Exploring Parametric Curves

 

By: Diana Brown

 


A parametric curve in the plane is a pair of functions

where the two continuous functions define ordered pairs (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t.


EXPLORATIONS:

 

Graph

x = cos (t)

y = sin (t)

for 0≤ t ≤ 30

 

 

 

 

How would you change the equations to explore other graphs?

 

For example: for various a and b of the graphs:

x = cos (at)

y = sin (bt)

for 0 ≤ t ≤ 30

 

First lets investigate different values of a and b, but keeping a = b.

a=2=b a=3=b

a=4=b a=10=b

All circles.

 

Now lets investigate when a = 1 and b varies

b = b = 2

 

b = 3 b = 4

 

Notice that when a = 1, b determines the number of loops

 

Now lets investigate when a = and b varies

 

 

b = 1 b = 2

This pair creates two loops This pair creates 4 loops

 

 

b = 3 b = 4

This pair creates 6 loops This pair creates 8 loops

 

Notice that when a = then the number of loops is determined by 2b.

 

Now lets investigate when a = 2 and b varies

 

b = b = 1

 

b = 3 b = 4

 

It now seems as though the number of loops is determined by b, lets try two more to see what happens:

 

 

b = 5 b = 6

It seems to be so.

 

 

Now lets hold b to be constant and vary a. For the following graphs b =1.

 

 

a = a = 2

 

a = 3 a = 4

 

 

It seems to do the same as above except about the y axis and when a is even it is open curves. Lets try two more to see if this is true.

 

a = 5 a = 6

It looks as though we will get something similar about the y axis when b is held constant and a is changing, just as the curves were changing above when a was held constant and b was changing.

 


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