Types of conic sections

Table of conics, Cyclopaedia, 1728

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a circular conical surface) with a plane. Cyclopaedia or A Universal Dictionary of Arts and Sciences ( folio, 2 vols Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed 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 three types of conics are the hyperbola, ellipse, and parabola. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The circle and the ellipse arise when the intersection of cone and plane is a closed curve. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone. If the cutting plane is parallel 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 (nappes) of the cone, producing two separate unbounded 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). In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume

Where the cone is a cylinder (the vertex is at infinity) cylindric sections are obtained. Although these yield mostly ellipses (or circles)^[1] as normal, a degenerate case of two parallel lines can also be produced.

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). In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. 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.

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. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae The distance from the center to a focus is .

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) is a Line segment associated with most Conic sections (that is with ellipses and

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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The equation will be of the form

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;
  • if we also have , the equation represents a rectangular hyperbola. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

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:
• 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,

• Circle: ,
• Ellipse: ,
• Parabola: ,
• Hyperbola: or . In Mathematics, parametric equations are a method of defining a curve
• Rectangular Hyperbola:

Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as:

A₁x² + A₂y² + A₃z² + 2B₁xy + 2B₂xz + 2B₃yz = 0. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

Or in matrix notation

The matrix is called the matrix of the conic section. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

is called the determinant of the conic section. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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 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. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the 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. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a A conic section is a circle if δ>0 and A₁ = A₂, it is an rectangular hyperbola if δ<0 and A₁ = -A₂. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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). In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. In Euclidean geometry, the Intersection of a line and a line can be the Empty set, a point, or a line If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating^[2].

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. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. 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). Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" "Ideal line" redirects here For the ideal line in racing see Racing line. 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. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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

,

where e is the eccentricity and l is the semi-latus rectum (see below). In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by 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.

Parameters

Various parameters can be associated with a conic section.

conic section	equation	eccentricity (e)	linear eccentricity (c)	semi-latus rectum (l)	focal parameter (p)
circle		0	0
ellipse	,
parabola	,	1	a		2a
hyperbola	,

conic parameters in the case of an ellipse

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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions e is called the eccentricity of the conic section. In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci). In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle

The latus rectum (2l) is the chord parallel to the directrix and passing through the focus (or one of the two foci). In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

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. Distance is a numerical description of how far apart objects are

The relation p = l / e holds.

Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. In Differential calculus, an inflection point, or point of inflection (or inflexion) is a point on a Curve at which the Curvature This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence. Laminar flow, sometimes known as streamline flow occurs when a fluid flows in parallel layers with no disruption between the layers In Fluid dynamics, turbulence or turbulent flow is a fluid 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. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Gravitation is a natural Phenomenon by which objects with Mass attract one another 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. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by

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, notable for its principle of duality. 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 It describes what

For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

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. In Mathematics, the matrix representation of conic sections is one way of studying a Conic section, its axis, vertices, foci, e. a 3x3 symmetric matrix which depends on six parameters. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}

The procedure to locate the intersection points follows these steps:

• given the two conics C₁ and C₂ consider the pencil of conics given by their linear combination λC₁ + μC₂
• identify the homogeneous parameters (λ,μ) which corresponds to the degenerate conic of the pencil. This can be done by imposing that det(λC₁ + μC₂) = 0, which turns out to be the solution to a third degree equation.
• given the degenerate cone C₀, 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 C₀
• 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. In Geometry, a nondegenerate Conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus Each

See also

References

• Akopyan, A. V. and Zaslavsky, A. A. (2007). Geometry of Conics. American Mathematical Society, 134. ISBN 0821843230.  

External links

• Derivations of Conic Sections at Convergence
• Conic sections at Special plane curves.
• Eric W. Weisstein, Conic Section at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• 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.