Understanding Parametric EquationsDate: 04/02/98 at 20:29:40 From: Kristen Demergian Subject: Parametric Equations Dear Dr. Math, I'm a junior at Granite Hills High School In El Cajon. For honors Pre-Calculus we are required to write an essay on an advanced calculus concept that relates to the real world. I have obtained information about parametric equations that relate them to scientific technology currently being used today. I have been researching the topic for about a week and am getting ready to write an outline. I understand that parametric equations are, in general, graphs and equations that use three variables, x, y, and t. However, I am confused on their basic principles. Everything I've come across has been written in advanced terminology, hence I am a little confused. I would appreciate it if you would explain, in general, what parametric equations are. Date: 04/03/98 at 08:08:59 From: Doctor Jerry Subject: Re: Parametric Equations Hi Kristen, Only relatively few curves in the plane can be described as the graph of an equation y = f(x). A circle can't be described in this way, just parts of it. Parametric equations can be used to describe circles, and much more. Parametric equations are like this: one gives the x and y coordinates of points on the curve in separate equations. For example, the parametric equations: x = cos(t) y = sin(t) describe a circle, as t varies over [0,2pi]. For each value of t, plot the point (x,y) = (cos(t),sin(t)). The moving point will trace a circle. Notice, for example, that x^2+y^2 = cos^2(t)+sin^2(t) = 1. Hence, (x,y) is on a circle with radius 1, centered at the origin. Another example: x = t*cos(t) y = t*sin(t) is a spiral, and: x = t-cos(t) y = 1-sin(t) is a cycloid curve. Parametric curves are used in three dimensions as well. The parametric equations: x = cos(t) y = sin(t) z = t describe a helix, curling around the z-axis. The variable t is a parameter, often describing time. The position of a particle at any time t might be: x = 4t-3 y = -t^2+7t+1 which is a parabolic path. Perhaps I've said enough for you to get the idea. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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