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Types of conic sections:
1. Parabola
2. Circle and ellipse
3. Hyperbola
Table of conics, Cyclopaedia, 1728

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Contents

History

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Menaechmus

It is believed that the first definition of a conic section is due to Menaechmus. This work does not survive, however, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today in that it requires the plane cutting the cone to be perpendicular to the line that generates the cone as a surface of revolution. Thus the shape of the conic is determined 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. Note that the circle cannot be defined this way and was not considered a conic at this time.

Euclid is said to have written four books on conics but these were lost as well. Archimedes is known to have studied conics, having determined the area bounded by a parabola and an ellipse. The only part of this work to survive is a book on the solids of revolution of conics.

Apollonius of Perga

The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga, whose eight volume Conic Sections summarized the existing knowledge at the time and greatly extended it. Apollonius's major innovation was to characterize a conic using properties within the plane and intrinsic to the curve; this greatly simplified analysis. With this tool, it was now possible to show that any plane cutting the cone, regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today.

Pappus is credited with discovering importance of the concept of a focus of a conic, and the discovery of the related concept of a directrix.

Omar Khayyám

Apollonius's work was translated into Arabic and much of his work only survives through the Arabic version. Muslims found applications to the theory; the most notable of these was the Persian mathematician and poet Omar Khayyám who used conic sections to solve algebraic equations.

Europe

Johann Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept of limits. Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and this help provide impetus for the study of this new field. In particular, Pascal discovered a theorem known as the hexagrammum mysticum from which many other properties of conics can be deduced. Meanwhile, René Descartes applied his newly discovered Analytic geometry to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra.

Types

The three types of conics are the hyperbola, ellipse, and parabola. The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone -- for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis 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).

Where the cone is a cylinder, i.e. with the vertex at infinity, cylindric sections are obtained. Although these yield mostly ellipses (or circles) as usual, a degenerate case of two parallel lines, known as a ribbon, can also be produced, and it is also possible for there to be no intersection at all.[1]

Eccentricity

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

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.

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 to F 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\text{ with }A, B, C\text{ not all zero.} \,

then:

  • if B2 − 4AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x2 + y2 + 10 = 0);
    • if A = C and B = 0, the equation represents a circle;
  • if B2 − 4AC = 0, the equation represents a parabola;
  • if B2 − 4AC > 0, 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 following sections.

In matrix notation the equation above becomes:

\begin{bmatrix}x & y \end{bmatrix} . \begin{bmatrix}A & B/2\\B/2 & C\end{bmatrix} . \begin{bmatrix}x\\y\end{bmatrix} +Dx +Ey+F= 0.

or

\begin{bmatrix}x & y & 1\end{bmatrix} . \begin{bmatrix}A & B/2 & D/2\\B/2 & C & E/2\\D/2&E/2&F\end{bmatrix} . \begin{bmatrix}x\\y\\1\end{bmatrix} = 0.

and

B^2 - 4AC = -4 \left|\begin{matrix}A & B/2\\B/2 & C\end{matrix}\right| .


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

  • Circle: x^2+y^2=r^2 \,
  • Ellipse: {x^2\over a^2}+{y^2\over b^2}=1 , {x^2\over b^2}+{y^2\over a^2}=1
  • Parabola: y^2=4ax ,\; x^2=4ay
  • Hyperbola: {x^2\over a^2}-{y^2\over b^2}=1,\; {x^2\over b^2}-{y^2\over a^2}=-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: (acosθ,asinθ),
  • Ellipse: (acosθ,bsinθ),
  • Parabola: (at2,2at),
  • Hyperbola: (asecθ,btanθ) or (\pm a\cosh u,b \sinh u).
  • Rectangular Hyperbola: \left(ct,{c \over t} \right)

Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as:

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

Or in matrix notation

\begin{bmatrix}x & y & z\end{bmatrix} . \begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix} . \begin{bmatrix}x\\y\\z\end{bmatrix} = 0.

The matrix M=\begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix} is called the matrix of the conic section.

 \Delta = \det(M) = \det\left(\begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix}\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 \begin{bmatrix}x & y & z\end{bmatrix} . \begin{bmatrix}1 & 0 & 0\\0 & -1 & 0\\0&0&0\end{bmatrix} . \begin{bmatrix}x\\y\\z\end{bmatrix} = 0 reduces to the union of two lines:

 \{ x^2 - y^2 = 0\} = \{(x+y)(x-y)=0\} = \{x+y=0\} \cup \{x-y=0\}.

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

\{x^2+2xy+y^2 = 0\} = \{(x+y)^2=0\}=\{x+y=0\} \cup \{x+y=0\} = \{x+y=0\}.

 \delta = \det\left(\begin{bmatrix}A_1 & B_1\\B_1 & A_2\end{bmatrix}\right) 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 and B1 = 0, 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.[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. 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 = { l \over {1 + e \cos \theta} },

where e is the eccentricity and l is the semi-latus rectum (see below). 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.

Development of the conic section as the eccentricity e increases

Parameters

Various parameters are associated with a conic section.

conic section equation eccentricity (e) linear eccentricity (c) semi-latus rectum () focal parameter (p)
circle x^2+y^2=r^2 \, 0 0  r \,  \infty
ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \frac{\sqrt{a^2-b^2}}{a} \sqrt{a^2-b^2} \frac{b^2}{a} \frac{b^2}{\sqrt{a^2-b^2}}
parabola y2 = 4ax 1 a\, 2a \, 2a\,
hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \frac{\sqrt{a^2+b^2}}{a} \sqrt{a^2+b^2} \frac{b^2}{a} \frac{b^2}{\sqrt{a^2+b^2}}
Conic parameters in the case of an ellipse

Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity.

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

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

The semi-latus rectum () 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 following relations hold:

  • p e = \ell \,
  • a e = c \,

Properties

Irreducible 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 or four possibly coincident intersection points. The best method of locating these solutions 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 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 conic 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.

See also

Notes

  1. ^ "MathWorld: Cylindric section". http://mathworld.wolfram.com/CylindricSection.html.  
  2. ^ Wilczynski, E. J. (1916), "Some remarks on the historical development and the future prospects of the differential geometry of plane curves", Bull. Amer. Math. Soc. 22: 317–329  .

References

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

External links


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