Consider a parametric curve given by equations
x = x(t),
y = y(t) for t in some interval
I.
Assume that the functions
Algorithm:
Step 1. Find out what happens at the "ends" of I. If an
endpoint of I is included in I, then substitute this time into
x and y to learn where the path starts or ends. If some
endpoint is not included, find the appropriate one-sided limit of x
and y with respect to t and then interpret this information.
Case "both limits proper": the curve goes toward the point given by the
limits.
Case "both limits improper": the curve goes toward corners depending on
the sign of the infinities in the limits, for instance if
Case
Case "limit of x improper,
Step 2. Find intercepts. The
Step 3. Find
Mark data from Step 1,2, and 3 in the plane. It is recommended to make a note
at each point to indicate at what time the path gets there.
The critical times split I into subintervals. For each subinterval
determine signs of
and ,
then determine signs of the spatial derivative
This is best done using a table. For each subinterval of I that we
obtain, the times from this subinterval give a certain part of the curve and
the increase or decrease of this part as a graph is given by the relevant sign.
It is now possible to connect points in the picture by temporary lines that
fit these trends.
Step 4. Find the second derivatives with respect to time of x
and y and calculate the second spatial derivative
Find times from the interior of I when the numerator or denominator
of this derivative is zero. Substitute these times into x and y
to find points where the curve might change spatial concavity. Mark these
points in the graph.
These times split I into subintervals. For each subinterval
determine signs of the second spatial derivative y′′,
this is best done using a table. For each subinterval of I that we
obtain, the times from this subinterval give a certain part of the curve and
the concavity of this part as a graph is given by the relevant sign of
y′′. We correct the temporary lines to reflect this data.
These are the main steps. If the curve has an end at some endpoint of
I (that is, when this endpoint was included in I or we get
both limits proper in Step 1), it is also a good idea to determine the
appropriate one-sided spatial derivative at that point, it tells us in which
direction the curve should start off. For instance, if it is the left
endpoint
For more details and another example see Sketching parametric functions in Theory - Implicit and parametric functions.
Example: Investigate the parametric curve
x = t⋅et,
y = t3 + 6t,
Solution: The interval
Thus the curve eventually disappears in the upper right corner. The limit of
Intercepts: The
The
We find
It remains to find
This derivative does not exist when t = −1 and we ignore
this point as before, and it is zero when t is the cubic root of −4,
which is less than −1 and hence not inside I, we also ignore it. We
see that for
The last piece of data is the one-sided derivative at the beginning of the curve:
The curve should therefore start vertically up. Since we do not have any
points beyond
For another example see Implicit and parametric functions in Solved Problems.