Conic Section and Parametric Equations

Topics: Conic section, Polar coordinate system, Coordinate system Pages: 2 (804 words) Published: November 12, 2006
A conic or conic section is a plane curve that can be obtained by intersecting a cone with a plane that does not go through the vertex of the cone. There are three possibilities, depending on the relative position of the cone and the plane. If no line of the cone is parallel to the plane, the intersection is a closed curve, called an ellipse. If one line of the cone is parallel to the plane, the intersection is an open curve whose two ends are asymptotically parallel; this is called a parabola. Finally, there may be two lines in the cone parallel to the plane; the curve in this case has two open pieces, and is a hyperbola. In mathematics, parametric equations bear slight similarity to functions: they allow one to use arbitrary values, called parameters, in place of independent variables in equations, which in turn provide values for dependent variables. A simple kinematical example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Abstractly, a relation is given in the form of an equation, and it is shown also to be the image of functions from items such as R. It is therefore somewhat more accurately defined as a parametric representation. It is part of regular parametric representation. Converting a set of parametric equations to a single equation involves solving one of the equations usually the simplest of the two for the parameter. Then the solution of the parameter is substituted into the remaining equation, and the resulting equation is usually simplified. It should be noted that the parameter is never present when the equation is in singular form; it must cancel out during conversion. Or, the process put simply: the simultaneous equations need to be solved for the parameter, and the result will be one equation. Additional steps need to be performed if there are restrictions on the value of the parameter. The arc length of a circle is the distance from one point on the circumference to...
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