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The purpose of this lab is to introduce you to curve computations
using Maple for parametric curves and vector-valued functions in three
To assist you, there is a worksheet associated with this lab that
contains examples. You can
copy that worksheet to your home directory with the following command,
which must be run in a terminal window, not in Maple.
cp ~bfarr/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
A parametric curve in three dimensions is a triple of functions
, , for in some interval .
A vector-valued function in
three dimensions is a function that associates a vector in
the plane with
each value of in its domain. Such a vector valued function can
written in component form as follows,
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. This happens most often when the quantity you want to
describe with the function is natural to think of as a vector, for
example, a force or a displacement. For our purposes, 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
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
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
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.
Give a brief explanation of how the values of and affect the
- , for
- , for
- , for
- Consider again the circular helix
- Compute the curvature.
- Compute the normal and tangential components of the acceleration.
- Consider the vector-valued function
- Plot the graph of this function for
. Explain why the graph might be called an elliptical helix. (Note
- the plot you get might make it look like a regular helix, that is
one whose projection in the plane is a circle, because of the
way Maple will scale the axes. The best way to overcome this is to
use the optional argument view. For example, try adding
to your VPlot command.)
- Find the minimum and maximum values of the speed.
- Find the maximum and the minimum values of the curvature.
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William W. Farr