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50-418: Parabolic usually refers to something in a shape of a parabola , but may also refer to a parable . Parabolic may refer to: Parabola One description of a parabola involves a point (the focus ) and a line (the directrix ). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of

100-405: A c = 0 , {\displaystyle b^{2}-4ac=0,} or, equivalently, such that a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} is the square of a linear polynomial . The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of

150-467: A corresponding increase in area. The quotients formed by the area of these polygons divided by the square of the circle radius can be made arbitrarily close to π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr , π being defined as the ratio of the circumference to the diameter (C/d). He also provided the bounds 3 +  / 71  <  π  < 3 +  / 70 , (giving

200-405: A function f ( x ) = a x 2  with  a ≠ 0. {\displaystyle f(x)=ax^{2}{\text{ with }}a\neq 0.} For a > 0 {\displaystyle a>0} the parabolas are opening to the top, and for a < 0 {\displaystyle a<0} are opening to the bottom (see picture). From

250-402: A graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape. An alternative proof can be done using Dandelin spheres . It works without calculation and uses elementary geometric considerations only (see the derivation below). Method of exhaustion The method of exhaustion ( Latin : methodus exhaustionis )

300-489: A parabola can then be transformed by the uniform scaling ( x , y ) → ( a x , a y ) {\displaystyle (x,y)\to (ax,ay)} into the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Thus, any parabola can be mapped to the unit parabola by a similarity. A synthetic approach, using similar triangles, can also be used to establish this result. The general result

350-419: A parabola is as a conic section , created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface. The graph of a quadratic function y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} (with a ≠ 0 {\displaystyle a\neq 0} )

400-420: A parabola is the inverse of a cardioid . Remark 2: The second polar form is a special case of a pencil of conics with focus F = ( 0 , 0 ) {\displaystyle F=(0,0)} (see picture): r = p 1 − e cos ⁡ φ {\displaystyle r={\frac {p}{1-e\cos \varphi }}} ( e {\displaystyle e}

450-495: A parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope . Designs were proposed in the early to mid-17th century by many mathematicians , including René Descartes , Marin Mersenne , and James Gregory . When Isaac Newton built the first reflecting telescope in 1668, he skipped using

500-456: A parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror . Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points ( locus of points ) in the Euclidean plane: The midpoint V {\displaystyle V} of the perpendicular from

550-681: Is V = ( 0 , 0 ) {\displaystyle V=(0,0)} , and its focus is F = ( p 2 , 0 ) {\displaystyle F=\left({\tfrac {p}{2}},0\right)} . If one shifts the origin into the focus, that is, F = ( 0 , 0 ) {\displaystyle F=(0,0)} , one obtains the equation r = p 1 − cos ⁡ φ , φ ≠ 2 π k . {\displaystyle r={\frac {p}{1-\cos \varphi }},\quad \varphi \neq 2\pi k.} Remark 1: Inverting this polar form shows that

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600-446: Is U-shaped ( opening to the top ). The horizontal chord through the focus (see picture in opening section) is called the latus rectum ; one half of it is the semi-latus rectum . The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p {\displaystyle p} . From the picture one obtains p = 2 f . {\displaystyle p=2f.} The latus rectum

650-401: Is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape . If the sequence is correctly constructed, the difference in area between the n th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for

700-403: Is a parabola with its axis parallel to the y -axis. Conversely, every such parabola is the graph of a quadratic function. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where

750-420: Is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p {\displaystyle p} is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p {\displaystyle p} ,

800-1063: Is not mentioned above. It is defined and discussed below, in § Position of the focus . Let us call the length of DM and of EM x , and the length of PM   y . The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE , we get B M ¯ ⋅ C M ¯ = D M ¯ ⋅ E M ¯ . {\displaystyle {\overline {\mathrm {BM} }}\cdot {\overline {\mathrm {CM} }}={\overline {\mathrm {DM} }}\cdot {\overline {\mathrm {EM} }}.} Substituting: 4 r y cos ⁡ θ = x 2 . {\displaystyle 4ry\cos \theta =x^{2}.} Rearranging: y = x 2 4 r cos ⁡ θ . {\displaystyle y={\frac {x^{2}}{4r\cos \theta }}.} For any given cone and parabola, r and θ are constants, but x and y are variables that depend on

850-402: Is proportional to the square of their diameters. Proposition 5 : The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. Proposition 10 : The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. Proposition 11 : The volume of a cone (or cylinder) of the same height is proportional to

900-436: Is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not. There are other simple affine transformations that map the parabola y = a x 2 {\displaystyle y=ax^{2}} onto

950-408: Is the distance of the focus from the directrix. Using the parameter p {\displaystyle p} , the equation of the parabola can be rewritten as x 2 = 2 p y . {\displaystyle x^{2}=2py.} More generally, if the vertex is V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} ,

1000-400: Is the eccentricity). The diagram represents a cone with its axis AV . The point A is its apex . An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola. A cross-section perpendicular to the axis of

1050-871: The eccentricity . If p > 0 , the parabola with equation y 2 = 2 p x {\displaystyle y^{2}=2px} (opening to the right) has the polar representation r = 2 p cos ⁡ φ sin 2 ⁡ φ , φ ∈ [ − π 2 , π 2 ] ∖ { 0 } {\displaystyle r=2p{\frac {\cos \varphi }{\sin ^{2}\varphi }},\quad \varphi \in \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}} where r 2 = x 2 + y 2 ,   x = r cos ⁡ φ {\displaystyle r^{2}=x^{2}+y^{2},\ x=r\cos \varphi } . Its vertex

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1100-467: The Parabola . The name "parabola" is due to Apollonius , who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved. The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus . Galileo showed that the path of a projectile follows

1150-407: The apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation. It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in

1200-449: The arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from

1250-451: The area of the base. Proposition 12: The volume of a cone (or cylinder) that is similar to another is proportional to the cube of the ratio of the diameters of the bases. Proposition 18 : The volume of a sphere is proportional to the cube of its diameter. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a sequence of polygons with an increasing number of sides and

1300-413: The area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The method of exhaustion typically required a form of proof by contradiction , known as reductio ad absurdum . This amounts to finding an area of a region by first comparing it to the area of a second region, which can be "exhausted" so that its area becomes arbitrarily close to

1350-446: The axis of symmetry. The same effects occur with sound and other waves . This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles . It is frequently used in physics , engineering , and many other areas. The earliest known work on conic sections

1400-399: The cone passes through the vertex P of the parabola. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius  r . Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a chord DE , which joins

1450-513: The directrix has the equation y = − 1 4 {\displaystyle y=-{\tfrac {1}{4}}} . The general function of degree 2 is f ( x ) = a x 2 + b x + c      with      a , b , c ∈ R ,   a ≠ 0. {\displaystyle f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0.} Completing

1500-752: The directrix has the equation y = − f {\displaystyle y=-f} , one obtains for a point P = ( x , y ) {\displaystyle P=(x,y)} from | P F | 2 = | P l | 2 {\displaystyle |PF|^{2}=|Pl|^{2}} the equation x 2 + ( y − f ) 2 = ( y + f ) 2 {\displaystyle x^{2}+(y-f)^{2}=(y+f)^{2}} . Solving for y {\displaystyle y} yields y = 1 4 f x 2 . {\displaystyle y={\frac {1}{4f}}x^{2}.} This parabola

1550-665: The equation uses the Hesse normal form of a line to calculate the distance | P l | {\displaystyle |Pl|} ). For a parametric equation of a parabola in general position see § As the affine image of the unit parabola . The implicit equation of a parabola is defined by an irreducible polynomial of degree two: 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,} such that b 2 − 4

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1600-441: The focus F {\displaystyle F} onto the directrix l {\displaystyle l} is called the vertex , and the line F V {\displaystyle FV} is the axis of symmetry of the parabola. If one introduces Cartesian coordinates , such that F = ( 0 , f ) ,   f > 0 , {\displaystyle F=(0,f),\ f>0,} and

1650-752: The focus F = ( v 1 , v 2 + f ) {\displaystyle F=(v_{1},v_{2}+f)} , and the directrix y = v 2 − f {\displaystyle y=v_{2}-f} , one obtains the equation y = 1 4 f ( x − v 1 ) 2 + v 2 = 1 4 f x 2 − v 1 2 f x + v 1 2 4 f + v 2 . {\displaystyle y={\frac {1}{4f}}(x-v_{1})^{2}+v_{2}={\frac {1}{4f}}x^{2}-{\frac {v_{1}}{2f}}x+{\frac {v_{1}^{2}}{4f}}+v_{2}.} Remarks : If

1700-638: The focus is F = ( f 1 , f 2 ) {\displaystyle F=(f_{1},f_{2})} , and the directrix a x + b y + c = 0 {\displaystyle ax+by+c=0} , then one obtains the equation ( a x + b y + c ) 2 a 2 + b 2 = ( x − f 1 ) 2 + ( y − f 2 ) 2 {\displaystyle {\frac {(ax+by+c)^{2}}{a^{2}+b^{2}}}=(x-f_{1})^{2}+(y-f_{2})^{2}} (the left side of

1750-468: The method of exhaustion so that it is no longer explicitly used to solve problems. An important alternative approach was Cavalieri's principle , also termed the method of indivisibles which eventually evolved into the infinitesimal calculus of Roberval , Torricelli , Wallis , Leibniz , and others. Euclid used the method of exhaustion to prove the following six propositions in the 12th book of his Elements . Proposition 2 : The area of circles

1800-398: The origin (0, 0) and the same semi-latus rectum p {\displaystyle p} can be represented by the equation y 2 = 2 p x + ( e 2 − 1 ) x 2 , e ≥ 0 , {\displaystyle y^{2}=2px+(e^{2}-1)x^{2},\quad e\geq 0,} with e {\displaystyle e}

1850-430: The origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola P {\displaystyle {\mathcal {P}}} can be transformed by a rigid motion to a parabola with an equation y = a x 2 ,   a ≠ 0 {\displaystyle y=ax^{2},\ a\neq 0} . Such

1900-587: The other by a similarity , that is, an arbitrary composition of rigid motions ( translations and rotations ) and uniform scalings . A parabola P {\displaystyle {\mathcal {P}}} with vertex V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} can be transformed by the translation ( x , y ) → ( x − v 1 , y − v 2 ) {\displaystyle (x,y)\to (x-v_{1},y-v_{2})} to one with

1950-491: The parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum " is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar . Parabolas have

2000-403: The parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola . This discussion started from the definition of a parabola as a conic section, but it has now led to a description as

2050-405: The perpendicular from the point V to the plane of the parabola. By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ , and angle PVF is complementary to angle VPF, therefore angle PVF is θ . Since the length of PV is r , the distance of F from the vertex of the parabola is r sin θ . It is shown above that this distance equals the focal length of

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2100-399: The points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point M. All the labelled points, except D and E, are coplanar . They are in the plane of symmetry of the whole figure. This includes the point F, which

2150-407: The positive y direction, then its equation is y = ⁠ x / 4 f ⁠ , where f is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is r sin θ . In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of

2200-410: The property that, if they are made of material that reflects light , then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (" collimated ") beam, leaving the parabola parallel to

2250-470: The section above one obtains: For a = 1 {\displaystyle a=1} the parabola is the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Its focus is ( 0 , 1 4 ) {\displaystyle \left(0,{\tfrac {1}{4}}\right)} , the semi-latus rectum p = 1 2 {\displaystyle p={\tfrac {1}{2}}} , and

2300-425: The square yields f ( x ) = a ( x + b 2 a ) 2 + 4 a c − b 2 4 a , {\displaystyle f(x)=a\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a}},} which is the equation of a parabola with Two objects in the Euclidean plane are similar if one can be transformed to

2350-454: The true area. The proof involves assuming that the true area is greater than the second area, proving that assertion false, assuming it is less than the second area, then proving that assertion false, too. The idea originated in the late 5th century BC with Antiphon , although it is not entirely clear how well he understood it. The theory was made rigorous a few decades later by Eudoxus of Cnidus , who used it to calculate areas and volumes. It

2400-435: The unit parabola, such as ( x , y ) → ( x , y a ) {\displaystyle (x,y)\to \left(x,{\tfrac {y}{a}}\right)} . But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola ). The pencil of conic sections with the x axis as axis of symmetry, one vertex at

2450-418: Was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction .) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of

2500-521: Was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle. The first use of the term was in 1647 by Gregory of Saint Vincent in Opus geometricum quadraturae circuli et sectionum . The method of exhaustion is seen as a precursor to the methods of calculus . The development of analytical geometry and rigorous integral calculus in the 17th-19th centuries subsumed

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