In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse. The circle 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 Apollonius of Perga undertook a systematic study of their properties.
Mathematics includes the study of such topics as quantity, structure, space, and change.
In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that its curvature need not be zero.
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. Each of those lines is called a generatrix of the surface.
The conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conicto be the set of those points whose distances to some particular point, called a focus , and some particular line, called a directrix, are in a fixed ratio, called the eccentricity . The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions.
In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed. For example, one or two foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola. In addition, two foci are used to define the Cassini oval and the Cartesian oval, and more than two foci are used in defining an n-ellipse.
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
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 real coordinates to admit complex coordinates, provides the means to see this unification algebraically.
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.
The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (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 degenerate conics 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.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
There are three types of conics, the ellipse, parabola, and hyperbola. The circle is a special kind of ellipse, although historically it had been considered as a fourth type (as it was by Apollonius). The circle and the ellipse arise when the intersection of the 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, see diagram, 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 of the cone, producing two separate unbounded curves.
In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The elongation of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to but less than 1.
In mathematics, a parabola is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves.
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
A property that the conic sections share is often presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F (called the focus of the conic) is a constant multiple (called the eccentricity, e) of the distance from P to a fixed line L (called the directrix of the conic). For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.
A circle is a limiting case and is not defined by a focus and directrix, in the plane (however, see the section on the extension to projective planes). The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane that is its directrix.
An ellipse and a hyperbola each have two foci and distinct directrices for each of them. The line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis, also called transverse axis in the hyperbola. The midpoint of this line segment is called the center of the conic. a denote the distance from the center to a vertex of an ellipse or hyperbola. The distance from the center to a directrix is a/ while the distance from the center to a focus is ae.Let
A parabola does not have a center.
The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.
If the angle between the surface of the cone and its axis is and the angle between the cutting plane and the axis is the eccentricity is
A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres.
Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, and the center in these cases is the midpoint of the line segment joining the foci. Some of the other common features and/or parameters of conics are given below.
The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci).
The latus rectum is the chord parallel to the directrix and passing through the focus (or one of the two foci). Its length is denoted by 2ℓ.
The semi-latus rectum (ℓ) is half of the length of the latus rectum.
The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix.
When an ellipse or hyperbola are in standard position (the principle axis is the x-axis and the center is the origin) the vertices of the conic have coordinates (−a, 0) and (a, 0), with a non-negative.
The semi-major axis is the value a.
The semi-minor axis is the value b in the standard Cartesian equation of the ellipse or hyperbola.
The following relations hold:
These parameters are related as shown in the following table, where the standard position is assumed. In all cases, a and b are positive.
|conic section||equation||eccentricity (e)||linear eccentricity (c)||semi-latus rectum (ℓ)||focal parameter (p)|
After introducing Cartesian coordinates the focus-directrix property can be used to produce equations that the coordinates of the points of the conic section must satisfy. x-axis as the principal axis and the origin (the point (0,0)) as the center. The vertices would have coordinates (±a, 0) and foci coordinates (±c, 0). Define b by the equations c2 = a2 − b2 for an ellipse and c2 = a2 + b2 for a hyperbola. For a circle, c = 0 so a2 = b2. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the origin. A special case of the hyperbola occurs when its asymptotes are perpendicular. This special case is called a rectangular or equilateral hyperbola. In this case, the standard form is obtained by taking the asymptotes as the coordinate axes and the line x = y as the principal axis. The foci would have coordinates (c, c) and (−c, −c).By means of a change of coordinates (a rotation of axes and a translation of axes) these equations can be put into standard forms. For ellipses and (general) hyperbolas a standard form would have the
The first four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and hyperbola), or about the x-axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines y = x and y = −x.
These standard forms can be written parametrically as,
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate), and all conic sections arise in this way. The most general equation is of the form
with all coefficients real numbers and A, B, C not all zero.
The above equation can be written in matrix notation as
The general equation can also be written as
This form is a specialization of the homogeneous form used in the more general setting of projective geometry (see below).
The conic sections described by this equation can be classified in terms of the value , called the discriminant of the equation. Thus, the discriminant is − 4Δ where Δ is the matrix determinant
If the conic is non-degenerate, then:
In the notation used here, A and B are polynomial coefficients, in contrast to some sources that denote the semimajor and semiminor axes as A and B.
The discriminant B2 – 4AC of the conic section's quadratic equation (or equivalently the determinant AC – B2/4 of the 2×2 matrix) and the quantity A + C (the trace of the 2×2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, as is the determinant of the 3×3 matrix above. :pp. 60–62 The constant term F and the sum D2+E2 are invariant under rotation only. :pp. 60–62
When the conic section is written algebraically as
the eccentricity can be written as a function of the coefficients of the quadratic equation. 4AC = B2 the conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given byIf
where η = 1 if the determinant of the 3×3 matrix above is negative and η = −1 if that determinant is positive.
It can also be shown p. 89 that the eccentricity is a positive solution of the equation:
where again This has precisely one positive solution—the eccentricity— in the case of a parabola or ellipse, while in the case of a hyperbola it has two positive solutions, one of which is the eccentricity.
In the case of an ellipse or hyperbola, the equation
can be converted to canonical form in transformed variables as
where and are the eigenvalues of the matrix — that is, the solutions of the equation
— and is the determinant of the 3×3 matrix above, and is again the determinant of the 2×2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form.
In polar coordinates, a conic section with one focus at the origin and, if any, the other at a negative value (for an ellipse) or a positive value (for a hyperbola) on the x-axis, is given by the equation
where e is the eccentricity and l is the semi-latus rectum.
As above, for e = 0, the graph is a circle, for 0 < e < 1the graph is an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.
The polar form of the equation of a conic is often used in dynamics; for instance, determining the orbits of objects revolving about the Sun.
Just as two (distinct) points determine a line, five points determine a conic. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.
Four points in the plane in general linear position determine a unique conic passing through the first three points and having the fourth point as its center. Thus knowing the center is equivalent to knowing two points on the conic for the purpose of determining the curve.
Furthermore, a conic is determined by any combination of k points in general position that it passes through and 5 – k lines that are tangent to it, for 0≤k≤5.
Any point in the plane is on either zero, one or two tangent lines of a conic. A point on just one tangent line is on the conic. A point on no tangent line is said to be an interior point (or inner point) of the conic, while a point on two tangent lines is an exterior point (or outer point).
All the conic sections share a reflection property that can be stated as: All mirrors in the shape of a non-degenerate conic section reflect light coming from or going toward one focus toward or away from the other focus. In the case of the parabola, the second focus needs to be thought of as infinitely far away, so that the light rays going toward or coming from the second focus are parallel.
Pascal's theorem concerns the collinearity of three points that are constructed from a set of six points on any non-degenerate conic. The theorem also holds for degenerate conics consisting of two lines, but in that case it is known as Pappus's theorem.
Non-degenerate conic sections are always "smooth". This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence.
It is believed that the first definition of a conic section was given by Menaechmus (died 320 BCE) as part of his solution of the Delian problem (Duplicating the cube).His work did not survive, not even the names he used for these curves, and is only known through secondary accounts. The definition used at that time differs from the one commonly used today. Cones were constructed by rotating a right triangle about one of its legs so the hypotenuse generates the surface of the cone (such a line is called a generatrix). Three types of cones were determined by their vertex angles (measured by twice the angle formed by the hypotenuse and the leg being rotated about in the right triangle). The conic section was then determined by intersecting one of these cones with a plane drawn perpendicular to a generatrix. The type of the conic is determined by the type of cone, that is, 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 (but only one branch of the curve).
Euclid (fl. 300 BCE) is said to have written four books on conics but these were lost as well.Archimedes (died c. 212 BCE) is known to have studied conics, having determined the area bounded by a parabola and a chord in Quadrature of the Parabola. His main interest was in terms of measuring areas and volumes of figures related to the conics and part of this work survives in his book on the solids of revolution of conics, On Conoids and Spheroids.
The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga (died c. 190 BCE), whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge. Apollonius's study of the properties of these curves made it possible to show that any plane cutting a fixed double cone (two napped), regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today. Circles, not constructible by the earlier method, are also obtainable in this way. This may account for why Apollonius considered circles a fourth type of conic section, a distinction that is no longer made. Apollonius used the names ellipse, parabola and hyperbola for these curves, borrowing the terminology from earlier Pythagorean work on areas.
Pappus of Alexandria (died c. 350 CE) is credited with expounding on the importance of the concept of a conic's focus, and detailing the related concept of a directrix, including the case of the parabola (which is lacking in Apollonius's known works).
An instrument for drawing conic sections was first described in 1000 CE by the Islamic mathematician Al-Kuhi.
Apollonius's work was translated into Arabic and much of his work only survives through the Arabic version. Persians found applications to the theory; the most notable of these was the Persianmathematician and poet Omar Khayyám, who used conic sections to solve algebraic equations.
Johannes Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept of limits. Kepler first used the term foci in 1604.
Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and this helped to 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.
René Descartes and Pierre Fermat both applied their newly discovered analytic geometry to the study of conics. This had the effect of reducing the geometrical problems of conics to problems in algebra. However, it was John Wallis in his 1655 treatise Tractatus de sectionibus conicis who first defined the conic sections as instances of equations of second degree.Written earlier, but published later, Jan de Witt's Elementa Curvarum Linearum starts with Kepler's kinematic construction of the conics and then develops the algebraic equations. This work, which uses Fermat's methodology and Descartes' notation has been described as the first textbook on the subject. De Witt invented the term directrix.
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.
The reflective properties of the conic sections are used in the design of searchlights, radio-telescopes and some optical telescopes.A parabolic mirror is used as the reflector, with a bulb at the focus, in a searchlight. The 4.2 meter Herschel optical telescope on La Palma, in the Canary islands, uses a primary parabolic mirror to reflect light towards a secondary hyperbolic mirror, which reflects it again to a focus behind the first mirror.
The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry. One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables (or equivalently, the zeros of an irreducible quadratic form). More technically, the set of points that are zeros of a quadratic form (in any number of variables) is called a quadric, and the irreducible quadrics in a two dimensional projective space (that is, having three variables) are traditionally called conics.
The Euclidean plane R2 is embedded in the real projective plane by adjoining a line at infinity (and its corresponding points at infinity) so that all the lines of a parallel class meet on this line. On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and removing it and all its points.
In a projective space over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one simply speaks of "a conic" without specifying a type. That is, there is a projective transformation that will map any non-degenerate conic to any other non-degenerate conic.
The three types of conic sections will reappear in the affine plane obtained by choosing a line of the projective space to be the line at infinity. The three types are then determined by how this line at infinity intersects the conic in the projective space. In the corresponding affine space, one obtains an ellipse if the conic does not intersect the line at infinity, a parabola if the conic intersects the line at infinity in one double point corresponding to the axis, and a hyperbola if the conic intersects the line at infinity in two points corresponding to the asymptotes.
In homogeneous coordinates a conic section can be represented as:
Or in matrix notation
The 3 × 3 matrix above is called the matrix of the conic section.
Some authors prefer to write the general homogeneous equation as
(or some variation of this) so that the matrix of the conic section has the simpler form,
but this notation is not used in this article.
If the determinant of the matrix of the conic section is zero, the conic section is degenerate.
As multiplying all six coefficients by the same non-zero scalar yields an equation with the same set of zeros, one can consider conics, represented by (A, B, C, D, E, F) as points in the five-dimensional projective space
Metrical concepts of Euclidean geometry (concepts concerned with measuring lengths and angles) can not be immediately extended to the real projective plane.They must be redefined (and generalized) in this new geometry. This can be done for arbitrary projective planes, but to obtain the real projective plane as the extended Euclidean plane, some specific choices have to be made.
Fix an arbitrary line in a projective plane that shall be referred to as the absolute line. Select two distinct points on the absolute line and refer to them as absolute points. Several metrical concepts can be defined with reference to these choices. For instance, given a line containing the points A and B, the midpoint of line segment AB is defined as the point C which is the projective harmonic conjugate of the point of intersection of AB and the absolute line, with respect to A and B.
A conic in a projective plane that contains the two absolute points is called a circle. Since five points determine a conic, a circle (which may be degenerate) is determined by three points. To obtain the extended Euclidean plane, the absolute line is chosen to be the line at infinity of the Euclidean plane and the absolute points are two special points on that line called the circular points at infinity. Lines containing two points with real coordinates do not pass through the circular points at infinity, so in the Euclidean plane a circle, under this definition, is determined by three points that are not collinear.
It has been mentioned that circles in the Euclidean plane can not be defined by the focus-directrix property. However, if one were to consider the line at infinity as the directrix, then by taking the eccentricity to be e = 0 a circle will have the focus-directrix property, but it is still not defined by that property. One must be careful in this situation to correctly use the definition of eccentricity as the ratio of the distance of a point on the circle to the focus (length of a radius) to the distance of that point to the directrix (this distance is infinite) which gives the limiting value of zero.
A synthetic (without the use of coordinates) approach to defining the conic sections in a projective plane was given by Jakob Steiner in 1867.
A perspective mapping of a pencil onto a pencil is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line , which is called the axis of the perspectivity .
A projective mapping is a finite sequence of perspective mappings.
As a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines, only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section.for the Steiner generation of a conic section, besides two points
By the Principle of Duality in a projective plane, the dual of each point is a line, and the dual of a locus of points (a set of points satisfying some condition) is called an envelope of lines. Using Steiner's definition of a conic (this locus of points will now be referred to as a point conic) as the meet of corresponding rays of two related pencils, it is easy to dualize and obtain the corresponding envelope consisting of the joins of corresponding points of two related ranges (points on a line) on different bases (the lines the points are on). Such an envelope is called a line conic (or dual conic).
In the real projective plane, a point conic has the property that every line meets it in two points (which may coincide, or may be complex) and any set of points with this property is a point conic. It follows dually that a line conic has two of its lines through every point and any envelope of lines with this property is a line conic. At every point of a point conic there is a unique tangent line, and dually, on every line of a line conic there is a unique point called a point of contact. An important theorem states that the tangent lines of a point conic form a line conic, and dually, the points of contact of a line conic form a point conic.
Karl Georg Christian von Staudt defined a conic as the point set given by all the absolute points of a polarity 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) bijection between the points and the lines of P that preserves the incidence relation. Thus, a polarity relates a point Q with a line q and, following Gergonne, 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 Steiner conic.
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. 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.
Another method, based on Steiner's construction and which is useful in engineering applications, is the parallelogram method, where a conic is constructed point by point by means of connecting certain equally spaced points on a horizontal line and a vertical line. x2/ + y2/ = 1, first construct the rectangle ABCD with vertices A(a, 0), B(a, 2b), C(−a, 2b) and D(−a, 0). Divide the side BC into n equal segments and use parallel projection, with respect to the diagonal AC, to form equal segments on side AB (the lengths of these segments will be b/ times the length of the segments on BC). On the side BC label the left-hand endpoints of the segments with A1 to An starting at B and going towards C. On the side AB label the upper endpoints D1 to Dn starting at A and going towards B. The points of intersection, AAi ∩ DDi for 1 ≤ i ≤ n will be points of the ellipse between A and P(0, b). The labeling associates the lines of the pencil through A with the lines of the pencil through D projectively but not perspectively. The sought for conic is obtained by this construction since three points A, D and P and two tangents (the vertical lines at A and D) uniquely determine the conic. If another diameter (and its conjugate diameter) are used instead of the major and minor axes of the ellipse, a parallelogram that is not a rectangle is used in the construction, giving the name of the method. The association of lines of the pencils can be extended to obtain other points on the ellipse. The constructions for hyperbolas and parabolas are similar.Specifically, to construct the ellipse with equation
Yet another general method uses the polarity property to construct the tangent envelope of a conic (a line conic).
Further unification is possible if one allows complex numbers as coefficients. In the complex projective plane the non-degenerate conics can not be distinguished from one another.
Over the complex numbers ellipses and hyperbolas are not distinct, since −1 is a square; precisely, the ellipse becomes a hyperbola under the substitution geometrically a complex rotation, yielding – a hyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Geometrically, this corresponds to intersecting the line at infinity in either 2 distinct points (corresponding to two asymptotes) or in 1 double point (corresponding to the axis of a parabola), and thus the real hyperbola is a more suggestive image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity.
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 one intersection point (possibilities: four distinct intersection points, two singular intersection points and one double intersection points, two double intersection points, one 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 is a hyperbola, if they are imaginary conjugated, the conic section is 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 (see circular points at infinity). If a conic section has one real and one imaginary point at infinity, or two imaginary points that are not conjugated then it is not a real conic section, because its coefficients cannot be real.
What should be considered as a degenerate case of a conic depends on the definition being used and the geometric setting for the conic section. There are some authors who define a conic as a two-dimensional nondegenerate quadric. With this terminology there are no degenerate conics (only degenerate quadrics), but we shall use the more traditional terminology and avoid that definition.
In the Euclidean plane, using the geometric definition, a degenerate case arises when the cutting plane passes through the apex of the cone. The degenerate conic is either: a point, when the plane intersects the cone only at the apex; a straight line, when the plane is tangent to the cone (it contains exactly one generator of the cone); or a pair of intersecting lines (two generators of the cone).These correspond respectively to the limiting forms of an ellipse, parabola, and a hyperbola.
If a conic in the Euclidean plane is being defined by the zeros of a quadratic equation (that is, as a quadric), then the degenerate conics are: the empty set, a point, or a pair of lines which may be parallel, intersect at a point, or coincide. The empty set case may correspond either to a pair of complex conjugate parallel lines such as with the equation or to an imaginary ellipse, such as with the equation An imaginary ellipse does not satisfy the general definition of a degeneracy, and is thus not normally considered as degenerated. The two lines case occurs when the quadratic expression factors into two linear factors, the zeros of each giving a line. In the case that the factors are the same, the corresponding lines coincide and we refer to the line as a double line (a line with multiplicity 2) and this is the previous case of a tangent cutting plane.
In the real projective plane, since parallel lines meet at a point on the line at infinity, the parallel line case of the Euclidean plane can be viewed as intersecting lines. However, as the point of intersection is the apex of the cone, the cone itself degenerates to a cylinder, i.e. with the apex at infinity. Other sections in this case are called cylindric sections.The non-degenerate cylindrical sections are ellipses (or circles).
When viewed from the perspective of the complex projective plane, the degenerate cases of a real quadric (i.e., the quadratic equation has real coefficients) can all be considered as a pair of lines, possibly coinciding. The empty set may be the line at infinity considered as a double line, a (real) point is the intersection of two complex conjugate lines and the other cases as previously mentioned.
To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let β be the determinant of the 3×3 matrix of the conic section—that is, β = (AC − B2/)F + BED − CD2 − AE2/; and let α = B2 − 4AC be the discriminant. Then the conic section is non-degenerate if and only if β ≠ 0. If β = 0 we have a point when α < 0, two parallel lines (possibly coinciding) when α = 0, or two intersecting lines when α > 0.
A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.
In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.
A pencil of conics can represented algebraically in the following way. Let C1 and C2 be two distinct conics in a projective plane defined over an algebraically closed field K. For every pair λ, μ of elements of K, not both zero, the expression:
represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the "conic C1". Another concrete realization would be obtained by thinking of C1 as the 3×3 symmetric matrix which represents it. If C1 and C2 have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.
The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections. In particular two conics may possess none, two or four possibly coincident intersection points. An efficient 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, where the conics are represented by matrices:
Conics may be defined over other fields (that is, in other pappian geometries). However, some care must be used when the field has characteristic 2, as some formulas can not be used. For example, the matrix representations used above require division by 2.
A generalization of a non-degenerate conic in a projective plane is an oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line.
Generalizing the focus properties of conics to the case where there are more than two foci produces sets called generalized conics.
The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. The classification mostly arises due to the presence of a quadratic form (in two variables this corresponds to the associated discriminant), but can also correspond to eccentricity.
Quadratic form classifications:
Eccentricity classifications include:
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.
In geometry, a degenerate conic is a conic that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers as the product of two linear polynomials.
In geometry and topology, the line at infinity is a projective line that 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. The line at infinity is also called the ideal line.
In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions.
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin spheres are also sometimes called focal spheres.
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.
A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom.
In geometry, the pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
In geometry, the director circle of an ellipse or hyperbola is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle.
The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.
In mathematics, a generalized conic is a geometrical object defined by a property which is a generalization of some defining property of the classical conic. For example, in elementary geometry, an ellipse can be defined as the locus of a point which moves in a plane such that the sum of its distances from two fixed points – the foci – in the plane is a constant. The curve obtained when the set of two fixed points is replaced by an arbitrary, but fixed, finite set of points in the plane is called an n–ellipse and can be thought of as a generalized ellipse. Since an ellipse is the equidistant set of two circles, the equidistant set of two arbitrary sets of points in a plane can be viewed as a generalized conic. In rectangular Cartesian coordinates, the equation y = x2 represents a parabola. The generalized equation y = xr, for r ≠ 0 and r ≠ 1, can be treated as defining a generalized parabola. The idea of generalized conic has found applications in approximation theory and optimization theory.
In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally. Parabolas possess only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally.
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