Parametric Curves

                              By Na Young Kwon

A parametric curve in the plane is a pair of functions


    x = f(x)

    y = g(x)


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 and your work with parametric equations should pay close attention the range of t . In many applications, we think of x and y "varying with time t " or the angle of rotation that some line makes from an initial location.


2. For various a and b, investigate

    x =  cos (at)

     y = sin (bt)                  for 0<= t <= 2¹


At first,  let a and b be the same number, say a=b=1.(0<t<2¹)

Picture 1


We can check this graph is a circle. Investigate the graph changing value.

When a and b have the same value the graph is always a circle regardless

of the value. (Only 0 < t < 2¹). We can check this truth algebraically.

For X=cosat , Y= sinat  for any value of a .

Then         by the rule of

trigonometric function.


Next, change the b value. To compare graphs we fix a value of a

 is 1. (0 < t < 2¹).

Picture 2



When the value of b is changing,  a number of curves increase.

I investigate graphs in the case that b is even and odd. Especially,

when b is odd, the graph passes a point (1,0) and (-1,0) and

the shape is similar to the graph of coskx and -coskx.

In this case we can find that each graph has concave curves

above the x-axis by b numbers. When b is even, the graph

passes an original point and the shape is similar to sinkx.

 In this case each graph has also concave curves above

 the x-axis by b numbers. And all the graph have the symmetry

 graph by x-axis and y-axis.

Picture 2-1                                                 


Picture 2-2        



Next, change the value of a. Fix b is 1(0 < t < 10).

Picture 3      


We can find an interesting point in this graph.

For example, when a value of a is equal to 3 and b=1, the graph

 is congruent to the graph which is rotated a graph of b=3 and a=1

 by 90 degrees. I found the result that a graph which is x=cos(at)

 and y=sin(bt) is rotated a graph which is x=cos(bt) and sin(at)

 by 90 degrees when a and b is odd. Strangely, when b=1 and

a value of a is changing, any graph doesnÕt pass an original point.

 In this case all graphs passes (0,1).



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