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Conic Sections

The term conic sections is used when discussing the derivation of a line that is a locus of points equal distance from either a line, a point, both a line and a point, two lines, etc. The term conic sections also can be used when discussing certain planes that are formed when they are intersected with a right circular cone. The planes, or lines as we know them, consist of the circle, the ellipse, the parabola, and the hyperbola. (West, 112) There are different ways to derive each separate curve, and many uses for them to be applied to as well. All of which are an important aspect to conic sections.

The cone is a shape that is formed when you have a straight line and a circle, and the straight line is moved around the circumference of the circle while also always passing through a fixed point at a distance away from the circle. The parts formed are labeled the upper nappe, the lower nappe, and the vertex, (Prime, 1) as described in the diagram below in diagram 1:

The cone is then used with the help of a right plane to form the different circles, parabolas, ellipses, and hyperbolas, as shown below in diagram 2 on the next page. Taking a flat plane that would be parallel to the base of the cone, and intercepting it with a single nappe of the cone produces the circle. The ellipse is formed by the intersection of the cone with a flat plane that intercepts one nappe of the cone, but is not parallel to the base, and is not parallel to any other side of the cone. The parabola is a curve produced by the intersection of the cone with a flat plane of a single nappe, parallel to the remaining slanted side. Finally, the hyperbola is produced when you take a flat plane and intercept it with the cone so that it passes through both nappes. It is nearly perpendicular to a base of the cone, but is more often than not, illustrated with a greater angle of the lower nappe being intercepted. (West, 114)

The easiest way to visually produce these shapes is to use a flashlight and have an empty wall in your house. By holding the flashlight parallel to the floor, you will produce a circle on the wall. In holding the flashlight slightly tilted, an ellipse is formed. When you hold the flashlight so that the side farthest away from the wall is parallel to the wall, a parabola is formed. Finally, when the flashlight is tilted so that it is almost parallel to the floor, you will get half of a hyperbola because the flashlight can only simulate a single cone, as the wall represents the plane. (West, 114-115)

To get into the details of conic sections it is necessary to break them up into each separate conic section. The circle can be considered the simplest conic section because it is the easiest to understand. A Greek Philosopher named Apollonius was studying the first forms of the circle back around 225 B. C., while he was writing Conic Sections, which was a series of eight different books analyzing all the primate forms of these...

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