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The purpose of this report is to investigate the conics, or conic sections. Conic sections include the circle, parabola, ellipse and hyperbola, all of which have applications to the world. Through this investigation, I hope to generate -D conics in excel , graph the conics on my graphics calculator and draw conics with pen to paper.
History of Conics
Conic sections, or conics, are among the oldest curves studied by mathematicians. They consist of the circle, ellipse, parabola and hyperbola. They were discovered around 50 BC by a Greek mathematician Menaechmus. His research was refined around 00 BC by Appollonius. Appollonius was fundamental in the study of conics since he 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. No important scientific applications were found for them until the 17th century, when Kepler discovered that planets move in ellipses and Galileo proved that projectiles travel in parabolas. Since then, conics are used in a number of things, including the study of optics.
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Description of Conics
A circle is formed when a cone is cut perpendicular to its axis. A parabola is formed when a cone is cut parallel to its generator. An ellipse is formed when a cone is cut at any angle between its generator and the perpendicular to its axis. A hyperbola is formed when a cone is cut parallel to its axis.
The eccentricity, e, is the ratio of the distance o a fixed point (focus) and a point to a fixed line (directrix). This is represented in the formula e = PF / PM
Plots of conics on a Cartesian plane, where (h ,k) is the centre.
Circle Ellipse Parabola Hyperbola
Equation (horizontally) (x-h) + (y-k) = r (x � h) / a + (y � k) / b = 1 4a(x � h) = (y � k) (x � h) / a � (y � k) / b = 1
Equation of Asymptote (y � k) = ± b/a(x � h)
Equation (vertically) (x-h) + (y-k) = r (y � k) / a + (x �h) / b = 1 4a(y � k) = (x � h) (y � k) / a � (x � h) / b = 1
Equations of Asymptote (x � h) = ± b/a(y � k)
Where… r = radius a = major axis (longest distance across the ellipse)
b= minor axis (longest distance across the ellipse that is perpendicular to the major axis)
b = a(1 - e) a = distance from vertex to focus (or directrix) a = 1/ length major axis
b = 1/ length minor axis
b = a(e � 1)
Eccentricity e = 0 0 e 1 e = 1 e 1
Generating �D and �D Conics
Cone � Using the distance formulae (sqrt(x + y), you are able generate distances from the horizontal axis. This ultimately graphs a cone. The horizontal distance to the origin is used as the x value.
Sphere � If you have a constant radius, simply rearrange the equation of the circle to get y = sqrt(r � x).
Ellipsoid � Again, rearrange the ellipse formula to get y = (sqrt(b(1 � x / a)).
Paraboloid � You already know the equation for a parabola, y = x, so simply substitute numbers in.
Hyperboloid � Again, with this, the easiest equation of a hyperbola is y = 1/x. So simply substitute the x values into the cells. Appendix 1 has all the -d conics.
For each conic section graphed on the graphics calculator, the center is (1, 1). The method used to graph this is rearranging, like the excel generations, the formulae to get y equals. Appendix 1 has all the graphs. You may notice with the graphs of the circle and ellipse that lines are not joining. This is because of the settings on the calculator.
Pen and Paper
You’ll see all the conics generated by pen and paper on appendix .
Circle � To generate a circle, simply use an instrument called a compass.
Ellipse � Fasten two thumbtacks in a piece of paper, a distance apart and fastening the ends of a piece of string to the thumbtacks. Now loop the string around a pencil and draw it taut.
Parabola � Have ten dots vertically, all equal spaces apart. Now put 10 ten dots horizontally, again equidistant apart. Join the top dot on the vertical to the first dot from the origin on the horizontal. Now join the seconds and so on.
Hyperbola � Using the ripple tank theory (see page 7), you are able to draw Hyperbolas where the two circles meet.
Occurrence of Conics
Physics is a science that takes advantage of these conic sections. Physicians are able to use conic sections to explain phenomena in nature, and also use the mathematics to construct many useful objects to humans.
There are two bridges that I’ll be discussing in this report; the suspension (parabola) and arch (circle) bridges. A suspension bridge spans much further distances than an arch bridge, up to 7 times in fact. Span is the distance between two bridge supports. The reason why the suspension bridge is able to span vast distances is because of two forces called compression and tension. Tension is a force that acts to expand or lengthen the thing it is acting on and compression is a force that acts to compress or shorten the thing it is acting on.
The circle was very important for the developing world. If the circle never existed, many things would seize to exist. Here are just a few of the important application of circles.
• Wheel � Let’s face it, the wheel is the most important invention man has made. They’re everywhere and are used to move objects that are heavier than man can lift. Without the wheel, societies wouldn’t form.
• Gears - The thing that is most important with gears is that it can produce a large amount of torque (rotating force). For this reason, gears are in just about everything that has spinning parts. On any gear, the ratio is determined by the distances from the center of the gear to the point of contact. For instance, if one gear is twice the size of the other, the ratio would be 1
• Arch Bridges - An arch bridge is a semicircular structure with abutments on each end. The design of the arch, the semicircle, naturally diverts the weight from the bridge deck to the abutments . For this reason, arch bridges were and still are very important. In arch bridges, tension is negligible; however, they are always under compression. This force is pushed outward, along the arch curve to the abutments.
• Bearings - Bearings are very important as they reduce friction. They do this because smooth metal balls roll easily through a smooth inner and outer metal surface. These balls bear the load, allowing the device to spin smoothly. Bearings have some really interesting uses. Bearings are used for earthquake “proof” buildings and very high-speed devices, using magnetic bearings.
A lot of things in nature that appear to be circles, but they really are ellipses. So it is not surprising that the ellipse is the most occurring curve in nature. Also, man uses some of the ellipse’s properties for real world applications.
• Orbits � The Orbits of planets were thought to have revolved around the sun in a circular path. However, in the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci. This is because each orbit is on a slight angle.
• Cylinders � Some examples of the above statement that every cylinder that is cut at an angle will reveal an ellipse. A few constructions will prove this, like the Tycho Brahe Planetarium in Copenhagen, below. When you tilt a glass of water, the resulting curve will also be an ellipse.
• Reflection of Waves � The ellipse has an important property when it reflects sound and light waves. The property is that when light or sound waves start at one focus, the waves will be reflected to the other focus. Lithotripsy, a medical procedure for the kidney stone, uses this principal. The patient is placed in an 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. The principle is also used in the construction of whispering galleries. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.
• Footballs � In the game of rugby, a ball is constructed as an ellipse. Unlike golf, tennis or cricket balls, rugby balls will bounce in unexpected directions (without spin).
The parabola has a lot of applications used by humans. It has important roles to help read books to driving at night to building bridges. It is a very important curve to humans.
• Projectile motion � When an object is projected in the air at an angle, the motion of the object will resemble a parabola path. This path is the result by the effect of gravity acting on the object. The parabolic shape of the object is because the gravity acting on the object has an acceleration of .81 m/s vertically and the object will have no acceleration horizontally.
• Optics � Parabolas exhibit useful reflective properties, aiding in the study of optics. If a light-bulb is placed at the focus of a parabolic mirror, concave, the light will be reflected in rays parallel to said axis . In this way, a straight beam of light is formed. Lenses are another example of the usefulness of the parabola. As you may already know, the angle of incidence is equal to the angle of reflection. Reading glasses, or convex lenses, will magnify an object. This is because the convex lens will focus light from a distant source to a point. The shorter the focus, the more powerful the lens.
• Suspension Bridges � A suspension bridge is when cables suspend the deck to form a bridge. Suspension bridges have two tall towers through which the cables are strung. The towers will support the majority of the roadways weight. Compression will push down on the deck, but since it is suspended roadway, the towers soak up all the force. This will dissipate the compression directly into the earth where they are firmly entrenched. The supporting cables are the recipients of the tension forces which will go to the anchorages (end of the bridge) and again dissipate to the earth.
The hyperbola has applications for sound. The occurrence of hyperbolas is not that common in the world, but still has important applications.
• Sonic Boom � When a sonic boom shock wave intersects the ground, the shape will be a hyperbola. The noise 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. The shape also explains why we are able to see the planes before we can hear it.
• Hyperboloids � In many nuclear power plants, the chimneys are shaped as a hyperboloid. This allows less waste into the atmosphere. It also looks attractive and therefore many buildings will look like the hyperbola.
• Ripple Tank - A ripple tank is a shallow glass tank of water with a motor powering two knobs. When powered simultaneously, the waves that are formed produce bright and dark lines, called antinodes and nodes respectively. The shape of these antinodes and nodes are hyperbolas. This concept also applies to sound waves where the high volume lines are antinodes and the low volume lines are nodes.
Conclusion � By investigating a small part of conics, I was amazed by the number of applications that involved the conics. I was able to see the full picture about using conics in real world applications. Also, generating -d and -d conics, my goal, proved to be quite easy. I was only able to generate part of the -d conic, though. Excel is a very powerful tool when it comes to graphing -d objects. As this investigation comes to and end, we are all able to see the importance of conics.
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