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64-464: The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to 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

128-410: A 2 − b 2 . {\textstyle c={\sqrt {a^{2}-b^{2}}}.} We define a number of related additional concepts (only for ellipses): The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity c (distance between the center of the ellipse and each focus) to the length of the semimajor axis a . The eccentricity is also the ratio of

192-401: A . (Here a is the semi-major axis defined below.) A parabola may also be defined in terms of its focus and latus rectum line (parallel to the directrix and passing through the focus): it is the locus of points whose distance to the focus plus or minus the distance to the line is equal to 2 a ; plus if the point is between the directrix and the latus rectum, minus otherwise. In addition to

256-414: A = b , with radius r = a = b . 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. For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are

320-440: A is the length of the semi-major axis and b is the length of the semi-minor axis. When the conic section is given in the general quadratic form the following formula gives the eccentricity e if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse , and not an imaginary ellipse: where η = 1 {\displaystyle \eta =1} if

384-541: A plane is a flat two- dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle \mathbb {R} ^{3}} . A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin. In two dimensions, there are infinitely many polytopes: the polygons. The first few regular ones are shown below: The Schläfli symbol { n } {\displaystyle \{n\}} represents

448-574: A plane , called the cutting plane , with the surface of a double cone (a cone with two nappes ). It is usually 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

512-402: A rectangular hyperbola is 2 {\displaystyle {\sqrt {2}}} . The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and

576-419: A regular n -gon . The regular monogon (or henagon) {1} and regular digon {2} can be considered degenerate regular polygons and exist nondegenerately in non-Euclidean spaces like a 2-sphere , 2-torus , or right circular cylinder . There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {n/m}. They are called star polygons and share

640-405: A change of coordinates ( rotation and translation of axes ) these equations can be put into standard forms . For ellipses and hyperbolas a standard form has the x -axis as principal axis and the origin (0,0) as center. The vertices are (± a , 0) and the foci (± c , 0) . Define b by the equations c = a − b for an ellipse and c = a + b for a hyperbola. For a circle, c = 0 so

704-403: A closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry . A conic is the curve obtained as the intersection of

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768-412: A constant ratio. That ratio is called the eccentricity, commonly denoted as e . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and

832-406: A focus; its half-length is the semi-latus rectum ( ℓ ). The focal parameter ( p ) is the distance from a focus to the corresponding directrix. The major axis is the chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the semi-major axis ( a ). When an ellipse or hyperbola are in standard position as in

896-470: 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

960-504: A measure of how far the ellipse deviates from being circular. If the angle between the surface of the cone and its axis is β {\displaystyle \beta } and the angle between the cutting plane and the axis is α , {\displaystyle \alpha ,} the eccentricity is cos ⁡ α cos ⁡ β . {\displaystyle {\frac {\cos \alpha }{\cos \beta }}.} A proof that

1024-400: A non-degenerate conic. There are three types of conics: the ellipse , parabola , and hyperbola . The circle is a special kind of ellipse, although historically Apollonius considered it a fourth type. Ellipses 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

1088-533: A region D in R of a function f ( x , y ) , {\displaystyle f(x,y),} and is usually written as: The fundamental theorem of line integrals says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. Let φ : U ⊆ R 2 → R {\displaystyle \varphi :U\subseteq \mathbb {R} ^{2}\to \mathbb {R} } . Then with p , q

1152-411: A right cone, this means the cutting plane is perpendicular to the axis. 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 : the plane intersects both halves of the cone, producing two separate unbounded curves. Compare also spheric section (intersection of a plane with

1216-410: A sphere, producing a circle or point), and spherical conic (intersection of an elliptic cone with a concentric sphere). Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points P whose distance to a fixed point F (called the focus ) is a constant multiple e (called the eccentricity ) of the distance from P to a fixed line L (called

1280-417: A vector A by itself is which gives the formula for the Euclidean length of the vector. In a rectangular coordinate system, the gradient is given by For some scalar field f  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U is defined as where r : [a, b] → C is an arbitrary bijective parametrization of the curve C such that r ( a ) and r ( b ) give

1344-472: A way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph . A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Eccentricity (mathematics) In mathematics ,

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1408-499: Is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines . It has also metrical properties induced by a distance , which allows to define circles , and angle measurement . A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane . The set R 2 {\displaystyle \mathbb {R} ^{2}} of

1472-408: Is a one-dimensional manifold . In a Euclidean plane, it has the length 2π r and the area of its interior is where r {\displaystyle r} is the radius. There are an infinitude of other curved shapes in two dimensions, notably including the conic sections : the ellipse , the parabola , and the hyperbola . Another mathematical way of viewing two-dimensional space

1536-618: 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 B 2 − 4 A C {\displaystyle B^{2}-4AC} , called the discriminant of the equation. Thus, the discriminant is − 4Δ where Δ is the matrix determinant | A B / 2 B / 2 C | . {\displaystyle \left|{\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right|.} If

1600-399: 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

1664-399: Is also called the semi-minor axis. The following relations hold: For conics in standard position, these parameters have the following values, taking a , b > 0 {\displaystyle a,b>0} . After introducing Cartesian coordinates , the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of

1728-437: Is characterized as being the unique contractible 2-manifold . Its dimension is characterized by the fact that removing a point from the plane leaves a space that is connected, but not simply connected . In graph theory , a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such

1792-430: Is defined as: A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by ‖ A ‖ {\displaystyle \|\mathbf {A} \|} . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B . The dot product of

1856-440: Is found in linear algebra , where the idea of independence is crucial. The plane has two dimensions because the length of a rectangle is independent of its width. In the technical language of linear algebra, the plane is two-dimensional because every point in the plane can be described by a linear combination of two independent vectors . The dot product of two vectors A = [ A 1 , A 2 ] and B = [ B 1 , B 2 ]

1920-424: Is given by an ordered pair of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the other axis. Another widely used coordinate system is the polar coordinate system , which specifies a point in terms of its distance from the origin and its angle relative to a rightward reference ray. In Euclidean geometry ,

1984-1537: 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 ( x y ) ( A B / 2 B / 2 C ) ( x y ) + ( D E ) ( x y ) + F = 0. {\displaystyle {\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}D&E\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+F=0.} The general equation can also be written as ( x y 1 ) ( A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ) ( x y 1 ) = 0. {\displaystyle {\begin{pmatrix}x&y&1\end{pmatrix}}{\begin{pmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}=0.} This form

Conic section - Misplaced Pages Continue

2048-419: The determinant of the 3×3 matrix is negative or η = − 1 {\displaystyle \eta =-1} if that determinant is positive. The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from

2112-448: The directrix ). 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 Euclidean plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane. The eccentricity of an ellipse can be seen as

2176-426: The eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: Two conic sections with the same eccentricity are similar . Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in

2240-519: The eigenvalues of the matrix ( A B / 2 B / 2 C ) {\displaystyle \left({\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right)} — that is, the solutions of the equation — and S {\displaystyle S} is the determinant of the 3 × 3 matrix above , and Δ = λ 1 λ 2 {\displaystyle \Delta =\lambda _{1}\lambda _{2}}

2304-405: The equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane). But: conic sections may occur on surfaces of higher order, too (see image). In celestial mechanics , for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to

2368-475: The pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipses, in Keplerian, i.e., 1 / r {\displaystyle 1/r} potentials. A number of classifications in mathematics use derived terminology from

2432-403: The second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called the numerical eccentricity . In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation . Three notational conventions are in common use: This article uses the first notation. Here, for the ellipse and the hyperbola,

2496-401: The above curves defined by the focus-directrix property are the same as those obtained by planes intersecting a cone is facilitated by the use of Dandelin spheres . Alternatively, an ellipse can be defined in terms of two focus points, as the locus of points for which the sum of the distances to the two foci is 2 a ; while a hyperbola is the locus for which the difference of distances is 2

2560-572: 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 x ~ , y ~ {\displaystyle {\tilde {x}},{\tilde {y}}} as or equivalently where λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} are

2624-485: The category of ellipses, then the eccentricity of an ellipse is strictly greater than 0. For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis . In the coordinate system with origin at the ellipse's center and x -axis aligned with the major axis, points on the ellipse satisfy the equation with foci at coordinates ( ± c , 0 ) {\displaystyle (\pm c,0)} for c =

Conic section - Misplaced Pages Continue

2688-424: The conic can be deduced from its equation. In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane , the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it

2752-488: 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 B – 4 AC of the conic section's quadratic equation (or equivalently the determinant AC – B /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

2816-403: The conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion . Euclidean plane In mathematics , a Euclidean plane is a Euclidean space of dimension two , denoted E 2 {\displaystyle {\textbf {E}}^{2}} or E 2 {\displaystyle \mathbb {E} ^{2}} . It

2880-475: The coordinate axes and the line x = y is the principal axis. The foci then have coordinates ( c , c ) and (− c , − c ) . 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

2944-438: The coordinate axes, as is the determinant of the 3 × 3 matrix above . The constant term F and the sum D + E are invariant under rotation only. When the conic section is written algebraically as the eccentricity can be written as a function of the coefficients of the quadratic equation. If 4 AC = B the conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming

3008-425: The discovery. Both authors used a single ( abscissa ) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify

3072-432: The eccentricity ( e ), foci, and directrix, various geometric features and lengths are associated with a conic section. The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center . A parabola has no center. The linear eccentricity ( c ) is the distance between the center and a focus. The latus rectum is the chord parallel to the directrix and passing through

3136-445: The eccentricity and l is the semi-latus rectum. As above, for e = 0 , the graph is a circle, for 0 < e < 1 the 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

3200-462: The endpoints of C and a < b {\displaystyle a<b} . For a vector field F  : U ⊆ R → R , the line integral along a piecewise smooth curve C ⊂ U , in the direction of r , is defined as where · is the dot product and r : [a, b] → C is a bijective parametrization of the curve C such that r ( a ) and r ( b ) give the endpoints of C . A double integral refers to an integral within

3264-399: The endpoints of the curve γ. Let C be a positively oriented , piecewise smooth , simple closed curve in a plane , and let D be the region bounded by C . If L and M are functions of ( x , y ) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is counterclockwise . In topology , the plane

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3328-538: The equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by 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 that the eccentricity is a positive solution of the equation where again Δ = A C − B 2 4 . {\displaystyle \Delta =AC-{\frac {B^{2}}{4}}.} This has precisely one positive solution—the eccentricity— in

3392-402: The equations below, with foci on the x -axis and center at the origin, the vertices of the conic have coordinates (− a , 0) and ( a , 0) , with a non-negative. The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis ( b ), the same value b as in the standard equation below. By analogy, for a hyperbola the parameter b in the standard equation

3456-467: The horizontal. For β = 0 {\displaystyle \beta =0} the plane section is a circle, for β = α {\displaystyle \beta =\alpha } a parabola. (The plane must not meet the vertex of the cone.) The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e ), is the distance between its center and either of its two foci . The eccentricity can be defined as

3520-608: The ideas contained in Descartes' work. Later, the plane was thought of as a field , where any two points could be multiplied and, except for 0, divided. This was known as the complex plane . The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot

3584-401: The maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis a , the eccentricity is given by which is the distance between the foci divided by the length of the major axis. The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of

3648-496: The ordered pairs of real numbers (the real coordinate plane ), equipped with the dot product , is often called the Euclidean plane or standard Euclidean plane , since every Euclidean plane is isomorphic to it. Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas , parallelism,

3712-427: The positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin . They are usually labeled x and y . Relative to these axes, the position of any point in two-dimensional space

3776-440: The ratio of the linear eccentricity to the semimajor axis a : that is, e = c a {\displaystyle e={\frac {c}{a}}} (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity . The eccentricity is sometimes called the first eccentricity to distinguish it from

3840-542: The same unit of length . Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat , although Fermat also worked in three dimensions, and did not publish

3904-404: The same vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with Schläfli symbols { n / m } for all m such that m < n /2 (strictly speaking { n / m } = { n /( n − m )}) and m and n are coprime . The hypersphere in 2 dimensions is a circle , sometimes called a 1-sphere ( S ) because it

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3968-583: The semimajor axis a to the distance d from the center to the directrix: The eccentricity can be expressed in terms of the flattening f (defined as f = 1 − b / a {\displaystyle f=1-b/a} for semimajor axis a and semiminor axis b ): (Flattening may be denoted by g in some subject areas if f is linear eccentricity.) Define the maximum and minimum radii r max {\displaystyle r_{\text{max}}} and r min {\displaystyle r_{\text{min}}} as

4032-432: The sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called Cartesian coordinate system , a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates , which are the signed distances from the point to two fixed perpendicular directed lines, measured in

4096-466: 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 which can be written in the form A x 2 + B x y + C y 2 + D x + E y + F = 0. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.} The geometric properties of

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