Parametric Equations

MAT 229, Spring 2021

Week 13

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 10.1: Parametric equations

Boelkins/Austin/Schlicker’s Active Multivariable Calculus
	Section 9.6: Vector-valued functions

Curves defined by parametric equations

Example

The location for Tara the Tyrannosaurus Rex at time t hours is given by
				(x,y)=(sin(t)-sin(2t),cos(t))
Time t is the parameter, and once it is known, the dinosaur’s position (x,y) is known.

Questions

What is Tara’s location at time t=0?

tara[t_]:={Sin[t]-Sin[2t],Cos[t]}
tara[0]

What is Tara’s location at time t=π/2?

tara[Pi/2]

When does Tara return to the location she was at when t was 0?

ParametricPlot[tara[t],{t,0, 2Pi}]

(Video)

Definition

Parametric equations are functional values for x and y coordinates
	x=f(t)
y=g(t)
usually presented as an ordered pair of coordinates in the plane: 
	(x,y) = (f(t),g(t))

Think of this as a curve (sometimes called a "space curve") in the plane, which is a function of t (generally thought of as time). This is an excellent way to represent a motion. And we can generalize this to (x,y,z) -- a curve in three-dimensional space. 

Like if an apple falls on your head, or something like that -- the motion of the apple can be captured in a space curve, given by parametric equations for the motion of the apple.

Questions

Consider the parametric equations x=cos(t),y=sin(t).

Make a table of values for these parametric equations for t=0,π/6,π/4,π/3,π/2,2π/3,3π/4,5π/6,π. Then connect these points.

x[t_]:=Cos[t]
y[t_]:=Sin[t]
TableForm[Table[{t,x[t],y[t]},{t,0,Pi,Pi/6}],
TableHeadings->{{},{"time(t)","x(t)","y(t)"}}]
positions=Table[{x[t],y[t]},{t,0,Pi,Pi/6}];
ListLinePlot[positions]

Using the Pythagorean identity, find an equation on x and y these parametric equations satisfy.

Simplify[x[t]^2+y[t]^2]

What curve do these parametric equations represent?

(Video)

Definition

A parametric curve is the set of all points represented by parametric equations x=f(t),y=g(t) for values of t over the given domain. 

Graphing calculators and software like Mathematica can draw curves represented by given parametric equations.

Example

To draw the curve in Mathematica given by the parametric equations
	x=sin(t)+(1/2)cos(5t)+(1/4)sin(13t)
y=cos(t)+(1/2)sin(5t)+(1/4)cos(13t)
for 0≤t≤2π enter

ParametricPlot[
{Sin[t]+1/2Cos[5t]+1/4Sin[13t],Cos[t]+1/2Sin[5t]+1/4Cos[13t]},
{t,0,2 Pi}
]

It’s kind of nice to watch the curve get traced out, which we can do with a “Manipulate” command in Mathematica.

x[t_]:=Sin[t]+1/2Cos[5t]+1/4Sin[13t]
y[t_]:=Cos[t]+1/2Sin[5t]+1/4Cos[13t]
Manipulate[
Module[{},
p1=ListPlot[{{x[end],y[end]}},PlotMarkers->{}];
(* I'm sorry that I don't have a dinosaur marker! :*)
p2=ParametricPlot[
{x[t],y[t]},
{t,0,2Pi},PlotRange->{{-1.5,1.5},{-1.5,1.5}}
];
Show[p2,p1]
],
{end,0,2Pi,.001}
]

Example: What’s the equation of Infinity?

x[t_]:=Cos[t]
y[t_]:=2Sin[t]Cos[t]
Manipulate[
Module[{},
p1=ListPlot[{{x[end],y[end]}},PlotMarkers->{}];p2=ParametricPlot[
{x[t],y[t]},
{t,0,2Pi},PlotRange->{{-1.5,1.5},{-1.5,1.5}}
];
Show[p2,p1]
],
{end,0,2Pi,.001}
]

Identifying curves defined by parametric equations

One may be able to identify a curve (like that infinity curve above) defined by parametric equations, by finding an equation in only x and y. Alternatively, you may want to create a motion that follows a given curve, given by x and y (e.g. you may want a robot in a bakery to trace a given curve, say laying down icing on a cake to wish someone a “Happy Birthday”).

Techniques

For the parametric equations x=f(t),y=g(t), two techniques for finding an equation on x and y:

Find an identity between f(t) and g(t).

Solve x=f(t) -- t=(f^(-1))(x) -- and plug this value of t into y=g(t)=g((f^(-1))(x)); or vice versa.

Questions

For parametric equations x=2cos(t),y=2sin(t), use the Pythagorean identity, (cos^2)(t)+(sin^2)(t)=1. What do you get?	(Video)

For parametric equations x=2t,y=1+(t^2), solve x=2t for t. Then relate x to y by replacing t in the equation for y.  What do you get?	(Video)

Questions

Identify the curves defined by the parametric equations.

x=2t+1,y=3t-4.		(Video)

x[t_]:=2t+1
y[t_]:=3t-4
ParametricPlot[{x[t],y[t]},{t,0,4}]

x=cos(t)-5,y=sin(t)+6.	(Video)

x[t_]:=Cos[t]-5
y[t_]:=Sin[t]+6
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]

x=2cos(t),y=3sin(t).		(Video)

x[t_]:=2Cos[t]
y[t_]:=3Sin[t]
ParametricPlot[{x[t],y[t]},{t,0,2Pi}]

Homework

IMath problems on Parametric Equations.