Math 1206CCH
Computer Lab #9
Parametric Equations

Preparation: Read 16.1-16.3 in Multivariable Calculus by McCallum, Hughes-Hallet, and Gleason, et al. and this assignment before going into the laboratory.

Goal:  To examine graphing of parametric equations and to investigate their use.

To sketch plane curves described by parametric equations, you will need the command ParametricPlot[{x[t],y[t]},{t, t1,t2}]. As with any Plot command, you can use many options to clarify your graph. (e.g. PlotRange, AxesLabel, PlotLabel, PlotStyle, etc.) Example 1. a) Below is the graph of x = t,  y = à(1-t2). As t changes from 0 to 7Pi/22, the ordered pairs (x,y) are graphed on the x and y axes.

ParametricPlot[{t,Sqrt[1-t^2]},{t,0,7*Pi/22},
AspectRatio->Automatic];

b) Note that different parametric equations can produce the same graph. We will graph x = cos(Pi/2*t) and y = sin(Pi/2*t) over the same domain.

ParametricPlot[{Cos[Pi/2*t],Sin[Pi/2*t]},
{t,0,7*Pi/22},AspectRatio->Automatic];

Both of the above equations give the same curve. The difference is in how the curves are traced out. Problem #1 will ask to look at the similarities and differences of graphs of several parametric equations.
Using parametric equations we can graph curves that would be complicated in Cartesian coordinates. Let's graph x = cos(3t), y = sin(4t). Graphs of the form x = cos(at), y = sin(bt) , a Ç b, are called Lissajous curves.

ParametricPlot[{Cos[3t],Sin[4t]},{t,0,2Pi}]

One curve that has a number of interesting applications is the cycloid.  A cycloid is the curve traced by a point P on the rim of a wheel as the wheel rolls along a straight line without slipping. You may click on the cell at right to see an animation of the cycloid being drawn. Use Animate Selected Graphics from the Graph menu. You may want to reset the number of frames per second. This is done by going to the Graph menu and selecting Animation; a window will pop up and you can reset the frames per second -- perhaps 2 or 3 per second.

AnimateCycloid[1,{0,5Pi}]

In 1696 Johann Bernoulli proposed the Brachistochrone problem which asks what shape a wire must be so that a bead slides down it in minimum time. The answer is an inverted cycloid. Even though a straight line is the shortest distance, the bead builds up velocity much more quickly on the cycloid and thus slides down faster than on the straight line.

p1=Plot[(-1.7519)*x,{x,0,1.414},
Axes->False,AspectRatio->Automatic,
PlotLabel-> Line,DisplayFunction->Identity]
d1=ListPlot[{{.5,-1.7519*.5}},PlotStyle->
{PointSize[.08],RGBColor[1,0,0]},
Axes->False,DisplayFunction->Identity]
s1=Show[p1,d1,DisplayFunction->$DisplayFunction]
p=ParametricPlot[{t-Sin[t],-1+Cos[t]},{t,0,Pi/2},
Axes->False,AspectRatio->Automatic,
PlotLabel->Cycloid,DisplayFunction->Identity]
d=ListPlot[{{Pi/4-Sin[Pi/4],-1+Cos[Pi/4]}},PlotStyle->
{PointSize[.08],RGBColor[1,0,0]},
Axes->False,DisplayFunction->Identity]
s=Show[p,d,DisplayFunction->$DisplayFunction]
Show[GraphicsArray[{s,s1}]]

Another interesting property of cycloids is that if A is the lowest point on the inverted cycloid, the time it takes a bead to slide down to A is the same no matter where the bead starts. In other words, if B1, B2, and B3 are beads in different positions on the cycloid, they will all reach A, the lowest point, at the same time.

p1=ParametricPlot[{t-Sin[t],-1+Cos[t]},{t,0,1.5Pi},
Axes->False,DisplayFunction->Identity]
b=ListPlot[{{2Pi/3-Sin[2Pi/3],-1+Cos[2Pi/3]},{Pi/2-Sin[Pi/2],
-1+Cos[Pi/2]},{Pi/3-Sin[Pi/3],-1+.5}},PlotStyle->
{PointSize[.05],RGBColor[1,0,0]},Axes->False,
DisplayFunction->Identity]
Show[p1,b,DisplayFunction->$DisplayFunction]

The equation for a cycloid can best be written parametrically as      x = a(t - Sin[t]), y = a(1 - Cos[t]). Let's graph the cycloid        when a = 2 and a = 1.

Clear[x,y,a,t]
x[t_]:=a(t-Sin[t])
y[t_]:=a(1-Cos[t])
a=2;
ParametricPlot[{x[t],y[t]},{t,0,4Pi},
AspectRatio->Automatic,Ticks->{{2Pi,4Pi,6Pi,8Pi},
{2,4}}];

a=1;
ParametricPlot[{x[t],y[t]},{t,0,4Pi},
AspectRatio->Automatic,Ticks->{{Pi,2Pi,3Pi,4Pi},
{1,2}}];

We can find the slope of parametric equations by using the chain rule dy/dt = (dy/dx)(dx/dt). Therefore the slope dy/dx = (dy/dt)/(dx/dt). Let's find the slope of the cycloid where a = 2 for the points x = p/2, p, 6. (By adding //N to the function we automatically get a numerical result. This is the same as using N[expression].)

Clear[x,y,m]
x[t_]:=2(t-Sin[t])
y[t_]:=2(1-Cos[t])
m[t_]:=y'[t]/x'[t]
m[Pi/2]//N
m[Pi]//N
m[6]//N

Over the interval [0,2p], where is the slope undefined?  Where is it equal to -1?

m[t]

The values of t where the slope is undefined are where cos(t) = 1. (m[t] = sin(t)/1-cos(t)) Those values are 0 and 2p. To find where the slope is -1, we must use the FindRoot command. We will first plot m[t] and -1 on the same graph and see where they intersect. Note that we are using as endpoints the values for which m[t] is undefined. The messages that are encountered after the Plot command let us know that values don't exist at these points.

Plot[{m[t],-1},{t, 0,2Pi},PlotRange->{-2,2}];

FindRoot[m[t]+1,{t,4.5}]

The value of t where the slope is -1 is 3p/2.

Problems (To be handed in)

1. a. Plot the following plane curves on separate coordinate axes for -5 æ t æ 5. (Note: et is written E^t in Mathematica.)
			i.   x = t,  y = t3
			ii.  x = t2, y = t6
			iii. x = et, y = e3t
			iv.  x = sint, y = sin3t
			
			b. Discuss the similarities and differences among the four graphs.
			
2.		The parametric equations: x = a*t - b*sint, y = a - b*cost are variations on the standard cycloid. If the point P lies inside the rolling circle, then b < a, and it is called a curtate cycloid.  If P lies outside the circle, then b > a and the curve is called a prolate cycloid.
  
  a.) Sketch the following cycloids on separate axes for 
  0 æ t æ 4p ( Use the PlotRange option to make the axes cross at the origin.)
     i)  a = 1 and b = 2 (prolate)
     ii) a = 2 and b = 1 (curtate)	
  
  b) Find the slope of the tangent line to the curtate cycloid above when t = 2.5 and 3p/2. Over the interval [0,2p], where does the slope = 0? Where is it undefined? where does it = .3?
  
3. Suppose Earth is moving on the path x[t]= cos2(t), y[t]= sint*cost. Simultaneously a meteor is moving along the path x[t]= (t - 1.6)/3, y[t]= .2. 
   a) Use ParametricPlot to plot each equation separately and then plot them together on the same graph. 
   b) What are the points of intersection of the two curves?
   c) Does the meteor collide with Earth?
   
4. Lab 2 discussed motion problems which included projectile problems.  We can now write the position of a projectile which is launched from height s0 with an initial velocity v0 at an angle A inclined upward from the horizontal t seconds after launching by the following parametric equations. x[t]=(v0*cosA)*t, 
y[t]=s0 + (v0*sinA)*t - 1/2g*t2 where g is the acceleration due to gravity (32 feet per second per second). Air resistance is neglected in these equations. 
   a) Eliminate the parameter t to obtain the equation of the projectile in xy - coordinates. What geometric figure is it?
   b) Suppose the projectile is a major league fastball which leaves the pitcher's hand traveling horizontally with initial speed 110 feet per second and initial height 7 feet. 
      i) Plot the graph of the path of the fastball. 
      ii)When does the ball cross the plate 60.5 feet away? 
      iii)If the strike zone is between a batter's armpits and knees,is the pitch a strike? 
      iv)If the batter, catcher and umpire all miss the ball, where will it land?
      v) What additional factors would need to be included if the pitch is a curve ball and we want to predict it's path?