Section 10.1: Parametric Equations

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?

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

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

Definition

Parametric equations are functional values for x and y coordinates
	x=f(t)
y=g(t)

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]
ListPlot[Table[{x[t],y[t]},{t,{0,Pi/6,Pi/4,Pi/3,Pi/2,2Pi/3,3Pi/4,5Pi/6,Pi}}],
PlotRange->{{-1,1},{-1,1}},AspectRatio->1]

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

What curve do these parametric equations represent?

Definition

A curve is defined by parametric equations, if all the points on the curve can be represented by parametric equations x=f(t),y=g(t) for different values of t. 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->{}];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 defined by parametric equations by finding an equation in only x and y.

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) for t and plug this value of t into y=g(t), 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?

For parametric equations x=2t,y=1+(t^2), solve x=2t for t. What do you get?

Questions

Identify the curves defined by the parametric equations.

x=2t+1,y=3t-4

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

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

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

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

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