Parametric functions

When we have a curve in the plane, whether given by a function or an implicit equation, we so far treated it as a shape. However, sometimes there is more information included in the situation, namely when the curve is actually a record of a movement, the path traced by, say, a bug.

The shape itself is not enough then, since two bugs can crawl along the same path in very much different ways (speed and even direction can vary, a bug can even turn back for a while and retrace the same path). This additional information has to be somehow stored and the most natural way is something called the parametric curve.

You obtain such a curve rather easily. At every time t that you are interested in you record the position in plane, that is, the x and y coordinate. Such a parametric curve is then described by two functions depending on time, x = x(t) and y = y(t) for t from some interval I. Here we will always assume that the two functions are continuous on I, so that the path is just one uninterrupted curve.

Such a parametric description carries complete information including instantaneous velocity at every time. Given the general way in which a bug can move, the curve described by a parametric equation can be very complicated and there is little hope of describing it also as a graph of a function. However, we can attempt to describe at least parts of it using functions so that we can use the arsenal we already have at our disposal, above all our skills in graph sketching. But first we look at some examples.

Example: Consider the parametric curve
x = 1 − t,
y = 2t for t real.

What shape does this curve have? Here it is actually simple, since we can eliminate the parameter t. From the first equation we get t = 1 − x, putting it into the second one we obtain y = 2 − 2x for x real. Thus the path described by the given parametric equations is actually a straight line (all of it, for t real also x varies throughout real numbers).

Example: Consider the parametric curve
x = 2t3,
y = 2 − 4t3 for t real.

What shape does this curve have? Here it is not necessary to actually get t itself from the first equation, we can write y = 2 − 2⋅2t3 = 2 − 2x. This parametric curve is therefore the same as the previous one, the only difference is the parametrization. From physics point of view these two curves describe different movements,

but from the point of view of mathematics, when we investigate them as curves, they are the same.

Example: Consider the parametric curve
x = 2t2,
y = 2 − 4t2 for t real.

We again have our straight line, but notice now that although t again runs through all real numbers, we cannot get negative x. Thus this curve is just the part of the straight line y = 2 − 2x given by the condition x ≥ 0.

Example: Consider the parametric curve
x = tet,
y = t3 + 6t for t ≥ −1.

We do not know how it looks like, since we cannot express t from the first nor the second equation. To get some idea of the shape we would have to somehow use the methods from function sketching, but for than we need to know how to differentiate such curves. For another look at this example we refer to the section Parametric functions in Derivative - Methods Survey - Graphing.

When trying to treat parametric curves as functions we encounter the same trouble we had with implicit functions, namely that sometimes it is not possible since the curve has multiple values for the same x. We also adopt similar strategy. We are not attempting any total conversion, we settle for local description. Just like with implicit function, we have a theorem about being able to locally describe a "nice" parametric curve using a function.

Theorem.
Consider a parametric curve given by x = x(t), y = y(t) for t from some interval I. Let t0 be from the interior of I. If (t0) ≠ 0, then there exists a neighborhood U of x0 = x(t0) and a function y = y(x) on U such that y(t0) = y(x0) and there is a neighborhood J of t0 such that the set of points

{(xy(x)) for x from U}

is equal to the set of points

{(x(t), y(t)) for t from J}.

To see how this theorem is applied to the example above that we could not figure out, see Parametric functions in Derivative - Theory - Implicit and parametric functions. Again, this gives existence but does not help with actually finding the function. In practice we either use advanced graphing techniques or try to eliminate t as we did above. One such example is in Solved Problems.

Remarks:
1. The notation x = x(t), y = y(t) is very suggestive, it tells us that the coordinates of points depend on time, but it can be also quite misleading, especially when y also starts depending on x locally. In situations where some confusion seems possible people therefore prefer to write x = f (t), y = g(t), then everything is clear.

2. When transforming a parametric curve into a function, we usually try to do it piecewise. That is, we take the largest possible piece and try to express it using a function, then we take a piece right next to it and try to express it etc. The intervals that we use come naturally when we try to solve one of the equations for t (usually the first one), since for that we need injectivity. If J is a subinterval of I on which x(t) is 1-1, then we can find the inverse function t = x−1(x) and substitute into y, thus obtaining the desired function y(x) that describes exactly the part of the parametric curve that corresponds to times from J. By the way, this shows why calling both the variable and the function "x" might be awkward, it looks much better if we write (see the previous remark) t = f−1(x), then we get the crucial formula

y = gf−1(x)).

The strategy therefore calls for splitting I into intervals on which the function f (or x(t) if you want) is 1-1 and then describing the corresponding pieces of the curve using the above function. Since the easiest way to find these intervals is to use monotonicity and derivative, we leave this topic to Parametric functions in Derivative - Theory.

3. We talked about transforming parametric curves into functions. Does it work the other way? When we have a graph of a function f, can we express it as a parametric curve? The answer is definitely positive, it is called parametrization of a curve and it is not unique, a given curve can be expressed in infinitely many different ways. For instance, above we had two parametric curves describing the same straight line, and we can change the cube power in the second example to any other positive odd integer power and get the same line again. In general, a graph of f for x from an interval I can be naturally parametrized as
x = t,
y = f (t) for t from I.

Bonus: Some famous curves.

Circle with radius r and center (a,b) is given by the equations
x = a + r⋅cos(t),
y = b + r⋅sin(t) for t real.

Note that the bug just goes around and around counterclockwise infinitely many times, at a constant speed. Thus we get the same curve by changing the bug's speed, for instance we can use cos(kt) and sin(kt) for any non-zero k (for negative k the bug circles clockwise), or for instance cos(et) and sin(et). We can also try another modification. It is pointless for the bug to go over this curve more times, so we can keep the formulas as given, but take t only from, say, [0,2π].

Ellipse with semi-major axis A, semi-minor axis B, and center (a,b) is given by the equations
x = a + A⋅cos(t),
y = b + B⋅sin(t) for t real.

Note that the bug keeps going around counterclockwise, we can therefore restrict time as above.

Remark: Note that in the equation of the circle the parameter t actually tells us the angle with respect to the x-axis at which the bug is at any given time, the second info is the distance r from the center. We can play with this distance a bit, for instance we can make it depend on time as well.

Example: Consider this parametric curve:
x = t⋅cos(t),
y = t⋅sin(t) for t ≥ 0.

If we try to substitute some prominent times (like π/4 etc), we should get a pretty good idea of what is going on. The bug circles and circles, starting from the origin and increasing the distance from the origin at a constant rate.

We obtained the Archimedes spiral. If we put a different function for radius, we can make other interesting shapes.

Remark: Note that t need not represent time. From mathematical point view we simply have some parameter (we may even use a different letter than t for it), the interpretation makes no difference. Moreover, in applications such a parameter may have an important meaning different from time, so once you understand parametric curves, it might be better not to get too hang up on the idea of time (especially since there are also parametric surfaces with more parameters, there the time idea falls through). However, the "time" point of view is so natural and above all helpful, that we decided to base our exposition of parametric curves on it. Of course, even if the parameter means something else, all methods covered here still work. And if you really run into trouble, you can at least pretend that the parameter is time and perhaps it helps (just don't tell anybody).