6

I am following the last module of Differential Calculus on Khan Academy, that deals with Parameteric equations.

Here are the parametric equations described in the lecture.

$x(t) = 5t + 10$

$y(t) = 50 - 5t^2/2$

However, I really don't understand what parametric equations really mean. How do they differ from normal equations. According to Wikipedia: "In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters." I really don't understand what this definition is trying to convey.

From what I observed, if two functions share a variable, it typically gets defined as a parametric equation. But that seems to be a loose definition.

Regarding my prerequisite knowledge, I have a Masters in Engineering. Therefore I understand the formulae of calculus quite well. I just never bothered to understand some of the underlying concepts. Therefore, I am revisiting it by through Khan Academy.

desert_ranger
  • 335
  • If you [edit] the question to show us a particular example of a parametric equation from that site we may be able to help. Don't expect us to check ourselves at Khan Academy. – Ethan Bolker Mar 27 '24 at 20:37
  • Thanks for the comment @EthanBolker. I have made my edits. – desert_ranger Mar 27 '24 at 20:39
  • I'm guessing that you have not yet studied multivariable calculus? – Lee Mosher Mar 27 '24 at 20:59
  • 1
    @LeeMosher I don't think you need multivariable calculus (i.e. partial derivatives) to study curves defined parametrically. – Ethan Bolker Mar 27 '24 at 21:03
  • @LeeMosher I have updated my question with my educational background. – desert_ranger Mar 27 '24 at 21:03
  • @EthanBolker: Yes, of course. I nonetheless wanted to get an idea of whether the OP had been exposed to an ordinary sequence of mathematical courses, in particular to a multivariable calculus course, which is sometimes the first place that people encounter this material. – Lee Mosher Mar 27 '24 at 21:07
  • Curious about your location, as this is standard high school material in the handful of countries I'm familiar with. – Servaes Mar 28 '24 at 11:20

3 Answers3

7

As the parameter $t$ varies the two equations tell you the position of a point $(x(t),y(t))$ as it wanders along a curve in the plane. (Try drawing a picture of that curve and marking each point with the corresponding value of $t$. The parameter in a system like this is often named "$t$" to suggest time.)

Then you can use some calculus to find the tangents to that curve and the speed with which the point traverses it. That's probably what the Khan Academy lesson is about.

The graph of a function can be thought of as a parametric curve for which the parameter is the value on the $x$-axis. Then the graph is the curve: the set of points $(x,f(x))$.

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
  • Thank you. This does clear up some of my confusion. I have a follow up question -

    Why is $t$ traversing a curve on the x-y plane? Perhaps it could be better represented in a 3-d coordinate, x-y-t plane?

     – desert_ranger Mar 27 '24 at 21:05
  • 2
    What you suggest is possible, but not at all "better" in most applications. For example, think about how the parametric equations $(\cos t, \sin t)$ describe uniform circular motion. What would you make of $(\cos t^2, \sin t^2)$? – Ethan Bolker Mar 27 '24 at 21:09
  • 2
    PS Convince yourself that $(\cos t, \sin t, t)$ is a helix. – Ethan Bolker Mar 27 '24 at 21:38
  • 1
    The other problem with the x-y-t coordinate system is that it's much harder to draw things out. – Teepeemm Mar 28 '24 at 15:29
  • @desert_ranger $t$ is not traversing a curve. $t$ is the parameter, the independent variable. $t$ is to curves (and line is a curve too) as $x$ is to functions or $n$ to series in first lectures on calculus. We are so used to $x$ and $y$ to be used as horizontal and vertical axes that we tend to miss the point. Sometimes it is meant that the curve is described by one equation like $y=f(x)$ but we miss one curve can be described by a set of equations like $x=f_1(t)$ and $y=f_2(t)$. – Crowley Mar 28 '24 at 20:30
  • @desert_ranger Also note that you cannot describe whole horizontal parabola by $y=f(x)$, because it is not a function (of $x$)! For every $x$, except for $x=0$ you need to define two values of $y$, which is a conflict with a definition of a function. On the other hand, parametrically you can do that easily. $y=t$, $x=t^2$ do the trick. – Crowley Mar 28 '24 at 20:36
3

Sometime it may be interesting to use physics in math.

You can treat the $t$ as time (which we call it parameter, is a variable). Then $x(t)$ and $y(t)$ is telling you the $x$- and $y$-coordinates of a moving object at time $t$. For example, in no air resistance case, a projectile motion with inclination angle $45^\circ$ can be described as $$x(t)=t,y(t)=t-\dfrac{1}{2}gt^2$$ ($g$ is some constant). This means at time $t=1$, the object is at $\left(1,1-\dfrac{g}{2}\right)$, as $t$ varies, you can imagine there is a parabola curve (which is physics is called trajectory) which is exactly the object passes.

Now in mathematics case, usually when we face some function curves, it may be tough to describe its changes in $x,y$ or area, blablabla. But with the above example, we can express the curve, in the sense by introducing a 'time'-like variable, which we call parameter, so that we can describe the $x$, $y$ coordinate in terms of this parameter. (Of course higher dimensional can be generalised) This allows us to like computing the arc length of the curve, surface area of a solid, with elegant formula.

Angae MT
  • 1,035
  • 1
  • 8
2

Late to the party, but for posterity, and deliberately (i) steering clear of technicalities and (ii) covering basics selectively:

Many objects of interest, such as curves (line segments, circles, parabolas, splines, ...), surfaces (polygons, spheres, tori, ...), and regions (solid rectangles and boxes, disks, balls, ...), are sets of points.

One way to describe an object is implicitly: By giving a "defining" equation or inequality, such as (in the plane) $y = mx + b$ for a non-vertical line, or $x^{2} + y^{2} = 1$ for the unit circle, or (in space) $$ (x - x_{0})^{2} + (y - y_{0})^{2} + (z - z_{0})^{2} \leq r^{2} $$ for the closed ball of radius $r$ and center $(x_{0}, y_{0}, z_{0})$.

Another way to describe an object is parametrically: By giving mapping whose domain is an open set in a Cartesian space (or the closure of an open set) and whose image is all or part of the object. The name is self-descriptive: We are describing points of our object in terms of parameters, variables that range independently over some region.

The prototypical example is arguably Cartesian coordinates, which parameterize a line in terms of a single real number, or a plane in terms of ordered pairs of real numbers, and so forth.

Among infinitely many other possibilities the unit circle can be described parametrically using

  • Trig functions, $x = \cos t$ and $y = \sin t$, with $t$ real. Every point on the circle has the form $(\cos t, \sin t)$ for some real $t$, and distinct values of $t$ specify the same point if and only if they differ by an integer multiple of $2\pi$.
  • Rational functions, $x = 2r/(r^{2} + 1)$ and $y = (r^{2} - 1)/(r^{2} + 1)$ for some real $r$. These formulas define a one-one correspondence between the real line and the circle with $(0, 1)$ removed.

For another example: The parametric curve $x = 5t + 10$ and $y = 50 - t^{2}/2$ can be written implicitly by eliminating the parameter, here solving the first equation for $t$ in terms of $x$, then substituting in the second equation. The end result is $y = 40 + 2x - \frac{1}{10}x^{2}$. This implicit equation defines precisely the same parabola as the parametric formulas, but contains strictly less information, namely the details on how the parabola is traced as $t$ varies. (Generally, eliminating the parameter is More Difficult than this; even when possible, eliminating the parameter may not define exactly the same set of points, or the parametrization may multiply-cover the object. The circle examples illustrate both possibilities.)

Conceptually, parameterization sets up an address (an ordered tuple of real numbers) for points (geometric locations).

Practically, a parameterization often reduces the study of an object to the study of intervals (parametric curves), regions in the plane (parametric surfaces), or regions in space (cylindrical and spherical coordinates, for example). In multivariable calculus, we almost always compute integrals over curves and surfaces by parameterizing the domain, which reduces us to computing an ordinary one-variable or two-variable integral.

Andrew D. Hwang
  • 78,195