How do you turn a parameterization into a curve so that you can start
drawing neat pictures too? A parameterization is simply 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 then
so
. 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 . Notice that
for all values of t. Thus, all points on this curve are
the same distance, namely three units, from the origin. Also note that
which is never
so the particle never stops moving. In fact,
.Thus, the particle moves at constant speed. The
acceleration vector is
so the
acceleration is exactly opposite to the position vector. Add to this the
fact that
and that
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.