EMT 668 - ASSIGNMENT 10

Parametric Curves

by
### Kimberly N. Bennekin

## EXPLORATIONS

### Graph

## x = cos(t)

y = sin(t)

**for t within the interval [0,2pi].**
### How would you change the parametric equations to explore other graphs?

First let us consider the original parametric curve for t within the interval
[0,2Pi]:

**x = cos (t)**

y = sin (t)

We can consider a number of variations to this curve. First let us consider
multiplying our parameter *t *by a constant in both x = cos
(t) and y = sin (t). Let's consider 2, 3 and -1.

### x = cos (2 t)

y = sin (2 t)

### x = cos (3 t)

y = sin (3 t)

x = cos (- t)

y = sin (- t)

Let us consider putting multiples on only one of
the funtions.

### x = cos (2 t)

y = sin (t)

### x = cos (3 t)

y = sin (t)

### x = cos (4 t)

y = sin (t)

### x = cos (5 t)

y = sin (t)

It seems that the constant multiple within **x = cos (a t),** for
even **a, **creates additional dips in our graph. For odd **a**, we
create additional complete cycles in the graph.

Multiplying by a constant within **y = sin (a t)** will
have a different effect. Consider the following graphs.
### x = cos (t)

y = sin (2 t)

### x = cos (t)

y = sin (3 t)

This manipulation creates a **"Bow Tie"** graph. The larger
the constant multiple, the more rotations we have.

Let's consider the addition of a constant to **t**. We
will not consider a parametric curve where the change in both **x = cos
(t)** and** y = sin (t)** are the same. Consider the following graphs:
### x = cos (t+1)

y = sin (t)

### x = cos (t+2)

y = sin (t)

### x = cos (t-1)

y = sin (t)

By adding or subtracting a constant to **t** within **x = cos (t
+ a)**, we create an ellipse. This investigation leads me to believe,
the larger the constant you add to **t**, the narrower your ellipse becomes.
Subtracting a constant reflects the graph about the y-axis.

Let's consider adding and subtracting a constant to **t**
within **y = sin (t +a)**. Consider the following:
### x = cos (t)

y = sin (t+1)

### x = cos (t)

y = sin (t-1)

It seems that adding 1 within **y = sin (t+a)** has the same effect
as subtracting 1 within

**x = cos (t - a)** and subtracting 1 within **y = sin (t - a)** has
the same effect as adding 1 within

**x = cos (t+ a)**.

These are just a few of the many explorations that can be done on this set
of parametric equations.