Most familiar to us all are cuves written as equations in rectangular form (y = f(x) ) where one variable is solely dependent on the other variable(s). There is, however, the need to describe the motion of a point,P, in the coordinate plane. To do this, we use Parametric curves defined to be : x = f(t) and y = g(t) for time, t. Therefore, the position of P as it moves through the coordinate plane is (f(t), g(t)) for time t. Time is the independent variable upon which both x, y and the position of Pare dependent.
The following explores the production of specific curves via parametric equations.
"Write the equation of the line through the point (7,5) with slope 3" is a fairly familiar problem when working in rectangular coordinates.
One could use the point-slope formula for the equation of a line.
How could we use parametric equations to obtain this same curve? (We can only define this curve for some given interval of time, t.)
We must define construct the equations f(t) and g(t) such that x=f(t) and y = g(t) yield the coordinates of the point P along this curve at some time, t.
In general x=f(t) = at + b and y = g(t) = ct + d. In order to find out which part of these equations determine the slope of the line, let's use the simple equation: slope = rise/run at two specific times, t(1) and t(2).
Therefore, we know specifically that a = 1 and c = 3, or...
Therefore, we are left to find the values of b and d. Because x=f(t) is responsible for the x-coordinate of the point P as it follows the path of the curve, the value of b moves the determines how far left or right of the origin, the point P lies. Similarly, because y=g(t) defines the y-coordinate of the point P at some time, t, the value d determines how far up or down from the origin the curve, P lies.
Because we want our curve to go through the point (7,5), this would mean that
Now that you're a pro at parametric curves, see if you can solve the following problem:
Any immediate guesses? Unlike the curve we just dealt with, this curve is obviously not linear. In fact, it involves the trigonometric functions cos (t) and sin (t).
Click here to see a portion of the graph of this parametric curve, the cycloid for .
for a Geometer's Sketch Pad script to construct a cycloid.
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