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