Michelle E. Chung

* EMAT6680 Assignment 10: Parametric Curves

About the graph of	About the graph of	About the graph of

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 y-axis.
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
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 3D-sine/cosine graph, which means its amplitude is 1 (here, we can see it on the x-axis) 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 x-axis...). The graph is always passing (1,0).
When b=2 and vary a
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 x-axis. The graph is always passing (1,0).
When a=1 and vary b
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 3D-sine/cosine graph, which means its amplitude is 1 (here, we can see it on the y-axis) 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 y-axis...). The graph is always passing (1,0).
When a=2 and b vary
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
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 x-axis and to the y-axis. 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 y-axis. 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 y-axis 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 y-axis 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 y-axis) 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, 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 bow-ties. The graph of looks like a bow-tie 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 bow-tie, 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 bow-ties. 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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