Parametric Functions and Motion

How Mathematica can help us understand parametric functions.

To do this, we need to see some Mathematica commands (you will learn Mathematica in your calculus class). You will not need to know any Mathematica in this class.

Built-in Command: ParametricPlot

x[t_]=3t+(1-t)(-4);
y[t_]=6t+(1-t)(-2);
curve1=ParametricPlot[{x[t],y[t]},{t,0,1},PlotStyle->{Thick},AxesOrigin->{0,0},PlotRange->{{-6,6},{-6,6}},AspectRatio->1]

Parametric Plots: determining direction curve is traversed via table of values

x[t_]=3t+(1-t)(-4);
y[t_]=6t+(1-t)(-2);
datapts=Table[{x[t],y[t]},{t,0,1,0.1}];
TableForm[datapts]
ListPlot[datapts,PlotStyle->PointSize[0.02],AxesOrigin->{0,0},PlotRange->{{-6,6},{-6,6}},AspectRatio->1]

Parametric Plots: animation of particle motion

In this parameterization, notice the particle moves with constant speed around the curve

x[t_]=3t+(1-t)(-4);
y[t_]=6t+(1-t)(-2);
curve1=ParametricPlot[{x[t],y[t]},{t,0,1},PlotStyle->{Thick},AxesOrigin->{0,0}];

point[x_,y_]:=Graphics[{Red,Disk[{x,y},{0.15,0.15}]}]
Manipulate[Show[curve1,point[x[t],y[t]]],{t,0,1,0.02}]

Parametric Plots: animation of particle motion: same curve, but different speed

Notice this is the same set of (x,y) points as above, but the particle traverses the curve at varying speed (slower at the start and faster at the end).

x[t_]=3(t^3)+(1-(t^3))(-4);
y[t_]=6(t^3)+(1-(t^3))(-2);
curve1=ParametricPlot[{x[t],y[t]},{t,0,1},PlotStyle->{Thick},AxesOrigin->{0,0}];

point[x_,y_]:=Graphics[{Red,Disk[{x,y},{0.15,0.15}]}]
Manipulate[Show[curve1,point[x[t],y[t]]],{t,0,1,0.02}]

FYI: Parametric Functions: Circles and Ellipses

Circle

A circle (((x-h))^2)+(((y-k))^2)=(r^2) is given parametrically as:

x=h+r cos(c t)
y=k-r sin(c t), 0≤t≤ 2π/c      where c is a constant which determines how a particle would move around the circle.

Typically you choose c=1. You can show this is a circle by substituting the parametric expressions for x and y into  (((x-h))^2)+(((y-k))^2)=(r^2) and showing it is identically true (true for all x and y).

h=3;
k=-1;
r=2;
c=5
x[t_]=h+r Cos[c t]
y[t_]=k+r Sin[c t]

plt[end_]:=ParametricPlot[{x[t],y[t]},{t,0,end},PlotRange->{{h-r,h+r},{k-r,k+r}},PlotStyle->{Thick,Orange}]

num=50;
step =( 2Pi/c)/num;
Manipulate[plt[t],{t,step,2Pi/c,step}]

Vary h,k,r in the above ans see that you always get a circle.

Ellipse

An ellipse is a generalization of the circle to ((((x-h))^2)/(a^2))+((((y-k))^2)/(b^2))=1 is given parametrically as:

x=h+a cos(c t)
y=k-b sin(c t), 0≤t≤ 2π/c      where c is a constant which determines how a particle would move around the circle.

Typically you choose c=1.

h=3;
k=-1;
a=2;
b=1
c=5
x[t_]=h+a Cos[c t]
y[t_]=k+b Sin[c t]

plt[end_]:=ParametricPlot[{x[t],y[t]},{t,0,end},PlotRange->{{h-a,h+a},{k-b,k+b}},PlotStyle->{Thick,Orange}]

num=50;
step =( 2Pi/c)/num;
Manipulate[plt[t],{t,step,2Pi/c,step}]

There are other parametric representations of circles and ellipses, but these are the most common.

FYI: Intrinsic Command: Eliminate (used to get an implicit representation of the function)

This is a complicated curve, easily described parametrically but it would be difficult to eliminate the parameter and determine the sketch by hand. Eliminating the parameter by handis possible since y is quadratic, but what a pain in the ass it would be to do! And once you had done that, you would have no luck identifying the curve since it is not one of our basic functions.

imp=Eliminate[{x==t^3+t^6,y==t^2+4},t]

paraplot=ParametricPlot[{t^3+t^6,t^2+4},{t,-1,1},PlotStyle->{Thick,Red}]

impplot=ContourPlot[Evaluate[imp],{x,-0.3,1},{y,4,5},ContourStyle->{Thick,Blue}]

Show[paraplot,impplot]

Notice the implicit plot is not nearly as accurate as the parametric plot near the point. Paramertic functions are generally better to plot than their associated implicit counterparts!

FYI: Beautiful Curves: Axes Surpressed in the Name Of Beauty

x[t_]=t+Sin[3t]
y[t_]=t+Sin[4t]
ParametricPlot[{x[t],y[t]},{t,-4Pi,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=Sin[t+Cos[50t]]
y[t_]=Cos[t+Sin[50t]]
ParametricPlot[{x[t],y[t]},{t,-2Pi,2Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=2Cos[t]+Sin[t+Cos[50t]]
y[t_]=2Sin[t]+Cos[t+Sin[50t]]
ParametricPlot[{x[t],y[t]},{t,0,2Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=t+Cos[3t]Sin[2t]
y[t_]=-Exp[-t^2]-Cos[Sin[t]]
ParametricPlot[{x[t],y[t]},{t,-2Pi,2Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=t-Exp[-t^2]+Sin[3t]
y[t_]=Exp[-t^2]-Sin[2t]
ParametricPlot[{x[t],y[t]},{t,-2Pi,2Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=t(Cos[t]+t Sin[t])
y[t_]=-8t Sin[2t]
ParametricPlot[{x[t],y[t]},{t,-4Pi,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=t(Cos[t]+t Sin[t])
y[t_]=-8t Sin[t]
ParametricPlot[{x[t],y[t]},{t,-4Pi,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

FYI: Beautiful Curves in Three Dimensions

You can even extend this to three dimensions. Click and drag the images to rotate them. The parametric representation extends very easily to higher dimensions.

x[t_]=t+Sin[3t]
y[t_]=t+Sin[4t]
z[t_]=Cos[t] 
ParametricPlot3D[{x[t],y[t],z[t]},{t,0,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=2Cos[t]+Sin[t+Cos[30t]]
y[t_]=2Sin[t]+Cos[t+Sin[30t]]
z[t_]=Cos[t]
ParametricPlot3D[{x[t],y[t],z[t]},{t,0,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]

x[t_]=Sin[t+Cos[100t]]
y[t_]=Cos[t+Sin[100t]]
z[t_]=Cos[t]
ParametricPlot3D[{x[t],y[t],z[t]},{t,0,4Pi},PlotStyle->{Red,Thick},Axes->False,AspectRatio->1,PlotRange->{All,All}]