Misplaced Pages

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76-611: The plaza, conceived of and designed by landscape architect Richard C. Bell, FASLA, with a state-appropriated budget of $ 160,000, is mostly covered with red and white bricks which have been organized into a decorative mosaic reminiscent of Piazza San Marco in Venice, where Bell once studied. There are two main green areas on the plaza, one on the east side and one on the west. The east green area features two concrete chairs which serve as parabolic reflectors , amplifying and focusing sound waves so that someone sitting in one chair can easily hear

152-415: A ≠ 0 {\displaystyle a\neq 0} ) 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

228-405: 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

304-415: A burning glass . Parabolic reflectors are popular for use in creating optical illusions . These consist of two opposing parabolic mirrors, with an opening in the center of the top mirror. When an object is placed on the bottom mirror, the mirrors create a real image , which is a virtually identical copy of the original that appears in the opening. The quality of the image is dependent upon the precision of

380-430: A hemisphere ( 2 3 π R 2 D , {\textstyle ({\frac {2}{3}}\pi R^{2}D,} where D = R ) , {\textstyle D=R),} and a cone ( 1 3 π R 2 D ) . {\textstyle ({\frac {1}{3}}\pi R^{2}D).} π R 2 {\textstyle \pi R^{2}}

456-535: A beam of light in flashlights , searchlights , stage spotlights , and car headlights . In radio , parabolic antennas are used to radiate a narrow beam of radio waves for point-to-point communications in satellite dishes and microwave relay stations, and to locate aircraft, ships, and vehicles in radar sets. In acoustics , parabolic microphones are used to record faraway sounds such as bird calls , in sports reporting, and to eavesdrop on private conversations in espionage and law enforcement. Strictly,

532-421: A cartesian coordinate system In mathematics , a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. 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

608-405: 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

684-416: A mirror that was parabolic would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes . The design he came up with bears his name: the " Gregorian telescope "; but according to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one. Isaac Newton knew about the properties of parabolic mirrors but chose

760-489: 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

836-420: 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}

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912-495: 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

988-456: 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

1064-583: A reflector must be correct to within about 20 nm. For comparison, the diameter of a human hair is usually about 50,000 nm, so the required accuracy for a reflector to focus visible light is about 2500 times less than the diameter of a hair. For example, the flaw in the Hubble Space Telescope mirror (too flat by about 2,200 nm at its perimeter) caused severe spherical aberration until corrected with COSTAR . Microwaves, such as are used for satellite-TV signals, have wavelengths of

1140-416: A spherical shape for his Newtonian telescope mirror to simplify construction. Lighthouses also commonly used parabolic mirrors to collimate a point of light from a lantern into a beam, before being replaced by more efficient Fresnel lenses in the 19th century. In 1888, Heinrich Hertz , a German physicist, constructed the world's first parabolic reflector antenna. The most common modern applications of

1216-681: 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

1292-446: 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

1368-462: Is a paraboloidal mirror which is rotated about axes that pass through its centre of mass, but this does not coincide with the focus, which is outside the dish. If the reflector were a rigid paraboloid, the focus would move as the dish turns. To avoid this, the reflector is flexible, and is bent as it rotates so as to keep the focus stationary. Ideally, the reflector would be exactly paraboloidal at all times. In practice, this cannot be achieved exactly, so

1444-539: Is called the " vertex " and is the point where 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

1520-420: 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} ,

1596-462: Is greatest, and where the axis of symmetry intersects the paraboloid. However, if the reflector is used to focus incoming energy onto a receiver, the shadow of the receiver falls onto the vertex of the paraboloid, which is part of the reflector, so part of the reflector is wasted. This can be avoided by making the reflector from a segment of the paraboloid which is offset from the vertex and the axis of symmetry. The whole reflector receives energy, which

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1672-1063: 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

1748-457: Is offset from the axis of rotation. To make less accurate ones, suitable as satellite dishes, the shape is designed by a computer, then multiple dishes are stamped out of sheet metal. Off-axis-reflectors heading from medium latitudes to a geostationary TV satellite somewhere above the equator stand steeper than a coaxial reflector. The effect is, that the arm to hold the dish can be shorter and snow tends less to accumulate in (the lower part of)

1824-401: Is part of a circular paraboloid , that is, the surface generated by a parabola revolving around its axis. The parabolic reflector transforms an incoming plane wave travelling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along

1900-421: Is rotated around axes that pass through the focus and around which it is balanced. If the dish is symmetrical and made of uniform material of constant thickness, and if F represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.8478 times F . The radius of

1976-436: 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

2052-467: Is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of 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

2128-681: Is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept. The area of the concave surface of the dish can be found using the area formula for a surface of revolution which gives A = π R 6 D 2 ( ( R 2 + 4 D 2 ) 3 / 2 − R 3 ) {\textstyle A={\frac {\pi R}{6D^{2}}}\left((R^{2}+4D^{2})^{3/2}-R^{3}\right)} . providing D ≠ 0 {\textstyle D\neq 0} . The fraction of light reflected by

2204-408: 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})} ,

2280-400: 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

2356-411: Is the focal length, D {\textstyle D} is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and R {\textstyle R} is the radius of the dish from the center. All units used for the radius, focal point and depth must be the same. If two of these three quantities are known, this equation can be used to calculate

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2432-570: Is then focused onto the receiver. This is frequently done, for example, in satellite-TV receiving dishes, and also in some types of astronomical telescope ( e.g. , the Green Bank Telescope , the James Webb Space Telescope ). Accurate off-axis reflectors, for use in solar furnaces and other non-critical applications, can be made quite simply by using a rotating furnace , in which the container of molten glass

2508-459: The Siege of Syracuse . This seems unlikely to be true, however, as the claim does not appear in sources before the 2nd century CE, and Diocles does not mention it in his book. Parabolic mirrors and reflectors were also studied extensively by the physicist Roger Bacon in the 13th century AD. James Gregory , in his 1663 book Optica Promota (1663), pointed out that a reflecting telescope with

2584-871: 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

2660-427: The natural logarithm of x , i.e. its logarithm to base " e ". The volume of the dish is given by 1 2 π R 2 D , {\textstyle {\frac {1}{2}}\pi R^{2}D,} where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder ( π R 2 D ) , {\textstyle (\pi R^{2}D),}

2736-467: 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

2812-479: The Scheffler reflector is not suitable for purposes that require high accuracy. It is used in applications such as solar cooking , where sunlight has to be focused well enough to strike a cooking pot, but not to an exact point. A circular paraboloid is theoretically unlimited in size. Any practical reflector uses just a segment of it. Often, the segment includes the vertex of the paraboloid, where its curvature

2888-407: 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

2964-449: 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

3040-401: The axis (or if the emitting point source is not placed in the focus), parabolic reflectors suffer from an aberration called coma . This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola. The precision to which a parabolic dish must be made in order to focus energy well depends on the wavelength of the energy. If

3116-446: 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

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3192-417: The axis. Parabolic reflectors are used to collect energy from a distant source (for example sound waves or incoming star light). Since the principles of reflection are reversible, parabolic reflectors can also be used to collimate radiation from an isotropic source into a parallel beam . In optics , parabolic mirrors are used to gather light in reflecting telescopes and solar furnaces , and project

3268-553: The campus walking areas from the University Plaza to the Memorial Tower. 35°47′14″N 78°40′13″W  /  35.78722°N 78.67028°W  / 35.78722; -78.67028 Parabolic reflectors A parabolic (or paraboloid or paraboloidal ) reflector (or dish or mirror ) is a reflective surface used to collect or project energy such as light , sound , or radio waves . Its shape

3344-399: 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

3420-513: 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

3496-752: 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

3572-474: The dish is wrong by a quarter of a wavelength, then the reflected energy will be wrong by a half wavelength, which means that it will interfere destructively with energy that has been reflected properly from another part of the dish. To prevent this, the dish must be made correctly to within about ⁠ 1 / 20 ⁠ of a wavelength. The wavelength range of visible light is between about 400 and 700 nanometres (nm), so in order to focus all visible light well,

3648-467: The dish, from a light source in the focus, is given by 1 − arctan ⁡ R D − F π {\textstyle 1-{\frac {\arctan {\frac {R}{D-F}}}{\pi }}} , where F , {\displaystyle F,} D , {\displaystyle D,} and R {\displaystyle R} are defined as above. The parabolic reflector functions due to

3724-480: The dish. The principle of parabolic reflectors has been known since classical antiquity , when the mathematician Diocles described them in his book On Burning Mirrors and proved that they focus a parallel beam to a point. Archimedes in the third century BCE studied paraboloids as part of his study of hydrostatic equilibrium , and it has been claimed that he used reflectors to set the Roman fleet alight during

3800-665: 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

3876-648: The equivalent: P = R 2 2 D {\textstyle P={\frac {R^{2}}{2D}}} ) and Q = P 2 + R 2 {\textstyle Q={\sqrt {P^{2}+R^{2}}}} , where F , D , and R are defined as above. The diameter of the dish, measured along the surface, is then given by: R Q P + P ln ⁡ ( R + Q P ) {\textstyle {\frac {RQ}{P}}+P\ln \left({\frac {R+Q}{P}}\right)} , where ln ⁡ ( x ) {\textstyle \ln(x)} means

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3952-441: 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

4028-752: 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

4104-638: 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

4180-406: The focus to the dish can be transmitted outward in a beam that is parallel to the axis of the dish. In contrast with spherical reflectors , which suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger, parabolic reflectors can be made to accommodate beams of any width. However, if the incoming beam makes a non-zero angle with

4256-410: The geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or " focus ". (For a geometrical proof, click here .) Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle. Similarly, energy radiating from

4332-523: The optics. Some such illusions are manufactured to tolerances of millionths of an inch. A parabolic reflector pointing upward can be formed by rotating a reflective liquid, like mercury, around a vertical axis. This makes the liquid-mirror telescope possible. The same technique is used in rotating furnaces to make solid reflectors. Parabolic reflectors are also a popular alternative for increasing wireless signal strength. Even with simple ones, users have reported 3 dB or more gains. Parabola#In

4408-451: The order of ten millimetres, so dishes to focus these waves can be wrong by half a millimetre or so and still perform well. It is sometimes useful if the centre of mass of a reflector dish coincides with its focus . This allows it to be easily turned so it can be aimed at a moving source of light, such as the Sun in the sky, while its focus, where the target is located, is stationary. The dish

4484-398: 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}

4560-530: The origin and its axis of symmetry along the y-axis, so the parabola opens upward, its equation is 4 f y = x 2 {\textstyle 4fy=x^{2}} , where f {\textstyle f} is its focal length. (See " Parabola#In a cartesian coordinate system ".) Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation: 4 F D = R 2 {\textstyle 4FD=R^{2}} , where F {\textstyle F}

4636-430: 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

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4712-587: 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

4788-403: 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

4864-503: The parabolic reflector are in satellite dishes , reflecting telescopes , radio telescopes , parabolic microphones , solar cookers , and many lighting devices such as spotlights , car headlights , PAR lamps and LED housings. The Olympic Flame is traditionally lit at Olympia, Greece , using a parabolic reflector concentrating sunlight , and is then transported to the venue of the Games. Parabolic mirrors are one of many shapes for

4940-405: 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

5016-654: The person in the other chair from a far distance. Harrelson Hall , a cylindrical classroom building was previously located on the southwest corner of the plaza, but was deconstructed and recycled in 2017. D. H. Hill Library is located on the north side of the Brickyard. The Brickyard consists of 588,060 red and white North Carolina bricks, most of which were donated by the North Carolina Bricklayers Association in 1966. Bell's original design called for large granite cobblestones that paved

5092-399: 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

5168-407: 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

5244-410: 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

5320-495: The rim is 2.7187  F . The angular radius of the rim as seen from the focal point is 72.68 degrees. The focus-balanced configuration (see above) requires the depth of the reflector dish to be greater than its focal length, so the focus is within the dish. This can lead to the focus being difficult to access. An alternative approach is exemplified by the Scheffler reflector , named after its inventor, Wolfgang Scheffler . This

5396-470: 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

5472-425: 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

5548-420: The third. A more complex calculation is needed to find the diameter of the dish measured along its surface . This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: P = 2 F {\textstyle P=2F} (or

5624-460: The three-dimensional shape of the reflector is called a paraboloid . A parabola is the two-dimensional figure. (The distinction is like that between a sphere and a circle.) However, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal . If a parabola is positioned in Cartesian coordinates with its vertex at

5700-435: 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

5776-418: 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

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