Page 1 of 10
PROJECT ON PROPERTIES AND APPLICATION OF
PARABOLA AND ELLIPSE
PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
https://sites.google.com/site/muragachhahighhsschool
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 2 of 10
PARABOLA PROPERTIES AND APPLICATIONS HISTORY
Parabolic compass designed by Leonardo da Vinci
The parabola was studied by Menaechmus who was a pupil of Plato and Eudoxus. He attempted to duplicate the cube, namely to find side of a cube that has a volume double that of a given cube. Hence he attempted to solve x3 = 2 by geometrical methods. In fact the geometrical methods of ruler and compass constructions cannot solve this (but Menaechmus did not know this). Menaechmus solved it by finding the intersection of the two parabolas x2 = y and y2 = 2x. Euclid wrote about the parabola and it was given its present name by Apollonius. The focus and directrix of a parabola were considered by Pappus. Pascal considered the parabola as a projection of a circle and Galileo showed that projectiles follow parabolic paths. The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. Gregory and Newton considered the properties of a parabola which bring parallel rays of light to a focus. 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 seventeenth 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 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.
DEFINITION A parabola is the set of all points in the plane equidistant from a given line L (the conic section directrix) and a given point F not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by p = 2a , where a is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.
Parabola as conic section.
In the graph, The focus of the parabola is at (a,0). The directrix is the line x = a. The focal distance is `|a|` (Distance from the origin to the focus and from the origin to the directrix. We take absolute value because distance is positive.) The point (x, y) represents any point on the curve. The distance d from any point (x, y) to the focus `(a, 0)` is the same as the distance from (x, y) to the directrix.
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 3 of 10
EQUATIONS OF PARABOLA
The general equation of PARABOLA derived from the general conic equation is : Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and the fact that, for a parabola, B2 = 4AC The equation for a general parabola with a focus point F(u, v) , and a directrix in the form ax + by + c = 0 is
PROPERTIES OF PARABOLA Reflective property of a parabola The reflective property states that, if a parabola can reflect light, then light which enters it travelling parallel to the axis of symmetry is reflected to the focus.
Tangent bisection property The diagram shows that the tangent BE bisects the angle FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus, and perpendicularly to the directrix.
Intersection of a tangent and perpendicular from focus The point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex.
Orthoptic property If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.
Perpendicular tangents intersect on the directrix
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 4 of 10
Lambert's theorem Three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle.
THE USES AND APPLICATIONS OF PARABOLAS The parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century.
The principle of parabolic reflector is used for the car headlight, torches etc. The light is placed in the focus of a parabolic mirror, as the light travels and meets the mirror, it is reflected in lines parallel to the axis (in straight lines ) as can be seen in the diagram. This is why the light beam from the headlights of cars and from torches is so strong.
How Parabolic Dish Antennas work? Point M is the point at which the ray hits the parabolic dish. “I” is the angle made by the incident ray and the normal (in red) which is perpendicular to the tangent (in blue) to the parabola at point M. r is the angle made by the reflected ray and the normal. According to the laws of reflection, angles “I” and “R” are equal. All reflected rays due to incident rays, at different positions, intercept the axis of the parabola y axis) at the same. In parabolic microphones, a parabolic reflector that reflects sound, but not necessarily electromagnetic radiation, is used to focus sound onto a microphone, giving it highly directional performance. Solar cooker with parabolic reflector
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 5 of 10
PARABOLIC SKIS A conventional ski which begins with a circular sidecut will deform under load to a less than perfect arc. The parabolic design will deform under load to a perfect arc resulting in a smooth turn. The result is that the ski only has to be tipped on edge to turn flawlessly with a minimum of skier exertion.
NATURAL OCCURRING PHENOMENA A jet of water, like that formed by a fountain, forms the shape of a parabola. When a ball is struck in the air, it travels along a path in the shape of a parabola.
Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope. Shown here is a rectangular container with fluid inside, the container is placed on a rotating table, when the table rotates the fluid inside take the shape of a parabola.
In all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.
Longperiod comets travel close to the Sun's escape velocity while they are moving through the inner solar system, so their paths are close to being parabolic. The path of Comet Kohoutek as it passed through the inner solar system, showing its nearly parabolic shape. The other orbit is the Earth's
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 6 of 10
ELLIPSE PROPERTIES AND APPLICATIONS HISTORY The ellipse was first studied by Menaechmus. Euclid wrote about the ellipse and it was given its present name by Apollonius. The focus and directrix of an ellipse were considered by Pappus. Kepler, in 1602, said he believed that the orbit of Mars was oval, then he later discovered that it was an ellipse with the sun at one focus. In fact Kepler introduced the word "focus" and published his discovery in 1609. The eccentricity of the planetary orbits is small (i.e. they are close to circles). The eccentricity of Mars is 1/11 and of the Earth is 1/60. In 1705 Halley showed that the comet, which is now called after him, moved in an elliptical orbit round the sun. The eccentricity of Halley's comet is 0.9675 so it is close to a parabola. The area of the ellipse is πab. There is no exact formula for the length of an ellipse in elementary Ellipse as conic section
functions and this led to the study of elliptic functions. Ramanujan, in 1914, gave the approximate length π(3(a + b) √[(a + 3b)(3a + b)]).
DEFINITION In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus). Ellipses have two mutually perpendicular axes A1A2 and B1B2 about which the ellipse is symmetric These axes intersect at the center C of the ellipse due to this symmetry. The larger of these two axes, which corresponds to the largest distance between antipodal points on the ellipse, is called the Major axis or Transverse diameter. The smaller of these two axes, and the smallest distance across the ellipse, is called the Minor axis or Conjugate diameter.
The four points where these axes cross the ellipse are the vertices and are marked as A1 , A2 , B1 and B2. In addition to being at the largest and smallest distance from the center, these points are where the curvature of the ellipse is maximum and minimum. The two foci (plural of focus and the term focal points is also used) of an ellipse are two special points F1 and F2 on the ellipse's major axis that are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis (PF1 + PF2 = 2a). The distance to the focal point from the center of the ellipse is sometimes called the linear eccentricity, f, of the ellipse. Here it is denoted by f, but it is often denoted by c. Due to the Pythagorean theorem and the definition of the ellipse explained in the previous paragraph: f2 = a2 −b2. The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 7 of 10
major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e < 1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus F1 is kept fixed as the other F2 is allowed to move arbitrarily far away. The eccentricity is also equal to the ratio of the distance from any particular point P on an ellipse to one of the foci F2 to the perpendicular distance to the directrix from the same point (line PD), e = PF2/PD.
EQUATIONS OF ELLIPSE Ellipses centered at the origin If the ellipse is centered on the origin (0,0) then the Cartesian equation is
And the parametric equations are
x = a cos t
where a, b are the radius on the x and y axes respectively, t is the parameter, which ranges from
y = b sin t
0 to 2π radians. The equations on this page are true only for ellipses that are aligned with the coordinate plane, that is, where the major and minor axes are parallel to the coordinate system.
Ellipses not centered at the origin If the Center of the ellipse is at ( h,k) then the Cartesian equation becomes And the parametric equations are Where a, b are the radius on the x and y axes respectively, t is the parameter, which ranges
x = h + a cos t
from 0 to 2π radians.
y = k + b sin t
The equation for a general ELLIPSE with a focus point F(u, v) , a directrix in the form ax + by + c = 0 and Eccentricity e (e<1) is e2(ax + by + c)2 = (a2 + b2)[(x – u)2 + (y – v)2] where 0
PROPERTIES OF ELLIPSE Focal property One of the definitions of ellipse is that it is a locus of points the sum of whose distances from the given two points is constant. The latter are called the foci of the ellipse. Under other definitions, the distance preservation becomes a property, among many others. As such, it is known as the Focal property of the ellipse. F1T + F2T = CONSTANT ( for any point T on the Ellipse)
Reflective (Optical) Property of an Ellipse In an ellipse, lightrays from one focus will reflect to the other focus .
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 8 of 10
Reflections in Ellipse Choose two points T1, T2 on an ellipse. Imagine a light source at T1 sending a ray of light in the direction of T2. Assuming the inner surface of the ellipse fully reflective, the ray will bounce off at T2 and reach the ellipse again at T3 where it will bounce again and then again at point T4 and so on.
From Foci to a Tangent in Ellipse The product of distances from the foci of an ellipse to any tangent is a constant (not depending on the particular tangent.
Parallel Chords in Ellipse Cross the ellipse by two parallel lines AB and CD, with points A, B, C, D on the ellipse. Find the midpoints M and N of the segments AB and CD. Line MN is incident to the center of the ellipse. Therefore by choosing a pair of parallel lines with a different direction, the center of the ellipse is found at the intersection of the two midlines.
Pascal in Ellipse Pascal's theorem which B. Pascal has famously discovered at the age of 16 states that if a hexagon is inscribed in a conic, then the three points at which the pairs of opposite sides meet are collinear. The universality of the diagram led to the introduction of the term Pascal's Mystic Hexagram that stuck around.
La Hire's Theorem in Ellipse Let there be two points A and B outside an ellipse. From points A and B draw tangents AC, AD, BE, BF to the ellipse. Then if B lies on CD, then A lies on EF
Gergonne in Ellipse The lines joining the points of tangency of the incircle with the opposing vertices of a triangle concur in a point known as the Gergonne point or Gergonne's center.
THE USES AND APPLICATIONS OF ELLIPSE PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 9 of 10
The Reflective Property of an Ellipse Now, apart from being mathematically interesting, what makes this property so fascinating? Well, there are several reasons. Most notable of which is its significance to physics, primarily optics and acoustics. Both light and sound are affected in this way. In fact there are many famous buildings designed to exploit this property. Such buildings are referred to as whisper galleries or whisper chambers. St. Paul's Cathedral in London, England was designed by architect and mathematician Sir Christopher Wren and contains one such whisper gallery. Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point. The effect that such a room creates is that if one person is standing at one of the foci, a person standing at the other focus can hear even the slightest whisper spoken by the other.
ELLIPTICAL BILLIARDS TABLE The elliptical billiards table uses the same principle as the whispering gallery above. The pocket is positioned at one of the focal points. If one hits the ball so that it goes through one focus, it will reflect off the ellipse and go into the hole which is located at the other focus.
SATELLITES All the planets orbiting the sun are satellites. These planets do not travel in a circular motion as many people believe they do, but they travel in elliptical orbits. The eccentricity of the earths orbit around the sun is approximately 0.0167, which is, as explained previously, almost circular. The planet Pluto has an orbit with an eccentricity of approximately 0.2481. The moon travels around the earth in an elliptical orbit also and so too do man made satellites.
LITHOTRIPSY Ellipses are used for a medical process called lithotripsy. If we imagine an ellipse as being made from a reflective material, than a ray emitted from one focus reflects off the ellipse and passes through the second; a fact true for all forms of energy, including shockwaves. In lithotripsy, which is the process of using ultrasound to shatter kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. Highenergy shock waves generated at the other focus are concentrated on the stone, pulverizing it.
Some tanks are in fact elliptical (not circular) in cross section. This gives them a high capacity, but with a lower centerofgravity, so that they are more stable when being transported. And they're shorter, so that they can pass under a low bridge. You might see these tanks transporting heating oil or gasoline on the highway
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
Page 10 of 10
Ellipses (or halfellipses) are sometimes used as fins, or airfoils in structures that move through the air. The elliptical shape reduces drag.
PROJECT ON PROPERTIES AND APPLICATION OF PARABOLA AND ELLIPSE PREPARED BY PRATAP C SAHA A.T. MURAGACHHA HIGH ( H.S.) SCHOOL
