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Points, Vectors, and Functions

Points, Vectors, and Functions

Parametrizations of the Unit Circle

One of the first shapes we learned as a toddlers was the circle. It probably didn't take long to realize that pizza is shaped this way. Now we'll discuss the unit circle, the circle with radius equal to 1, in terms of pizza.

When we're young, we only eat cheese pizza. Plain, boring, but still delicious, the cheese pizza satisfied every inner desire. Even now, it still sometimes hits the spot.

Next, we learned to relate trigonometric functions to the circle. Sine and cosine became the pepperoni as we added our first pizza topping.

For example, if we erase the arrows that give direction and the labels that say which values of t go with which points, the following three parameterizations all produce the pepperoni unit circle:

  • The normal parameterization
    x(t) = cos t
    y(t) = sin t
    for 0 ≤ t ≤ 2π.
    This parameterization starts at the point (1, 0) when t = 0 and travels the unit circle once in a counterclockwise direction.
  • The parameterization

    This starts at the point (-1, 0) and travels the unit circle once in a counterclockwise direction.
  • The graph of the parametric equations

    for 0 ≤ t ≤ 8π.
    This starts at the point (1,0) and travels the unit circle twice in a counterclockwise direction.


If we erase the arrows and the labels that say what values of t belong to which points, we can't tell the difference between these two graphs. Both graphs would just be the unit circle.

If the graph doesn't have labels and arrows, we can't tell how quickly or how many times the circle is traced. We also don't know where the stylus started.

All we can see is the circle. Put another way, it's still a pizza.

Now that we're a little older, our tastes have refined, and we prefer other pizza toppings. We invite our friends over for a pizza dinner, but no three people can agree on a topping set. Fortunately, there are infinite choices of pizza toppings, and these are unit pizzas. They're small enough for everyone to have their own. Each different set of toppings changes the way a pizza tastes.

Similarly, there are infinitely many parameterizations of the unit circle. Different parameterizations may affect

  • the starting point on the circle,
  • the speed at which the circle is drawn,
  • how many times the circle is traced, and
  • whether the circle is drawn clockwise or counterclockwise.

A typical parameterization of the unit circle is
x(t) = cos t
y(t) = sin t

for 0 ≤ t ≤ 2π.

We can tweak this to find new parameterizations that meet certain criteria.

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