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Conicsections dpsvk


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Conicsections dpsvk

  1. 1. • Appolonious of Perga, a 3rd century B.C.Greek geometer, wrote the greatest treatise onthe curves, his work "Conics" was the first toshow how all three curves, along with thecircle, could be obtained by slicing the sameright circular cone at continuously varyingangles• Conics was discovered by Menaechmus, tutorto Alexander the Great, over 2000 years ago APPOLONIUS APPOLONIUS’ OPTICS
  2. 2. •Euclid’s works, “Optics” and “Elements” alsoindicated knowledge of conics EUCLID’S OPTICS•Descarte and Fermats co-ordinate geometryincluded the many concepts introduced by theancient greeks•Kepler’s laws of planetary motion brought conicsinto new light, from being taken for granted, tobeing marveled at
  3. 3. The conic sections are the non-degenerate curves generated by theintersections of a plane with one ortwo napes of a cone.For a plane perpendicular to the axisof the cone, a circle is producedFor a plane which is notperpendicular to the axis and whichintersectsonly a single nape, the curveproduced is either an ellipse or aparabolaThe curve produced by a planeintersecting both napes is ahyperbolaThe ellipse and hyperbola are know
  4. 4. Given a line D and a point F not on D,conics is the locus of points P such that:the distance from P to F divided by the distance from Pto D is a constant. That is, distance[P,F]/distance[P,D] == e .F is called the focus of the conic, D the directrix, and ethe eccentricity . If 0 < e < 1, the conics is an ellipse if e=0,it is a circle If e == 1, it is a parabola If e > 1, it is a hyperbola
  5. 5. A circle is a symmetrical figure, which can be formedwhen a plane cuts a right circular cone parallel to itsbase.A circle is actually a special case of an ellipse, witheccentricity = 0, that is, as if the two foci of theellipse are one.For a circle at centre (x1,y1) and radius r, we cangive the equation as:(x – x1) 2 + (y – y1) 2 = r 2 EQUATION (FROM x2+y2=r2 CENTRE) ECCENTRICITY 0 RELATION TO FOCUS P=0
  6. 6. An ellipse is formed when a plane cuts a cone at an angleThe eccentricity e of an ellipse is defined as e := c/a, where cis half the distance between foci.For any ellipse, 0 < e < 1.Any cylinder sliced on an angle will reveal an ellipse in cross-section (as seen in the Tycho Brahe Planetarium inCopenhagen).For an ellipse with centre at the origin, with c and d as anyconstants, the equation is: x2 + y2 = 1 c2 d2
  7. 7. In the 17th century, Keplerdiscovered that planets moved aboutthe sun in elliptical orbits, with thesun at one of their foci.The orbits of  the moon and ofartificial satellites of the earth arealso elliptical as are the paths ofcomets in permanent orbit around thesunHalley’s comet takes about 76 yearsto orbit the sun in its elliptical orbit.This was correctly predicted byEdmund Halley. The electrons of the atom travel in approximately elliptical orbits with the nucleus as a focus.
  8. 8. Any light or signal that starts at one focus ofan ellipse will be reflected to the other focus. This principle is used in lithotripsy , a medicalprocedure for treating kidney stones.The principle is also used in the constructionof "whispering galleries" where if a personwhispers near one focus, he can be heard atthe other focus, although he cannot be heardat many places in between. Equation (from centre) x2 + y2 = 1 c2 d2 Eccentricity 0<e<1 Relation to focus a2 - b2 = c2
  9. 9. •Pool shots are easier on anelliptical table.•The surface of the liquid ina cylindrical glass willappear elliptical when it istilted at an angle.
  10. 10. If the plane cuts a cone so that it lies at the sameangle as the slope of the cone then the intersectionis a parabolaA common definition defines it as the locus ofpoints P such that the distance from a line (calledthe directrix) to P is equal to the distance from P toa fixed point F (called the focus).Parabola has eccentricity e:=1A segment of a parabola is called a Lissajouscurve. Equation (from 4px = y 2 centre) Eccentricity c/a = 1 Relation to focus p = p
  11. 11. A body falling under the pull of gravity is a verycommon approximation of a parabolic pathIn the case of water from a water fountain,every molecule of water follows a parabolic pathThis discovery by Galileo in the 17th centurymade it possible for cannoneers to work out thekind of path a cannonball would travel if it werehurtled through the air at a specific angle This knowledge is used today in military,aeronautics and sports to improve performance
  12. 12. •Parabolic reflectors have interesting properties. They can transformany light from the focus to a straight beam.• They are thus used in car lights.Satellite dishes and antennae work on the reverse principal.They can focus parallel rays of light on a focus to provide highconcentration.Solar Furnaces use parabolic reflectors which focus sunlight suchthat the temperature reaches very high extremes.
  13. 13. A plane which cuts a double ended codethrough both its napes projects ahyperbolaHyperbola is commonly defined as thelocus of points P such that the differenceof the distances from P to two fixedpoints F1, F2 (called foci) are constant.That is, Abs[ distance[P,F1] -distance[P,F2] ] == 2 a, where a is a constant.The eccentricity of a hyperbola is c/a > 1(where c is the distance between the foci) Equation x2 - y2 = 1 (from centre) c2 d2 Eccentricity e>1 Relation to a2 - b2 = c2 focus
  14. 14. •A hyperbola can be observed in physicalsituations like the light projected by a lamp,or while sharpening a pencil•A sonic boom shock wave has the shape ofa cone, and it intersects the ground in part ofa hyperbola.•A hyperbola revolving around its axis formsa surface called a hyperboloid. CertainBuildings like nuclear reactors areconstructed as hyperboloids.