Visualizing Space Curves as Time-Dependent Vectors

Create a trajectory of a point or a particle

ParametricPlot3D allows us to visualize the entire trajectory at once.

Here is a function to create a graphic with a variable end - point. We will have the function remember when it has already computed a graphic, trading memory for a possible speed-up.

Use manipulate on the graphics function to visualize how the curve develops with its parameter

However, we need to fix the length scale between frames, so we use the last graphic to infer what PlotRange should be.

Next, we add a graphic element to show the vector, drawn from the origin, for each end-point.

Created by Wolfram Mathematica 6.0  (17 September 2007)