Circle illustration This article is about the shape and mathematical concept of circle. ... For other uses, see Ellipse (disambiguation). ... For other uses, see Curve (disambiguation). ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...


Degenerate cases

There are a number of degenerate cases, in which the plane passes through the apex of the cone. The intersection in these cases can be a straight line (when the plane is tangential to the surface of the cone); a point (when the angle between the plane and the axis of the cone is larger than this); or a pair of intersecting lines (when the angle is smaller). There is also a degenerate where the cone is a cylinder (the vertex is at infinity) which can produce two parallel lines. A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...


Eccentricity

Ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with fixed focus F and directrix.

The four defining conditions above can be combined into one condition that depends on a fixed point F (the focus), a line L (the directrix) not containing F and a nonnegative real number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. Image File history File links Download high resolution version (540x960, 27 KB)Chart of the eccentricity of the major objects in our solar system. ... Image File history File links Download high resolution version (540x960, 27 KB)Chart of the eccentricity of the major objects in our solar system. ... (This page refers to eccentricity in mathematics. ...


For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is a / e, where is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is . The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...


In the case of a circle, the eccentricity e = 0, and one can imagine the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity.


The eccentricity of a conic section is thus a measure of how far it deviates from being circular.


For a given , the closer is to 1, the smaller is the semi-minor axis. In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ...


Cartesian coordinates

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. The equation will be of the form Fig. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

with , , not all zero.

then:

  • if , the equation represents an ellipse (unless the conic is degenerate, for example );
    • if and , the equation represents a circle;
  • if , the equation represents a parabola;
  • if , the equation represents a hyperbola;

Note that A and B are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the previous sections. For other uses, see Ellipse (disambiguation). ... Circle illustration This article is about the shape and mathematical concept of circle. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...


Through change of coordinates these equations can be put in standard forms:

  • Circle:
  • Ellipse: ,
  • Parabola:
  • Hyperbola:
  • Rectangular Hyperbola:

Such forms will be symmetrical about the x-axis and for the circle, ellipse and hyperbola symmetrical about the y-axis.
The rectangular hyperbola however is only symmetrical about the lines and . Therefore its inverse function is exactly the same as its original function.


These standard forms can be written as parametric equations, Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

  • Circle: ,
  • Ellipse: ,
  • Parabola: ,
  • Hyperbola: or .
  • Rectangular Hyperbola:

Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as: In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius, allow affine transformations to be easily represented by a matrix. ...

A1x2 + A2y2 + A3z2 + 2B1xy + 2B2xz + 2B3yz = 0.

Or in matrix notation In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

The matrix is called the matrix of the conic section.


is called the determinant of the conic section. If Δ = 0 then the conic section is said to be degenerate, this means that the conic section is in fact a union of two straight lines. A conic section that intersects itself is always degenerate, however not all degenerate conic sections intersect themselves, if they do not they are straight lines. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...


For example, the conic section reduces to the union of two lines:


.


Similarly, a conic section sometimes reduces to a (single) line:


.


is called the discriminant of the conic section. If δ = 0 then the conic section is a parabola, if δ<0, it is an hyperbola and if δ>0, it is an ellipse. A conic section is a circle if δ>0 and A1 = A2, it is an rectangular hyperbola if δ<0 and A1 = -A2. It can be proven that in the complex projective plane CP2 two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection points and there is always 1 intersection point (possibilities: 4 distinct intersection points, 2 singular intersection points and 1 double intersection points, 2 double intersection points, 1 singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating[1]. In algebra, the discriminant of a polynomial is a certain expression in the coefficients of the polynomial which equals zero if and only if the polynomial has multiple roots in the complex numbers. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... For other uses, see Ellipse (disambiguation). ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ... In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ... Intersecting airplane trails. ... For other uses, see tangent (disambiguation). ...


Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is said to be tangent and it is called the tangent line. Because every straight line intersects a conic section twice, each conic section has two points at infinity (the intersection points with the line at infinity). If these points are real, the conic section must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic section has one double point at infinity it is a parabola. If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle. If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated it is neither a parabola nor an ellipse nor a hyperbola. A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... For other uses, see Infinity (disambiguation). ... In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... For other uses, see Ellipse (disambiguation). ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... Circle illustration This article is about the shape and mathematical concept of circle. ...


Polar coordinates

Semi-latus rectum in the case of an ellipse

The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. Image File history File links Elps-slr. ... Image File history File links Elps-slr. ... In mathematics, the latus rectum of a conic section is the chord parallel to the directrix and passing through the single focus, or one of the two foci. ...


In polar coordinates, a conic section with one focus at the origin and, if any, the other on the x-axis, is given by the equation A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...

.

As above, for e = 0, we have a circle, for 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.


Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence. Plot of y = x3 with inflection point of (0,0). ... Laminar flow (bottom) and turbulent flow (top) over a submarine hull. ... In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ...


Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. For other uses, see Astronomy (disambiguation). ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... Gravity is a force of attraction that acts between bodies that have mass. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... This article is about the n-body problem in classical mechanics. ...


In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. Projective geometry is a non-metrical form of geometry. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ...


For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola. Circle illustration This article is about the shape and mathematical concept of circle. ... For other uses, see Ellipse (disambiguation). ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...


Dandelin spheres

See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix. Dandelin Spheres&#8212;graphics by Hop David In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus: Each Dandelin sphere touches, but does not cross, both the plane and the cone. ...


See also

The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic section. ... In geometry, the focus (pl. ... A Lambert conformal conic projection (LCC) is a conic map projection, which is often used for aeronautical charts. ... In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... f(x) = x2 - x - 2 A quadratic function, in mathematics, is a polynomial function of the form , where . ... Rotation of Axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counter-clockwise direction with respect to the origin (0, 0) through a scalar quantity denoted by θ. With the exception of the degenerate cases, if a general second-degree equation has... Dandelin Spheres&#8212;graphics by Hop David In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus: Each Dandelin sphere touches, but does not cross, both the plane and the cone. ...

External links

Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

References

  1. ^ E. J. Wilczynski, Some remarks on the historical development and the future prospects of the differential geometry of plane curves, Bull. Amer. Math. Soc. 22 (1916) pp. 317-329.

  Results from FactBites:
 
Conic Sections (1860 words)
Conic sections are among the oldest curves, and is an oldest math subject studied systematically and thoroughly.
The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone.
The directrix of the conics is the intersection of the cutting plane and the plane of the tangent circle.
conic section (403 words)
A conic section is an algebraic curve of the 2nd degree (and every 2nd degree equation represents a conic).
The conic section can be defined as the collection of points P for which the ratio 'distance to F / distance to l' is constant.
A conic section is used, together with a hexagon, to show Pascal's theorem (or its dual, Brianchon's theorem).
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