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History of Conic Sections
Transcript of History of Conic Sections
--Wikipedia. Wikimedia Foundation, n.d. Web. 13 Feb. 2013.
--"The Conic Sections." The Conic Sections. N.p., n.d. Web. 13 Feb. 2013.
--Verial, Damon. "Importance of Hyperbolas in Life." EHow. Demand Media, 05 June 2011. Web. 19 Feb. 2013.
Mathematics Department. N.p., n.d. Web. 13 Feb. 2013.
--"Conic Sections - Math Tables, Facts and Formulas - Hoxie High School Mathematics Department." Conic Sections - Math Tables, Facts and Formulas - Hoxie High School
--"Apollonius of Perga." Apollonius Biography. N.p., n.d. Web. 21 Feb. 2013.
--"Circles." Circles. N.p., n.d. Web. 23 Feb. 2013.
--"History of Conics." History of Conics. N.p., n.d. Web. 23 Feb. 2013.
--"Conics: Ellipses: Word Problems." Conics: Ellipses: Word Problems. N.p., n.d. Web. 23 Feb. 2013. The conics were first defined as the intersection of a right circular cone of varying angle; a plane perpendicular to an element of the cone. Appollonius was the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. Appolonious of Perga, a 3rd century B.C. Greek geometer, wrote the greatest treatise on the curves. His work "Conics" was the first to show how all three curves, along with the circle, could be obtained by slicing the same right circular cone at continuously varying angles. Appollonius (c. 262-190 BC) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Discovered by Menaachmus tutor to Alexander the Great. They were conceived in an attempt to solve the three famous construction problems of trisecting the angle, doubling the cube, and squaring the circle. There were also applications made by Apollonius, using his knowledge of conics, to practical problems. He developed the hemicyclium, a sundial which has the hour lines drawn on the surface of a conic section giving greater accuracy -Past world example: This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle. -Real world: ~ Tilt a glass of water and the surface of the liquid acquires an elliptical outline.
~ Salami is often cut obliquely to obtain elliptical slices which are larger.
~ The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it. -Real world examples: Focus lenses designs which will determine focal points depending on your curve design. OR
- Cell phones-- Satellite systems make heavy use of hyperbolas and hyperbolic functions. When scientists launch a satellite into space, they must first use mathematical equations to predict its path. Because of the gravity influences of objects with heavy mass, the path of the satellite is skewed even though it may initially launch in a straight path. Using hyperbolas, astronomers can predict the path of the satellite to make adjustments so that the satellite gets to its destination. -Real World example: - The wheels on a bicycle or the wheels on a car. The odometer in a car knows how far the car has traveled because it uses the circumference of the circular tire. OR
-The information on the hard drive on your computer or on a DC or DVD is stored in concentric circles. -Past World example: ~Statuary Hall in the U.s. Capital building is elliptical. 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, easly eavesdropping on the private conversations of other House members located near the other focal point. Real World Example -Past World example: A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path. -Past World example- wagon wheals OR
- Bits for horses -Satellites can be put into elliptical orbits if they need only sometimes to be in high- or low- earth orbit, thus avoiding the need for propulsion and navigation in low-earth orbit and the expense of launching into high-earth orbit. Suppose a satellite is in an elliptical orbit, with a=4420 and b=4416, and with the center of the Earth being at one of the foci of the ellipse. Assuming the Earth has a radius of about 3960 miles, find the lowest and highest altitudes of the satellite above Earth.
The lowest altitude will be at the vertex closer to the Earth; the highest altitude will be at the other vertex. since i need to measure these altitude from the focus, I need to find the value of c.
c^2 = a^2 – b^2 = 4420^2 – 4416^2 = 35,344 Then c = 188. If I set the center of my ellipse at the origin and make this a wider-than-tall ellipse, then I can put the Earth's center at the point (188, 0).
The vertex closer to the end of the ellipse containing the Earth's center will be at 4420 units from the ellipse's center, or 4420 – 188 = 4232 units from the center of the Earth. Since the Earth's radius is 3960 units, then the altitude is 4232 – 3960 = 272. The other vertex is 4420 + 188 = 4608 units from the Earth's center, giving me an altitude of 4608 – 3960 = 648 units