Parametric Curves in 3-space

Here is a parametric curve in space

F[t_] := {
2*E^t*
          Cos[10*t]/(1 + E^(2  t)), 
2*E^t*
          Sin[10*t]/(1 + E^(2  t)), 
1 - 
          2/(1 + E^(2  t))
}

Mathematica writes it a little nicer:

F[t]

Here is how to evaluate the function at a point

F[0]

F[1]

The N[ ] function allows us to evaluate an expression numerically

N[F[1]]

Now we attempt to plot it:

ParametricPlot3D[F[t], {t, -10, 10}]

Use the option PlotPoints->n to plot n points. It uses 75 as a default, but often more are needed.

PlotRange->All makes sure Mathematica plots all points on the curve, rather than the points it wants to.

ViewPoint->{x,y,z} allows you to change the viewpoint of the plot.

List all available options and their default settings: (More information on each can be found in help)

Options[ParametricPlot3D]

Here is a new plot using the PlotPoints and PlotRange options

ParametricPlot3D[F[t], {t, -10, 10}, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t, -10, 10}, PlotRange -> All, 
      PlotPoints -> 3000]

We change perspectives using the ViewPoint option. {1,2,1} is the coordinates of "your eye"

ParametricPlot3D[F[t], {t, -10, 10}, PlotPoints -> 3000, 
      PlotRange -> All, ViewPoint -> {4, 2,  .5}]

Set P to be the point F(1)

P = F[1]

This gives us access to the first coordinate of P

P[[1]]

The second and third coordinates

P[[2]]
P[[3]]

We define a function r2 which gives us the square of the distance from the origin to P

r2[P_] := P[[1]]^2 + P[[2]]^2 + P[[3]]^2

We can compose our two functions

r2[F[t]]

The Simplify[ ] command attempts to reduce an expression to a simpler form:
Behold: the following shows that our curve lies on the surface of the unit sphere!

Simplify[r2[F[t]]]

Another view of the same curve:

ParametricPlot3D[F[t], {t, -10, 10}, PlotPoints -> 3000, 
      PlotRange -> All, 
      ViewPoint -> {-2, 1, - .2}]

Here we have a bunch of viewpoints using a For statement:

For[k = 0, k < 40, k = k + 1, 
      ParametricPlot3D[F[t], {t, -10, 10}, PlotPoints -> 3000, 
        PlotRange -> All, 
        ViewPoint -> {2 Cos[k/40*2*Pi], 2 Sin[k/40*2*Pi], 
            1}]]

You can animate the above by first selecting the group of images by selecting the blue bracket on the right containing them, then clicking on the menu 
Cell->Animate Selected Graphics
Its lots of fun, try it!

Limits

Set F(t)=(cos(1/t^2)t, sin(1/t^2)t, (Sqrt[t]))
What is the limit of F(t) as t->0?

F[t_] := {Cos[1/t^2] t, Sin[1/t^2] t, Sqrt[t]}

As written by mathematica:

F[t]

Here we evaluate it at a few points

F[0]

F[ .11]

A first look:

ParametricPlot3D[F[t], {t, 1, 10}, PlotPoints -> 3000, 
      PlotRange -> All]

Some more plots, changing the interval we plot

ParametricPlot3D[F[t], {t,  .5, 10}, PlotPoints -> 3000, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t,  .25, 10}, PlotPoints -> 3000, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t,  .1, 1}, PlotPoints -> 3000, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t,  .001, 1}, PlotPoints -> 3000, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t,  .001, 10}, PlotPoints -> 3000, 
      PlotRange -> All]

A tornando or a whirlpool? But the limit as t->0 appears to be the origin!

However the following parametric curve does not have a limit as t->infinite

F[t_] := {Cos[t], Sin[t], 1/Log[Log[t]]}

F[t]

Mathematica defines Log[ ] to be the natural log. Note: You refer to the constant e as E:

Log[E]

Here are some plots of our curve

ParametricPlot3D[F[t], {t, 3, 10}, PlotPoints -> 3000, 
      PlotRange -> All]

ParametricPlot3D[F[t], {t, 4, 100}, PlotPoints -> 3000, 
      PlotRange -> All]

Time as the third dimension

Take a parametric curve in the plane:

G[t_] := {Sin[5 t],  Cos[7 t]}

G[t]

Here we plot our curve
(AspectRatio->1 gives us our curve in a square)

ParametricPlot[G[t], {t, 0, 2*Pi}, AspectRatio -> 1]

Options[ParametricPlot]

We can make a G into a parametric curve in space by setting z=t:

F[t_] := {Sin[5 t],  Cos[7 t], t}

Note: G is periodic with period 2 Pi

G[t]

G[t + 2*Pi]

Simplify[G[t + 2*Pi]]

This means that F should be invariant under a vertical translation of 2 Pi

ParametricPlot3D[F[t], {t, 0, 2 Pi}, PlotPoints -> 3000, 
      PlotRange -> All]

Can you see that the top half is the same curve as the bottom half?

ParametricPlot3D[F[t], {t, 0, 4 Pi}, PlotPoints -> 3000, 
      PlotRange -> All]

We plot the curve from several different viewpoints

For[k = 0, k < 2 Pi,  k += Pi/20, 
      ParametricPlot3D[F[t], {t, 0, 2 Pi}, PlotPoints -> 3000, 
        PlotRange -> All, 
        ViewPoint -> {2 Sin[k], 2  Cos[k], 1.1}]]

Recall to animate select the graphics, then click menu CELL->ANIMATE SELECTED GRAPHICS

It jumps around and the box is annoying. We can clean it up by playing with some options. Here they all are:

Options[ParametricPlot3D]

Here it is fixed up a lot

For[k = 0, k < 2 Pi,  k += Pi/20, 
      ParametricPlot3D[F[t], {t, 0, 2 Pi}, PlotPoints -> 3000, 
        PlotRange -> All, ViewPoint -> {2 Sin[k], 2  Cos[k], 1.1}, 
        Axes -> False, Boxed -> False, 
        SphericalRegion -> True]]

You should animate the above sequence of pictures!