Parametric Curves

by Venessa Brown

The goal of this assignment is to investigate a pair of functions in a plane:

x = f(t)

y = g(t)

I chose to exlore,

For various a and

b, investigatex = a cos(t);y = b sin(t)for0≤t≤2π.

__Notes on cosine and sine functions__

Both the cosine and sine functions move sinuously from t=0 to t=2π, with cosine starting 1 and sine starting at 0 when t= 0 and both ending at their starting point of 1 and 0 respectively when t=2π. For both functions, the crest (highest point) is 1 and the trough (lowest point) is -1.

Specific to the equations of **x = a cos(t)** and **y = b sin(t)** for 0≤ **t **≤2π, we can note the following:

aandbare scalars that adjust the amplitude (magnitude of the crest and trough) but otherwise has no impact on the cycle (or period) of the function.- these cosine and sine functions move sinuously from t=0 to t=2π, with cosine starting at
aand sine starting at 0 when t= 0 and both ending at their starting point ofaand 0 respectively when t=2π. For the cosine function, the crest (highest point) isaand the trough (lowest point) is -a, while for the sine function, the crest (highest point) isband the trough (lowest point) is -b.

__Signs of a and b__

Considering t=0 as the starting point and t=2π as the ending point, we can note the following in the Cartesian plane:

- the starting point will be (
**a**, 0), and - x will move sinuously from
**a**to 0, while y will sinuously move from 0 to**b**.

So, when **a** and **b** are positive, the function will start at (**a**,0) and then move (counter-clockwise) in order through quadrants I, II, III, & IV. When **a** is positive and **b** is negative, the function will start at (**a**,0) and then move in order (clockwise) through quadrants IV, III, II, & I. When **a** and **b** are negative, the function will start at (-a,0) and then move (counter-clockwise) in order through quadrants III, IV, I, & II. When **a** is negative and **b** is positive, the function will start at (-a,0) and then move in order (clockwise) through quadrants II, I, IV, & III.

However, it is important to note that due to the sinuously values for cosine and sine, the equations **x = a cos(t)** and **y = b sin(t)** for 0≤ **t **≤2π will go through exactly the same points for any given values of **a** and **b**, irrespective of the sign of **a** or **b**.

E.g. See the graph below has the overlapping values of the parametric equations **x = a cos(t)** and **y = b sin(t)** for 0≤ **t **≤2π when:

**a**= 4 and**b**= 2,**a**= 4 and**b**=-2,**a**= -4 and**b**= -2, and**a**= -4 and**b**= 2

__Observation__

- When
**a**=**b**and are nonzero, the graph produces a perfect circle with radius =**a**.

- When
**a**≠**b**and are nonzero the graph produces an ellipse with height 2**b**and width 2**a**.

- When
**a**=0 and**b**is nonzero, the graph produces a line on the y-axis between the points (0,–**b**) and (0,**b**) and when**b**=0 and a is nonzero, the graph produces a line on the x-axis between the points (–**a**, 0) and (**a**,0)

- As already mentioned, the graphs are identical for a given
**a**and**b**irrespective of the sign of**a**or**b**.

__Relating x and y__

From trigonometric properties, we know that

cos²t + sin²t = 1

So,

Now, since

x =acos(t) and y =bsin(t)for 0≤t≤2π

then for -1≤ x ≤1 and -1≤ y ≤1

This was displayed in the graphs, this is an equation for an ellipse around the origin (x=0,y=0).

The vertices are (-a,0) and (a,0) along the x-axis and (0,-b) and (0,b) along the y-axis.