Drawing a Curve

How do you turn a parameterization into a curve so that you can start drawing neat pictures too? A parameterization is simply $\vec{r}(t) = x(t) \hat{i} + y(t) \hat{j}, a \le t \le b$ in two dimensions, so if we can solve x = x(t) for t = g(x) we can plug this into y = y(t) = y(g(x)) to get y in terms of x and then graph the result from the point (x(a), y(a)) to (x(b), y(b)).

Example. If $\vec{r}(t) = (1 - t)\hat{i} + \cos 2\pi t \hat{j}, 0 \le t \le 1$ then $x = 1 - t \Rightarrow t = 1 - x$ so $y = \cos 2 \pi t = \cos 2 \pi (1 - x)$. Thus, this curve is simply a shifted cosine curve, with period 1 that passes from (x(0), y(0)) = (1,1) to (x(1), y(1)) = (0,1).

Another method is to simply plot the points that the curve passes through and then connect them based on common sense.

Example. Suppose the following graphs give x =x(t) and y = y(t). From this, we can create the following table of points along the curve:

t x y

0 1 1

1 0 1

2 1 -1

3 0 -1

4 1 1

From this table we can construct the graph of the curve, using the fact that x and y vary smoothly from point to point, but discontinuously at integer values of t.

At other times, you may simply know what the curve should look like from a little experience and advanced knowledge.

Example. Graph the curve parameterized by $\vec{r}(t) = 3 \cos t \hat{i} + 3 \sin t \hat{j}, 0 \le t \le 2\pi$. Notice that $\vert\vert \vec{r}(t)\vert\vert = 3$ for all values of t. Thus, all points on this curve are the same distance, namely three units, from the origin. Also note that $\vec{v}(t) = 3(-\sin t \hat{i} + \cos t \hat{j})$ which is never $\vec{0}$so the particle never stops moving. In fact, $\vert\vert \vec{v}(t)\vert\vert = 3$.Thus, the particle moves at constant speed. The acceleration vector is $\vec{a} = d^2 \vec{r}/dt^2 = -3 \vec{r}(t)$ so the acceleration is exactly opposite to the position vector. Add to this the fact that $\vec{a}(t) \cdot \vec{v}(t) = 0$ and that $x^2 + y^2 = (3 \cos t)^2 + (3 \sin t)^2 = 9$ and we have that the path is a circle, centered at the origin with radius 3. Since (x(0), y(0)) = (3,0) and y is initially increasing, the particle is moving counterclockwise.