# User Contributed Dictionary

#### Derived terms

# Extensive Definition

wikibooks Conic
sections

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.

## Types of conics

The five types of conics are the circle, hyperbola, ellipse, parabola, and rectangular hyperbola. The circle and the ellipse 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.)### Degenerate cases

There are multiple 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 tangential); or a pair of intersecting lines (when the angle is smaller).Where the cone is a cylinder (the vertex is at
infinity) cylindric sections are obtained. Although these yield
mostly ellipses (or circles) as normal, a degenerate case of two
parallel lines can also be produced.

### Eccentricity

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 1 a hyperbola.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 a \ 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
ae \ .

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 a \ , the closer e \ is to 1, the
smaller is the semi-minor
axis.

## 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- Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\; with A \ , B \ , C \ not all zero.

- if B^2 - 4AC , the equation represents an ellipse (unless the conic is
degenerate, for example x^2 + y^2 + 10 = 0 \ );
- if A = C \ and B = 0 \ , the equation represents a circle;

- if B^2 - 4AC = 0 \ , the equation represents a parabola;
- if B^2 - 4AC > 0 \ , the equation represents a hyperbola;
- if we also have A + C = 0 \ , the equation represents a rectangular hyperbola.

Note that A and B are just polynomial
coefficients, not the lengths of semi-major/minor axis as defined
in the previous sections.

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

- Circle: x^2+y^2=r^2\,
- Ellipse: +=1 \ , +=1 \
- Parabola: y^2=4ax\, \ , x^2=4ay\, \
- Hyperbola: -=1 \ , -=-1 \
- Rectangular Hyperbola: xy=c^2 \

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 y = x\ and y = -x\ . Therefore its inverse function is
exactly the same as its original function.

These standard forms can be written as parametric
equations,

- Circle: (a\cos\theta,a\sin\theta)\,,
- Ellipse: (a\cos\theta,b\sin\theta)\,,
- Parabola: (a t^2,2 a t)\,,
- Hyperbola: (a\sec\theta,b\tan\theta)\, or (\pm a\cosh u,b \sinh u)\,.
- Rectangular Hyperbola: (ct,)\,

## Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as:- A_1x^2 + A_2y^2 + A_3z^2 + 2B_1xy + 2B_2xz + 2B_3yz = 0.

Or in matrix
notation

- \beginx & y & z\end . \beginA_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end . \beginx\\y\\z\end = 0.

The matrix M=\beginA_1 & B_1 & B_2\\B_1
& A_2 & B_3\\B_2&B_3&A_3\end is called the matrix
of the conic section.

\Delta = \det(M) = \det\left(\beginA_1 & B_1
& B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end\right)
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.

For example, the conic section \beginx & y
& z\end . \begin1 & 0 & 0\\0 & -1 &
0\\0&0&0\end . \beginx\\y\\z\end = 0 reduces to the union
of two lines:

\ = \ = \ \cup \.

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

\ = \=\ \cup \ = \.

\delta = \det\left(\beginA_1 & B_1\\B_1 &
A_2\end\right) is called the discriminant of the conic
section. If δ = 0 then the conic section is a parabola, if δ0, it
is an ellipse. A conic
section is a circle if
δ>0 and A1 = A2, it is an rectangular
hyperbola if δ1 = -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.

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.

## Polar coordinates

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- r = ,

## Parameters

Various parameters can be associated with a conic section.For every conic section, there exist a fixed
point F, a fixed line L and a non-negative number e such that the
conic section consists of all points whose distance to F equals e
times their distance to L. e is called the eccentricity
of the conic section.

The linear
eccentricity (c) is the distance between the center and the
focus
(or one of the two foci).

The latus rectum (2l) is the chord parallel to
the directrix and
passing through the focus (or one of the two foci).

The semi-latus rectum (l) is half the latus
rectum. The focal parameter (p) is the distance from the focus (or one
of the two foci) to the directrix.

The relation p = l/e holds.

## 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.## 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.In projective
geometry, the conic sections in the projective plane are
equivalent to each other up to projective
transformations.

For specific applications of each type of conic
section, see the articles circle, ellipse, parabola, and hyperbola.

## Intersecting two conics

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 sections. In particular two conics may possess none, two, four possibly coincident intersection points. The best method to locate these solutions is to exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.The procedure to locate the intersection points
follows these steps:

- given the two conics C_1 and C_2 consider the pencil of conics given by their linear combination \lambda C_1 + \mu C_2
- identify the homogeneous parameters (\lambda,\mu) which corresponds to the degenerate conic of the pencil. This can be done by imposing that det(\lambda C_1 + \mu C_2) = 0, which turns out to be the solution to a third degree equation.
- given the degenerate cone C_0, identify the two, possibly coincident, lines constituting it
- intersects each identified line with one of the two original conic; this step can be done efficiently using the dual conic representation of C_0
- the points of intersection will represent the solution to the initial equation system

## 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.## See also

- Focus (geometry), an overview of properties of conic sections related to the foci
- Lambert conformal conic projection
- Matrix representation of conic sections
- Quadrics, the higher-dimensional analogs of conics
- Quadratic function
- Rotation of Axes
- Dandelin spheres
- Projective conics

## References

- Geometry of Conics

## External links

- Derivations of Conic Sections at Convergence
- Conic sections at Special plane curves.
- Determinants and Conic Section Curves
- Occurrence of the conics. Conics in nature and elsewhere.
- Conics. An essay on conics and how they are generated.
- See Conic Sections at cut-the-knot for a sharp proof that any finite conic section is an ellipse and Xah Lee for a similar treatment of other conics.

directrix in Afrikaans: Keëlsnit

directrix in Arabic: قطوع مخروطية

directrix in Bengali: কনিক

directrix in Bulgarian: Конично сечение

directrix in Catalan: Cònica

directrix in Czech: Kuželosečka

directrix in Danish: Keglesnit

directrix in German: Kegelschnitt

directrix in Modern Greek (1453-): Κωνική
τομή

directrix in Spanish: Sección cónica

directrix in Esperanto: Koniko

directrix in French: Conique

directrix in Korean: 원뿔 곡선

directrix in Hindi: शांकव

directrix in Indonesian: Irisan kerucut

directrix in Italian: Sezione conica

directrix in Hebrew: חתכי חרוט

directrix in Lithuanian: Kūgio pjūvis

directrix in Hungarian: Kúpszelet

directrix in Dutch: Kegelsnede

directrix in Japanese: 円錐曲線

directrix in Norwegian: Kjeglesnitt

directrix in Norwegian Nynorsk:
Kjeglesnitt

directrix in Polish: Krzywa stożkowa

directrix in Portuguese: Cónica

directrix in Romanian: Conică

directrix in Russian: Коническое сечение

directrix in Slovak: Kužeľosečka

directrix in Slovenian: Stožnica

directrix in Finnish: Kartioleikkaus

directrix in Swedish: Kägelsnitt

directrix in Tamil: கூம்பு வெட்டு

directrix in Thai: ภาคตัดกรวย

directrix in Vietnamese: Đường cô-nic

directrix in Turkish: Konikler

directrix in Urdu: تکونی قطعات

directrix in Chinese: 圆锥曲线