Parametric Curves

by Venessa Brown

I chose to exlore,

For various a and b, investigate   x = a cos(t) ; y = b sin(t)   for  0≤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:

• a and b are 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 a and sine starting at 0 when t= 0 and both ending at their starting point of a and 0 respectively when t=2π. For the cosine function, the crest (highest point) is a and the trough (lowest point) is -a, while for the sine function, the crest (highest point) is b and 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 2b and width 2a.

• 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 = a cos(t) and y = b sin(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.

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