Ways of Representing Curves in the Plane
(with an Emphasis on Parametrized Curves)

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Overview: Ways of Representing Curves in the Plane

Our goal here is to explore the parametric approach to representing a curve in the plane. In order to appreciate the various aspects of the parametric approach, it's helpful to contrast it with the familiar approaches to representing curves studied previously: namely, the explicit and implicit approaches.

The page includes two Java applets and a link to a third. A "Java applet" is a small interactive program that runs within a web browser (e.g. Internet Explorer or Firefox). The Java applets included herein allow you to enter your own examples (i.e. functions or equations) and the see the curves drawn on your computer screen. An applet is provided for graphing curves expressed in each of the three forms discussed: explicit, implicit, and parametric.

A highlight of this web page is the inclusion of a special type animation for parametrized curves. A description of how to interpret and understand this type of animation is provided in Example 1 of the section entitled "Parametric Form".

At the bottom of this web page, links are provided to three other web pages with interesting information, graphics, and animations for three interesting types of curves: cycloids, hypercycloids, and epicycloids.

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Explicit Form

Ask a first semester calculus student: "How would you represent a curve in the x y-plane in an analytic/symbolic fashion?" They would likely respond: "As the graph of function of the form  y = f(x)". This is a natural response, since we work so frequently with equations of this form, due to their utility in the study of calculus. Indeed, many curves can be described by an equation of the form y = f(x), where fis a real-valued function of a real variable x. The function f(x)can be any arbitrary (typically continuous) function, for instance: a polynomial, a trig function, an algebraic function, etcetera. Some possible examples are shown below.

[Graphics:HTMLFiles/index_7.gif]

A curve expressed in the form y = f(x)is said to be given in explicit form, since the y-coordinate of each point on the curve is given explicitly as a function of the x-coordinate of the point. That is, the curve consists of the set of all points in the plane corresponding to the ordered pairs (x, f(x)) ∈ ÷µ^2, as xvaries over the domain of the function (or perhaps a specified subset of the domain).

A Java applet for graphing explicitly defined curves appears directly below this paragraph. The applet is configured so that the four graphs shown above are "preloaded" as examples. Initially, the polynomial function appears graphed in the viewing window. To view the other preloaded examples, you can click on the pop-up menu labelled "A Fifth Degree Polynomial", and you will see a list of three other functions. Select any of these and the click the "Load Example" button at the top left of the applet window. Once a graph is visible, you can drag the slider beneath the graph with your mouse and you will see a cross-hair trace out points on the graph. As the slider is moved, both the x-coordinate and the corresponding y = f(x)value are displayed below the graph.

If you would like to experiment with graphing curves of your own choice (given in explicit form), you can do so using the above applet. To enter your choice of function, use your mouse to select whatever appears in the input field to the right of the label "f(x)=", at the bottom left of the screen. Then type in the formula for your chosen function. To see the curve y = f(x) graphed in the main viewing screen, press the return key or click the "New Function" button.

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Implicit Form

The explicit method of representing a curve is often convenient, but it also has its drawbacks. For one, only curves which pass the vertical line test can be obtained in this way. Moreover, there are many curves of interest which it may be difficult, or even impossible to represent in explicit form. There are other ways to represent curves in the plane. We will discuss two more ways below. The first, with which you should be familiar from Calculus I, expresses the relationship between the xand ycoordinates of points on the curve implicitly by specifying a equation in xand ywhich is not necessarily solved for yin terms of x.

Recall that any equation in the variables xand ycan be written in the form g(x, y) = 0, where g(x, y)is a formula involving xand y. To any such equation there corresponds a set of points in the x y-plane, namely the set of all solutions of the equation, i.e. S = {(x, y) | g(x, y) = 0}. This set is typically a "curve", and a curve expressed in this way is said to be in implicit form. Such a curve need not pass the vertical line test, and it may have self-intersections. Moreover, it may consist of more than one connected piece--or even of none (e.g. if the equation has no real number solutions, as in x^2 + y^2 + 1 = 0). Below are some examples, where in each case we've taken g(x, y)to be a polynomial in xand y.

6 graphs of functions in explicit form]

If we consider examples where g(x, y)involves transcendental functions we can get yet more interesting "curves". Here are two examples.

Two graphs of implicitly defined functions

The class of curves which can be represented in implicit form includes all curves that can be represented in explicit form, and is much larger. (Any curve represented in explicit form as y = f(x)can also be represented in the implicit form g(x, y) = 0by setting g(x, y) ≡ y - f(x).) Calculus can be used to study various aspects of curves given in implicit form—this is explored a bit in first semester calculus when the topic of implicit differentiation is discussed. However, for many purposes, curves represented in implicit form are quite cumbersome and are typically difficult to work with. There is yet another way to represent curves analytically, the Parametric Method, which is generally more tractable, and thus more useful. (The Parametric method is discussed in the next section.)

Here is a link to a Java applet that will draw the graphs of curves given in implicit form.  You can draw up to four implicitly defined graphs, each shown in a different color. Enter the equations at the bottom of the applet window. For best results, check both of the boxes labeled "X Scan" and "Y Scan". [Note:  This applet behaved well for the equations I tested it with. However, be aware that many computational methods for rendering graphs of curves in implicit form are prone to computational errors from round-off and other problems—especially for certain functions. Always take a computer-generated graph as an approximation which may be anywhere from slightly to grossly inaccurate in portraying certain parts (or perhaps all) of a graph!]

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Parametric Form

The parametric form is a very flexible approach for representing a curve in the plane. Here we think of the points (x, y)on the curve as being swept out as a "parameter" tvaries over an interval [a, b]. That is, to specify a curve, we simply provide a pair of real-valued functions  { x = x (t) y = y (t)  for t ∈ [a, b]. We often tend to think of the parameter tas "time"--for each value of t, we have a point in the plane, namely:  (x(t), y(t)) ∈ ÷µ^2. The curve consists of all points generated in this way as tvaries throughout the "parameter interval" [a, b]. There is an interesting aspect of this approach to specifying a curve. Namely, the same curve (i.e. set of points in the plane) can be represented by many (in fact, infinitely many) different parametrizations. These parametrizations differ in the manner in which the points on the given curve are "swept out" by the different parametrizations, say x = x _ 1(t)and y = y _ 1(t)with a _ 1 <= t <= b _ 1and x = x _ 2(t)and y = y _ 2(t)with a _ 2 <= t <= b _ 2. This is illustrated by Examples 1 and 2 below.

Example 1

Consider the case where { x = cos ( t) y = sin ( t)for t ∈ [0, 2 π]. Here is a table of values for x(t)and y(t)with t = (k π)/6, for 1 <= k <= 12

t 0 π/6 π/3 π/2 (2 π)/3 (5 π)/6 π (7 π)/6 (4 π)/3 (3 π)/2 (5 π)/3 (11 π)/6 2 π
x(t) 1 3^(1/2)/2 1/2 0 -1/2 -3^(1/2)/2 -1 -3^(1/2)/2 -1/2 0 1/2 3^(1/2)/2 1
y(t) 0 1/2 3^(1/2)/2 1 3^(1/2)/2 1/2 0 -1/2 -3^(1/2)/2 -1 -3^(1/2)/2 -1/2 0

If we plot these 12 points in the x y-plane we get the following picture:

12 points in the preceding table plotted in xy plane

It appears that these points all lie on the unit circle (which shouldn't be surprising if you consider the definitions of sine and cosine in terms of the unit circle!). In this relatively simple (and standard) example we can use some algebra to show that for every parameter value t ∈ [0, 2 π], the point with coordinates { x = cos ( t) y = sin ( t)must lie on the unit circle. This follows from the simple observation that x^2 + y^2 = cos^2(t) + sin^2(t) = 1. In other words, we have x^2 + y^2 = 1, which is familiar to us as the equation of the circle centered at the origin with radius r = 1. Thus, we see that no matter which value ttakes on, the point (x(t), y(t))must lie on this circle. The process of starting with the parametric equations { x = x (t) y = y (t)and finding a familiar equation involving only xand yis called eliminating the parameter.

To understand a parametrized curve in the xy-plane, it is often helpful to examine the graphs of the coordinate functions x = x(t)in the t x-plane, and y = y(t)in the t y-plane. In this example, they look like this:

Graphs of x and y as a function of t

If we use these graphs, in conjunction with our observation that (x(t), y(t))must lie on the unit circle we can see that:

From all this, it follows that the parametrization { x = cos ( t) y = sin ( t)  for t ∈ [0, 2 π]sweeps out the unit circle once counterclockwise as tvaries through the parameter interval [0, 2 π].

The animation below illustrates some of the relationships between the parameter interval t, the graphs of the coordinate functions  { x = cos ( t) y = sin ( t) , and the final graph of the parametrized curve in the xy-plane. To understand what is happening in this animation read the comments appearing below.

An animated GIF showing relationship of coordinate functions and curve in xy-plane

Example 2

Consider the case where { 3 x = cos ( π t ) 3 y = sin ( π t )for  0 <= t <= 2. Again, we can eliminate the parameter tby noting that x(t)^2 + y(t)^2 = cos^2(π t^3) + sin^2(π t^3) = 1     ==> x^2 + y^2 = 1. Thus we know that no matter what value ttakes on, the point (x(t), y(t))must lie on the unit circle.  But how does this parametrization of the unit circle differ from that of the last example? Considering the graphs of the coordinate functions may help understand the situation.

A pair of graphs:\n those of x=x(t) and y=y(t)

From the above coordinate graphs we can surmise that:

Thus, for this parametrization, the circle is swept out counterclockwise exactly four as tvaries from 0to 2--and it is swept out faster and faster as tincreases from 0to 2. Here's an animation of this parametrization.

An animated GIF showing relationship of coordinate functions and curve in xy-plane

Example 3

Consider the parametrized curve { x = cos ( t)    y = sin ( 2 t)for  0 <= t <= 2 π.  In this example, we cannot eliminate the parameter tto obtain a familiar equation for a curve. This is because the curve resulting from this parametrization is not familiar to us from our previous study of analytic geometry in rectangular coordinates. However, we can still obtain an animation diagram like those of the previous example to illustrate both the curve (as a set of points in the x y-plane) and its parametrization (i.e. the manner in which the curve is "swept out" as tmoves through the parameter interval [0, 2 π].

An animated GIF showing relationship of coordinate functions and curve in xy-plane

Example 4

Consider the parametrized curve { x = cos ( 3 t) y = sin ( t)     for  0 <= t <= 2 π. In this example, we cannot eliminate the parameter tto obtain a familiar equation for a curve. This is because the curve resulting from this parametrization is not familiar to us from our previous study of analytic geometry in rectangular coordinates. However, we can still obtain an animation diagram like those of the previous example to illustrate both the curve (as a set of points in the x y-plane) and its parametrization (i.e. the manner in which the curve is "swept out" as tmoves through the parameter interval [0, 2 π].

An animated GIF showing relationship of coordinate functions and curve in xy-plane

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A Java Applet for Drawing Parametrized Curves

The applet below will draw the graph of a parametrized curve { x = x(t) y = y ( t)for t ∈ [a, b]. Enter the formulas for the coordinate functions x(t)and y(t)in the fields at the bottom of the applet window, then press return on your keyboard. Be sure to use the letter tfor the parameter (not x!). After the graph is rendered on the computer screen, you can press the "Trace Curve!" button to see a cursor "sweep out" the curve (with the parameter tincreasing from t = ato t = bat a constant rate).

Here are some tips for using the above applet:

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Exploring Other Interesting Parametrized Curves: Some Links

The "cycloid" is a parametrized curve that arises by following the motion of a fixed point on a circle as that circle rolls (without slipping) along a flat line. Furthur information on the "cycloid" (and a nice animation of the rolling circle generating the cycloid) can be found on the cycloid page of the MathWorld website.

"Hypercycloids" and "Epicycloids" are variations on the cycloid idea. They arise by following the motion of a fixed point P on a circle as that circle rolls without slipping inside ("hyper") or outside ("epi") another circle. More information (including nice animations) is available on the hypercycloid and epicycloid webpages of the MathWorld website.

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Copyright © 2004 Aaron Schusteff
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