Examples for Section 9.4
Parameterizations of Plane Curves

p.741

SetOptions[ParametricPlot, AspectRatio -> Automatic];

3.

x = sin(2  π(1 - t)), y = cos(2  π(1 - t)), 0 ≤ t ≤ 1.

The Cartesian equation is \(TraditionalForm
      \`x\^2 + \ y\^2 = 
        \(\(sin\^2\)(2 \( π(1 - t)\)) + \(cos\^2\)(2 \( π(1 - t)\)) = 
          1\)\) so the curve lies on a circle. As the parameter t goes from 0 to 1, the angles in the trig functions go from 2  π to 0, and the curve is traced clockwise, starting at the point (0,1).

ParametricPlot[{Sin[2  π (1 - t)], Cos[2  π (1 - t)]}, {
        t, 0, 1}]

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[ParametricPlot[{Sin[2  π (1 - t)], 
          Cos[2  π (1 - t)]}, {t, 0, u},  
        PlotRange -> {{-1, 1}, {-1, 1}}], {u, 0.1, 1.0,  .05}]

21.

x = (t^2), y = (Sqrt[(t^4) + 1]), t ≥ 0.

The Cartesian equation is \(TraditionalForm
      \`y\  = \ \(\@\(\((t\^2)\)\^2 + 1\) = \@\(x\^2 + 1\ \)\)\). Thus, the curve lies on the hyperbola (y^2) -  (x^2) =  1. Since x  ≥ 0, y ≥ 1, the curve only traces half od the upper branch.

ParametricPlot[{(t^2), (Sqrt[(t^4) + 1])}, {t, 0, 2}, 
      PlotRange -> {0, 5}]

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[ParametricPlot[{(t^2), (Sqrt[(t^4) + 1])}, {t, 0, u}, 
        PlotRange -> {{0, 4}, {0, 5}}], {u, 0.1, 2.0,  .1}]

29.

The witch of Agnesi

ParametricPlot[{2  Cot[t], 2  Sin[t]^2}, {t,  .3, π -  .3}]

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[ParametricPlot[{2  Cot[t], 2  Sin[t]^2}, {t,  .3, u}, 
        PlotRange -> {{-6, 6}, {0, 2.5}}], {u, 0.4, 
        π -  .3,  .1}]

33.

Hypocycloids

\(b = 1/3; 

    ParametricPlot[{\((1 - b)\) Cos[t] + b\ Cos[\(\((1 - b)\)\/b\) t], 
        \((1 - b)\) Sin[t] - b\ Sin[\(\((1 - b)\)\/b\) t]}, {t, 0, 2  π}]
      \)

\(b = 1/4; 

    ParametricPlot[{\((1 - b)\) Cos[t] + b\ Cos[\(\((1 - b)\)\/b\) t], 
        \((1 - b)\) Sin[t] - b\ Sin[\(\((1 - b)\)\/b\) t]}, {t, 0, 2  π}]
      \)

\(b = 1/5; 

    ParametricPlot[{\((1 - b)\) Cos[t] + b\ Cos[\(\((1 - b)\)\/b\) t], 
        \((1 - b)\) Sin[t] - b\ Sin[\(\((1 - b)\)\/b\) t]}, {t, 0, 2  π}]
      \)

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[ParametricPlot[{(1 - b) Cos[t] + b Cos[(((1 - b))/b) t], 
          (1 - b) Sin[t] - b Sin[(((1 - b))/b) t]}, {t, 0, u}, 
        PlotRange -> {{-1, 1}, {-1, 1}}], {u, 0.1, 
        2  π,  .1}]

Example

Cycloid

\(p\  = \ 
      ParametricPlot[{t\  - \ Sin[t], \ 1\  - \ Cos[t]}, \ {t, 0, 4  π}, 
        
	PlotStyle -> RGBColor[1, 0, 0], 
        PlotRange -> {{\(-1\), 13}, {0, 2}}, 
	
        Ticks -> {{0, π, 2  π, 3  π, 4  π}, {0, 1, 2}}]
        \)

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[Show[p, 
        Graphics[{Circle[{t, 1}, 1],  PointSize[ .02], 
            Point[{t  -  Sin[t],  1  -  Cos[t]}]}], 
        AspectRatio -> Automatic], {t, 0, 4  π}]

Example

Just adding together various trig functions usually give an interesting curve. Here's a random example.

ParametricPlot[{Cos[2  t] + Sin[3  t], Sin[t] - Cos[5  t]}, {t, 0, 
        2  π}]

Select the graphics cell and select Animate Selected Graphics under the Cell menu.

Do[ParametricPlot[{Cos[2  t] + Sin[3  t], Sin[t] - Cos[5  t]}, {t, 0, 
          u}, PlotRange -> {{-2, 2}, {-2, 2}}], {u, 0.1, 
        2  π,  .1}]