Parametric Equations The Point?

[DecayProduct]

I have found a lot of information about how to solve and graph parametric equations, but what I want to know is, what is the point, or why do they exist at all? The way I have seen it explained is that both x and y are stated in terms of another "term", usually t. But if I can state x and y in terms of t, shouldn't I be able to state t in terms of x, and therefore y in terms of x, like we are used to with Cartesian coordinates?

I read something on Wikipedia that described the parameter, t, something like this:

"the speed of a car is akin to the dependent variable, and the gas pedal is the independent variable. Press the pedal, the car goes a certain speed. Change the mechanical linkages, and you have changed a parameter by which the dependent relates to the independent"

Ok, so that sounds like, y = x in one setting, and changing the "parameters" would be like changing how y relates to x: like y = 2x. I'm having trouble actually articulating the question! So, what's the point? Why do they exist? What makes it any different than just using x to define the function(s)?

 

[mbisCool]

They are used when you have multiple depedent variables.

A simple example would be two dimensional projectile motion. Both the horizontal and verticle displacement depend on the parameter time. A text on multivariable calculus should provide a sufficient explanation.

 

[Office_Shredder]

Good example: x=cos(t), y=sin(t). There is no 'nice' way to write y in terms of x such that you still get a full circle. When you do stuff like differentiation on a curve like a circle, you don't want to have to deal with the case of whether y is positive or negative, and a third case in case y=0

 

[hotvette]

Parametric curves can fold, wrap, or cross. Try plotting

x = 26t^3 – 40t^2 + 15t - 1
y = -4t^2 +3t

where t is between 0 and 1. It would be a chore to solve the 2nd equation for t and substitute into the 1st. Much easier to leave it in parametric form. Also, parametric equations can have vertical slope. They are used for many purposes, such as CAD programs and for representing fonts.

http://en.wikipedia.org/wiki/Bézier_curve