Parametric Curves in 3-space

Here is a parametric curve in space

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Mathematica writes it a little nicer:

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Here is how to evaluate the function at a point

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

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Now we attempt to plot it:

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Options for ParametricPlot3D

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)

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Here is a new plot using the PlotPoints and PlotRange options

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

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Set P to be the point F(1)

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This gives us access to the first coordinate of P

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The second and third coordinates

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

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We can compose our two functions

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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!

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Another view of the same curve:

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

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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, )
What is the limit of F(t) as t->0?

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As written by mathematica:

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Here we evaluate it at a few points

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A first look:

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Some more plots, changing the interval we plot

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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

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

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Here are some plots of our curve

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Time as the third dimension

Take a parametric curve in the plane:

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

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

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Note: G is periodic with period 2 Pi

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

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

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We plot the curve from several different viewpoints

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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:

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Here it is fixed up a lot

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You should animate the above sequence of pictures!

Created by Mathematica  (September 14, 2004)