The parametric curve represented by: x = t* cos(2t);  y = t* sin(t) over the interval (0, 4Pi)	The parametric curve represented by: x = 20t cos(.2t);  y = 10 sin(t) over the interval (0, 80Pi)

Sometimes, especially in science, it is convenient to describe a curve in terms of a parameter. "t" is often the parameter as it often represents time. The ability to describe curves parametrically allows for vastly different types of curves with simple parametric equations. The equations are simpler, but there is now a seperate equation for each variable. Because the equations are simpler, an equation that would be very complex using the normal Cartesian equation, can be amazingly simple using parametric equations.

This separate equation for x and y can also be easier to use for computer aided manufacturing devices, like robotic arms. (See below under the definition for Bezier curves.)

Below you see a table of values constructed for the parametric equations shown in the heading of the table. The parameter in this case is the angle,  .  Using a spreadsheet, this process can be automated and plotted using the scatterplot function.. The top graph below illustrates this. A parametric plot in Maple is also shown, with the accompanying syntax. Please note the minor difference between multiple plots in Maple, and parametric plots in Maple. The only difference is the position of the square brackets. In multiple plots, the square brackets enclose only the functions to be plotted. In parametric plots, the square brackets are around the entire expression. In both cases, the rounded parentheses are on the outside.
 

0.25 0.494808 0.968912 0.5 0.958851 0.877583 0.75 1.363278 0.731689 1 1.682942 0.540302 1.25 1.897969 0.315322 1.5 1.99499 0.070737 1.75 1.967972 -0.17825 2 1.818595 -0.41615 2.25 1.556146 -0.62817 2.5 1.196944 -0.80114 2.75 0.763322 -0.9243 3 0.28224 -0.98999 3.25 -0.21639 -0.99413 3.5 -0.70157 -0.93646 3.75 -1.14312 -0.82056 4 -1.5136 -0.65364 4.25 -1.78998 -0.44609 4.5 -1.95506 -0.2108 4.75 -1.99859 0.037602 5 -1.91785 0.283662 5.25 -1.71787 0.512085 5.5 -1.41108 0.70867 5.75 -1.01656 0.861192 6 -0.55883 0.96017 6.25 -0.06636 0.999449 6.5 0.43024 0.976588 6.75 0.900088 0.893006 7 1.313973 0.753902 7.25 1.646162 0.567924 7.5 1.876 0.346635

> plot([2*cos(t),sin(t),t=0..100]);

Below is an animated graph of the following parametric equations:
x = cos(3ct), y = sin(5ct)

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 1.7A: Parametric Curves. This module will let you explore with parametric equations and get a better feel for how the two seperate functions combine.

Definitions

Parameter	A third variable which x and y are described in terms of. (t is a frequently used parameter representing time).
Parametric Equations	Two equations, each representing the x and y coordinates of a Cartesian graph expressed in terms of a third variable , or parameter.
Parametric Curve	A Cartesian graph or curve represented by two parametric equations.
Initial Point	The initial point of a parametric curve is the point which represents the x and y values when the parameter takes on the lowest value in its domain .
Terminal Point	The terminal point of a parametric curve is the point which represents the x and y values when the parameter takes on the greatest value in its domain .
Bezier Curves	Special parametric curves that are often used in manufacturing, and in describing the shape of characters sent to laser printers. Bezier curves "smoothly" connect a set of points.
Cycloid	A curve traced out by a point on the circumference of a circle as the circle rolls along a straight line.

Parametric Equations

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 1.7B: Families of Cycloids. This module shows how the cycloid is formed as the circle moves.

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