**Assignment
10**

**Parametric
Curves**

by

Behnaz Rouhani

A parametric curve in the plane is a pair of functions

This assignment is divided into three parts:

**I. First we
will explore the equation**

**II. For the
second part we will investigate the parametric curve**

for different values of a and b.

**III.** **Special
explorations**

**Case I: Let us explore the
equation**

We will set to demonstrate what happens to the curve of the parametric equation as the exponent for t increases. While viewing the following curves do you see an interesting phenomenon?

**x = cos (at)**

**y = sin (bt)**

**for different values of
a and b.**

(cost, sin3t) ---- **Green**

Observation: When a =1, the number of hoops is the
same as b.

(cos2t, sin 4t) --- **magenta**

(cos2t, sin8t) --- **Navy
blue**

Observation: When a = 2, the number of hoops is the
same as b/2.

**Part C: a
< b, a =3**

(cos3t, sin4t) --- Red

(cos3t, sin7t) ---- Magenta

Observation: When a = 3, the number of hoops is the
same as b.

**Part D: a
< b, a = 4**

(cos4t, sin6t)** **---** Green**

(cos4t, sin8t)** **---** Yellow**

These graphs are similar to graphs of (cos2t, sin3t)
and (cos2t, sin4t) respectively. Can you tell why we get the same graphs?

**Part E: a
< b, a = even number**

(cos4t, sin5t)** --- **Brown

(cos6t, sin7t)** --- **Green

What kind of conjecture could we make about this case?

(cos10t, sin5t) ---- **brown**

Observation: As long as coefficient of b is half
of a, the graph is a parabola with vertex at (1,0).

**Part G: a
> b, a = even number, b = odd number , difference between a and b is one**

(cos4t, sin3t) --- **cyan**

(cos6t, sin5t) --- **Brown**

(cos14t, sin13t) ---** Blue**

Observation: As the value of a and b increases, with a = even number, b = odd number and the difference between a and b is one, the graphs seem to be getting filled up.

**Part H: a
> b, a = odd number, b = even number, difference between and b is one**

(cos3t, sin2t) --- Green

(cos5t, sin4t) --- Red

(cos7t, sin6t) --- Blue

This interesting pattern occurs when simple curves are displayed as the "envelope" of a more complicated function. In practice such things happen when low frequency waves ride carrier waves broadcast from a radio station.

Notice the two blue graphs in part G and H, don't they look almost alike?

**Part I: a
> b, a =odd number, b = odd number, and difference between a and b is more
than one**

(cos7t, sin3t) --- Blue

(cos11t, sin3t) --- Green

Observation: Number of hoops in each case is the same as a.

**Case III: Special Investigations**

We will try to explore the following special parametric curves.

**Part A:**

This is the graph of **Astroid**
for different values of a = b. Here is the color coding for the graph.

Maroon graph: a = b = 1

Blue graph: a = b = 2

Red graph: a = b = 4

Yellow graph a = b =
6

**Part B:**

( t - sint, 1 - cost)

This is the graph of **Cycloid**.

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