1.09 Parametric Plotting
 LITERACY

Off[General::spell1]; Off[General::spell]; Off[Plot::plnr]; \(Needs["Graphics`Colors`"]; 

    Graphics`Colors`GosiaGreen = RGBColor[0, \ 0.392187, \ 0]; \)

L.1)

Put x[t] = Cos[t] and y[t] = Sin[t]. 
Sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to (π/2).

Show[Graphics[Line[{{-1, 0}, {1, 0}}]], 
      PlotRange -> {-1, 1}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

Next, sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to π.

Show[Graphics[Line[{{-1, 0}, {1, 0}}]], 
      PlotRange -> {-1, 1}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

Next, sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to 2  π.

Show[Graphics[Line[{{-1, 0}, {1, 0}}]], 
      PlotRange -> {-1, 1}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

L.2)

Put x[t] = 2 Cos[t] and y[t] = Sin[t]. 
Sketch with a pencil the curve traced out by {x[t], y[t]} as t advances from 0 to (π/2). Next, sketch the curve traced out by {x[t], y[t]} as t advances from 0 to π. Finally, sketch the curve traced out by {x[t], y[t]} as t advances from 0 to 2 π.
Here are some axes for you to use:

\(Show[Graphics[Line[{{\(-2.5\), 0}, {2.5, 0}}]], 
      Graphics[Text[
          FontForm["t from 0 to π/2 ", {"Palatino", 10}], {1.8, 
            2}]], PlotRange -> {\(-2.5\), 2.5}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}]; 

    Show[Graphics[Line[{{\(-2.5\), 0}, {2.5, 0}}]], 
      Graphics[Text[
          FontForm["t from 0 to π ", {"Palatino", 10}], {1.8, 
            2}]], PlotRange -> {\(-2.5\), 2.5}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}]; 

    Show[Graphics[Line[{{\(-2.5\), 0}, {2.5, 0}}]], 
      Graphics[Text[
          FontForm["t from 0 to 2π ", {"Palatino", 10}], {1.8, 
            2}]], PlotRange -> {\(-2.5\), 2.5}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}]; 
    \)

L.3)

Put x[t] = 2 Cos[t] and y[t] = 3 Sin[t]. 
Sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to (π/2).

Show[Graphics[Line[{{-3, 0}, {3, 0}}]], 
      PlotRange -> {-3, 3}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

Next, sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to π.

Show[Graphics[Line[{{-3, 0}, {3, 0}}]], 
      PlotRange -> {-3, 3}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

Finally, sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to 2 π.

Show[Graphics[Line[{{-3, 0}, {3, 0}}]], 
      PlotRange -> {-3, 3}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

L.4)

Put x[t] = t and y[t] = Sin[t]. 
Sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to 2 π.

Show[Graphics[Line[{{0, 0}, {2 π, 0}}]], 
      PlotRange -> {-1.5, 1.5}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

L.5)

Put x[t] = 1 + t and y[t] = (t/2). 
Sketch on the axes below the curve traced out by {x[t], y[t]} as t advances from 0 to 2.

Show[Graphics[Line[{{0, 0}, {4, 0}}]], 
      PlotRange -> {-1.5, 1.5}, Axes -> True, 
      AspectRatio -> Automatic, AxesLabel -> {"x", "y"}];

L.6)

When you go with 
       x[t] = 1.2 t + Sin[t] and y[t] = 2 (1 + Cos[(t/2)]), 
then as t advances from 0, {x[t], y[t]} sweeps out a curve:

\(Clear[x, y, t]; 
x[t_] = 1.2\ t + Sin[t]; 

    \(y[t_] = 2\ \((1 + Cos[t\/2])\); \)\)ParametricPlot[{x[t], y[t]}, {t, 0, 2 π}, 
      PlotStyle -> Thickness[0.01], 
      AxesLabel -> {"x", "y"}];

As you can see, this curve is the graph of a function 
           y = f[x]. 
But it would be a whale of a lot of trouble (in fact, it would be impossible) to find a clean formula for f[x].  In spite of this, calculate f'[x] by hand for:
i) x = x[(π/2)] = ((1.2 π)/2) + 1:

N[((1.2 π)/2) + 1]

ii) x = x[π] = 1.2 π

N[1.2 π]

For your information: Sin[(π/4)] = Cos[(π/4)] = (1/(Sqrt[2])) ,
Sin[(π/2)] = 1, Cos[(π/2)] = 0, Sin[π] = 0, Cos[π] = -1.

L.7)

Come up with the specific values of the coordinates {x, y} of the highest point on the curve parameterized by
       {x[t], y[t]} = {t (e^t), 1 - (t^2)}  with  -∞ < t < ∞.

L.8)

Put {x[r, t], y[r, t], 
          z[r, t]} = {1, 0, 1} + {r Cos[t], r Sin[t], 0}. 
Describe what you get when you plot {x[r, t], y[r, t], z[r, t]} for 0 ≤ r ≤ 2 and 0 ≤ t ≤ 2 π .  
Is this a curve or a surface?

Put {x[t], y[t], z[t]} = {1, 0, 1} + {2 Cos[t], 2 Sin[t], 0}. 
Describe what you get when you plot {x[t], y[t], z[t]} for  0 ≤ t ≤ 2 π.  
Is this a curve or a surface? 
What relation does it have to what you said immediately above?

L.9)

Curves can be thought of as one-dimensional creatures living in two or three dimensions.
How many parameters do you need to plot a curve in three dimensions?
Surfaces can be thought of as two-dimensional creatures living in three dimensions.
How many parameters do you need to plot a surface in three dimensions?

L.10)

You are given functions x[t], y[t], and z[t]. When you plot {x[t], y[t], z[t]} in three dimensions for 0 ≤ t ≤ 4, you get a curve. What would you plot in three dimensions to get the tube consisting of the edges of all circles of radius 0.5 parallel to the xy - plane and centered on the curve?
What would you do to plot a sample of a horn consisting of edges of all circles parallel to the xy-plane and centered on the curve?

L.11)

Comment on the statement:
"Parametric plotting gives you a lot more freedom than nonparametric plotting."