Parametric Curves

EMT 668

By Kelly Pierce

Parametric equations are quite different from conventional equations like
y = cos (x). You can think of parametric points as representing positions
of an object, and of t as representing time in seconds. Evaluating the parametric
equations for a value of t gives us the coordinates of the position of the
object after t seconds have passed.

Now, let's start our investigation by graphing the parametric curve that
is formed by the equations : x = cos (t) and y = sin (t).

The curve as you probably guessed is a circle. Let's look at the table below
to define some important points.

t |
x |
y |

0 |
1 |
0 |

p/4 |
Ö 2/2 |
Ö 2/2 |

p/2 |
0 |
1 |

3p/4 |
-Ö 2/2 |
Ö 2/2 |

p |
-1 |
0 |

5p/4 |
-Ö 2/2 |
-Ö 2/2 |

6p/4 |
0 |
-1 |

7p/4 |
Ö 2/2 |
-Ö 2/2 |

2p |
1 |
0 |

If you compare the tables coordinates (x, y) to the graph, you will find the coordinates match. It is easy to see how the t variable comes into play with the x and y coordinates. I do not want you to think this is all the values of t. The variable t can be any number between 0 and 2p.

To pursue this investigation, I would like to change the equation to x = cos (t) and y = 2sin (t) and see how this effects the parametric curve. Below you will find the curve and the table that corresponds with the curve.

t |
x |
y |

0 |
1 |
0 |

p/4 |
Ö 2/2 |
Ö 2 |

p/2 |
0 |
2 |

3p/4 |
-Ö 2/2 |
Ö 2 |

p |
-1 |
0 |

5p/4 |
-Ö 2/2 |
-Ö 2 |

6p/4 |
0 |
-2 |

7p/4 |
Ö 2/2 |
-Ö 2 |

2p |
1 |
0 |

The graph of the curve changes to an oblong shape. Look on the table. The
y coordinates are all multiplied by 2, but the x coordinates all remain
the same. Below, is a graph of the curve above and the new equation x
= 2 cos (t) and y
= 2 sin (t).

t |
x |
y |

0 |
2 |
0 |

p/4 |
Ö 2 |
Ö 2 |

p/2 |
0 |
2 |

3p/4 |
-Ö 2 |
Ö 2 |

p |
-2 |
0 |

5p/4 |
-Ö 2 |
-Ö 2 |

6p/4 |
0 |
-2 |

7p/4 |
Ö 2 |
-Ö 2 |

2p |
2 |
0 |

The curve formed by the above equation, constructed the original curved
multiplied by two. Mathematically, this makes since, because we essentially
multiplied each coordinate by two. Unlike the previous example, where we
multiplied just one coordinate by two and the curve changed to an ellipse,
this example stayed a circle and its radius changed to two. With the two
above examples we have demonstrated what happens when you multiply the parametric
equations by positive integers.

What do you think would happen if we multiplied the parametric equations
by negative integers? Let's graph the equation x
= -2 cos (t) and y
= sin (t):

t |
x |
y |

0 |
-2 |
0 |

p/4 |
-Ö 2 |
Ö 2/2 |

p/2 |
0 |
1 |

3p/4 |
Ö 2 |
Ö 2/2 |

p |
2 |
0 |

5p/4 |
Ö 2 |
-Ö 2/2 |

6p/4 |
0 |
-1 |

7p/4 |
-Ö 2 |
-Ö 2/2 |

2p |
-2 |
0 |

From the curve, it looks identical to x = 2 cos (t) and y = sin (t), but
if you look at the tables you would see the coordinates do not match. For
example at t = 0, the positive 2 curve's x coordinate is 2, while the negative
2 curve's x coordinate is -2. Since we are doing a full cycle of t values
(from 0 to 2 p), then
eventually all the coordinates are covered and make a complete circle.

Now, let's try to make a circle with the negative coefficients in front
of the parametric equations. Let's try to graph x
= -2 cos (t) and y
= -2 sin (t) for t = 0..2p.

t |
x |
y |

0 |
-2 |
0 |

p/4 |
-Ö 2 |
-Ö 2 |

p/2 |
0 |
-2 |

3p/4 |
Ö 2 |
-Ö 2 |

p |
2 |
0 |

5p/4 |
Ö 2 |
Ö 2 |

6p/4 |
0 |
2 |

7p/4 |
-Ö 2 |
Ö 2 |

2p |
-2 |
0 |

Again, we form a circle, but the manner in which the circle is formed is
different from the positive 2 curve. The positive curve starts at point
(2, 0) and goes counterclockwise to form the circle, but the negative curve
starts at (-2,0) and goes counterclockwise. So the positive curve forms
the top half of the circle and then forms the bottoms half, but the negative
curve does the opposite.

For the next exploration, I would like to graph the parametric curve x =
1/2 cos (t) and y = 1/2 sin (t).

t |
x |
y |

0 |
.5 |
0 |

p/4 |
Ö 2/4 |
Ö 2/4 |

p/2 |
0 |
-.5 |

3p/4 |
-Ö 2/4 |
Ö 2/4 |

p |
.5 |
0 |

5p/4 |
-Ö 2/4 |
-Ö 2/4 |

6p/4 |
0 |
.5 |

7p/4 |
Ö 2/4 |
-Ö 2/4 |

2p |
.5 |
0 |

The formation of the curve was exactly what I suspected, a circle with radius
equal to .5 and the formation starting at the point (.5,0).

To summarize this investigation, when the |a| is equal to |b| , then the
curve forms a circle with a radius of a. If |a| is greater than |b|, then
the curve forms an ellipse with the major axis on the x axis. Finally, if
|a| is less than |b|, then the curve forms an ellipse with the major axis
on the y axis.

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