
Michelle E. Chung 
*
EMAT6680 Assignment 10: Parametric Curves 

1. Graph
for
How do you change the equations to explore other graph?


The equation is a parametric equation. So, if you want to draw it, you have to use the relationship
among the variables.
First, since we know that , we can substitude x=cos t and y=sin t to it.
Then we would have .
As we know, this is the equation of the circle that has (0,0) as its center and
1 as its radius and the following is the graph:

2. For various a and b, investigate
for


When
a=1, b=1



 As we see from the first case, if a=b=1, the graph of the parametric equation is a
circle that has (0,0) as its center and 1 as its radius.

When a=2,
b=1


 However, if a=2 and b=1, the graph looks like a parabola that (1,0) as its vertex
and p is about 1/2.

When a=2, b=1


 As we see, the sign of value of 'b' dosen't matter to the graph. Even though
we have b=1 instead of b=1, we still have the same graph.

When
a=2, b=1


 As we see, the sign of value of 'a' also dosen't matter to the graph. Even though
we have a=2 instead of a=2, we still have the same graph.
 So, now we know that the sign of 'a' and 'b' don't matter to the graph.

When a=1,
b=2



 When a=1 and b=2, the graph of the equation looks like a bow tie.
 The graph is symmetric to the yaxis.

When a=2,
b=2



 When a=2 and b=2, the graph of the equation looks like a circle, which has (0,0)
as its center and 1 as its radius, again.

When b=1
and vary a






 As we see, when b=1 and 'a' veries, the graph starts from a circle, which has
(0,0) as its center and 1 as its radius, and is changing to
a kind
of
3Dsine/cosine graph, which means its amplitude is 1 (here,
we can see it on the xaxis) and its period is decreasing as
the
value
of 'a' is increasing. Also, it forms a circle on the yz plane
(if we can imagine..., like, we are looking at it from xaxis...).
 The graph is always passing (1,0).

When b=2
and vary a






 This looks similar to the graph of when b=1 if the value of 'a' is big enough,
but the starting is a little bit different.
The part of it looks like a sine/cosine graph on the xaxis.
 The graph is always passing (1,0).

When a=1
and vary b






 As we see, when a=1 and 'b' veries, the graph starts from a circle, which has
(0,0) as its center and 1 as its radius, and is changing to
a kind
of
3Dsine/cosine graph, which means its amplitude is 1 (here,
we can see it on the yaxis) and its period is decreasing as
the
value
of 'b' is increasing. Also, it forms a circle on the xz plane
(if we can imagine..., like, we are looking at it from yaxis...).
 The graph is always passing (1,0).

When a=2 and
b vary 





 This looks similar to the graph of when a=1.
 The graph is always passing (1,0).

When a=3 and
vary b 





 This looks similar to the graph of when a=1, too. Maybe its period is a little
shorter that the other.
 The graph is always passing (1,0).


11. Inveatigation.
Consider the parametric equations
for .
Graph these for .
Describe fully.
You may have to increase the range of t for the larger
fractions.
This class of parametric curves are called the Lissajous
curves. Compare with


Let's investgate the graph of
.
When a=1, b=2
(i.e. a/b = 1/2) 


 When a=1 and b=2, the graph looks like a bow tie.
 It is symmetric to the xaxis and to the yaxis.
 It passes the origin, (0,0).

When a=1, b=4
(i.e. a/b = 1/4) 


 When a=1 and b=4, the graph still looks like a bow tie, but with more bow.
It seems to have more bows as the value of 'b' is increasing.
(when 'a' is fixed!)
 It is still symmetric figure to the x and yaxis.
 It still passes the origin, (0,0).

When a=2, b=3
(i.e. a/b = 2/3) 


 When a=2 and b=3, it looks like a little bit weird bow tie. ^^;
 However, it is still symmetric to the x and yaxis and passes the origin, (0,0).

When a=12, b=13
(i.e. a/b = 12/13) 


 Now , it looks like a rectangle with some curved lines.
 However, it is still symmetric to the x and yaxis and passes the origin, (0,0).

When a=1 and vary b



 The graph of this function looks like a combination of and .
 It always passes the origin, (0,0).

When a=1, b=0 



When a=1, b=1 


 When a=1 and b=1, the graph is a straight line.

When a=2 and vary b
(interval [10,10]) 


 The graph looks similar to the one when a=1.

When a=2, b=0.4 


 When a=2 and b=0.4, the graph looks like a 90 degrees rotated sine/cosine graph.
 It passes the origin, (0,0).

When a=2, b=0.4 


 When a=2 and b=0.4, i.e. the value of 'b' is opposite, the graph is a reflection
of the one when a=2 and b=0.4.
 It also passes the origin, (0,0).

When a=2, b=6 


 When a=2 and b=6, the graph looks like a reflected (to the yaxis) sine graph.
 It passes the origin, (0,0).

When a=2, b=6 


 When a=2 and b=6, the graph looks like a sine graph and is also a reflection
of the one when a=2 and b=6.


Let's compare the graphs of
.
When a=1, b=1





 When a=1 and b=1, the graph of is a line passing through (0,0) and (1, 0.75); however, the graph of is a line segment, which is part of the line y=x between x=1 and x=1.

When a=1, b=2
(i.e. a/b = 1/2) 

 When a=1 and b=2, i.e. a/b=1/2, the both graphs look like bowties.
 The graph of looks like a bowtie that passes through (4,0), (0,0), and (4,0), and the maximum value of the graph is 3 and the minimum is 3.
 The graph of also looks like a bowtie, but it is much smaller. It passes through (1,0), (0,0), and (1,0), and its maximum is 1 and minimum is 1.

When a=1, b=4
(i.e. a/b = 1/4) 


 When a=1 and b=4, i.e. a/b=1/4, the both graphs look like double bowties.
 The graph of has four parts that look like the combination of part of sine graph and sine graph; however, it still passes through (4,0), (0,0), and (4,0), and its maximum and minimum are still 3 and 3.
 The graph of also four parts that look like the combination of part of sine graph and sine graph, but it is much smaller. It still passes through (1,0), (0,0), and (1,0), and its maximum and minimum are still 1 and 1.

When a=2, b=3
(i.e. a/b = 2/3) 


 When a=2 and b=3, i.e. a/b=2/3, the both graphs look like two boomerangs.
 The graph of doesn't passes through (4,0) and (4,0) anymore but still passes through (0,0). Also, its maximum and minimum are still 3 and 3.
 The graph of doesn't passes through (1,0) and (1,0) anymore but still passes through (0,0). Also, its maximum and minimum are still 1 and 1.

When a=12, b=13
(i.e. a/b = 12/13) 


 When a=12 and b=13, i.e. a/b=12/13, the graphs look like rectangle made out of curve lines.
 The graph of doesn't passes through (4,0) and (4,0) anymore but still passes through (0,0). The graph starts from (0,0) and go around y=3/4 so that form the rectangular shape. Also, its maximum and minimum are still 3 and 3.
 The graph of doesn't passes through (1,0) and (1,0) anymore but still passes through (0,0). Also, its maximum and minimum are still 1 and 1.


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