cp /math/calclab/MA1024/Curves3D_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Curves3D_start.mws`, and go through it
carefully. Then you can start working on the exercises

where , , and are functions defined on some interval . From our definition of a parametric curve, it should be clear that you can always associate a parametric curve with a vector-valued function by just considering the curve traced out by the head of the vector. However, there are lots of situations where a vector-valued function is more appropriate. We will focus on the case of motion of a particle in three dimensions. That is, we have a vector-valued function that gives the position at time of a moving point . The velocity of this point is given by the derivative and the acceleration is given by the second derivative, . If the velocity, , is never zero, then we can define the unit tangent vector and the curvature the same way we did in two dimensions by

and

If the curvature is never zero for a particular curve, then we can
define another intrinsic property of curve, the unit normal vector
by the following equation.

It can be shown that at each point on the curve the vector defined by this equation is a unit vector that is always perpendicular to the tangent vector at that point. Furthermore, the unit normal vector always points in the direction of the centripetal acceleration required to keep a particle moving on the curve. One way to see this is to compute the acceleration by differentiating both sides of the equation

Using the chain rule and the definition of the curvature and the normal vector one obtains the following important equation.

To see why this equation is useful, recall that is the speed, so is the rate of change of the speed. That is, this term measures whether the particle is speeding up or slowing down. Because this component of the acceleration is in the direction of the tangent vector it is often called the tangential acceleration, denoted by the symbol . The component of the acceleration in the direction of the normal vector is called the normal acceleration, denoted . In the case of motion on a circular path, the curvature is the reciprocal of the radius, so this term should be easily recognizable as the centripetal acceleration.

Computing these quantities is generally not an
easy task. The **Getting started** worksheet for this lab
describes commands from the `CalcP` package that simplify these
calculations and provides examples for you to work from.

- Consider the circular helix
. Plot the graphs
for the following sets of values of and using the
`VPlot`command. You should also look at animations of the plots by using the`ParamPlot3D`command, though they won't appear in your printout.- , for .
- , for .
- , for .

- Consider again the circular helix
. Compute the speed and the curvature as functions of the parameters and and relate your results to your findings from the first exercise.
- Consider the function
that appeared as one of the examples in the background.
- Compute the curvature, and plot it for .
- Compute the normal acceleration, and plot it for .
- Can you explain why the normal acceleration is increasing, even though the curvature is decreasing?

2004-11-03