Next: About this document ...
The purpose of this lab is to introduce you to curve computations
using Maple for parametric curves and vector-valued functions in the
To assist you, there is a worksheet associated with this lab that
contains examples and even solutions to some of the exercises. 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/Curves2D_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 Curves2D_start.mws, and go through it
carefully. Then you can start working on the exercises.
By parametric curve in the plane, we mean a pair of equations
and for in some interval . A vector-valued function in
the plane 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. However, there can be situations when
the quantity is not a vector, but it is convenient to think of it as a
vector to simplify the notation or the calculations. The idea to keep
in your mind is that one of the motivations of vector notation is to
simplify and clarify calculations or other manipulations.
Many applications of calculus to engineering or science involve
parametric curves or vector valued functions, including the examples
- The kinematics of mechanisms, like internal combustion engines.
- Population models involving more than one species.
- Chemical kinetics in systems with more than one chemical
- Forced or transient responses of electrical circuits.
For the applications we will deal with, we only need to consider
curvilinear motion of a point in the plane, That is, we have a
vector-valued function that gives the position at time
of a moving point in the plane. The velocity of this point is
given by the derivative
and the acceleration is given
by the second derivative,
. If the velocity,
, is not the zero vector, then it is clear from the
way it is defined that
is a vector that is tangent to
the curve at the point . A simple example of
curvilinear motion is when the velocity is constant. That is, there is
a fixed vector such that
In this case, the motion is along a straight line, with
where is the position at .
In many applications of
curvilinear motion, we need to know the magnitude of the velocity, or
the speed. This is easy to compute - just take the magnitude
. If and are the component functions of
, then the speed is given by
Using our idea of the parametric curve associated with
and recalling the definition of arc length, we arrive
at a different interpretation of speed as the rate of change of
arc length, or
where is arc length.
This should make sense, if you recall that speed is the rate of change
of distance with time and arc length is distance measured along a
If the speed is not zero for any value of in the interval ,
then it is possible to define a unit vector, that is
tangent to the curve as follows.
Using this definition, you can write the velocity in the following form.
Now, it is natural for the student to ask why one would want to do
this. Computing the velocity using the right hand side of the equation
above is certainly more difficult than taking the derivative of
, so this equation is not really useful from a
computational viewpoint. The answer to this is that the unit tangent
vector at a particular point on the curve is an intrinsic property
of the curve, at least up to a sign change. To use an analogy, think of the
curve as being a one-way road. Different drivers can drive at different
speeds, but at a particular point on the road, each car is
pointed in the same direction, i.e. the direction of the unit tangent
vector at that point. Now suppose that the road allows traffic in
both directions. If you think about it a little, it should be clear
that if the tangent vector at a particular point on the road for
traffic going one direction is , then the tangent vector at the
same point for traffic going the opposite direction is .
Now suppose, to carry our analogy a little further, that you wanted to
analyze the acceleration of a car as it traversed the road. On
sections where the road is straight, acceleration depends only on
whether the car is speeding up or slowing down. In a curve, however,
there is an additional acceleration, called the centripetal
acceleration, that is needed to keep the car on the road. The
magnitude of this acceleration depends on the speed of the car and the
sharpness of the curve. It turns out that you can quantify the
sharpness of a curve with an intrinsic property of the curve called
the curvature, usually denoted , defined by the following
That is, the curvature is the magnitude of the rate of change of the
tangent vector with respect to arc length. For example,
the curvature of a straight line is zero and it can be shown that the
curvature of a circle of radius is the same for every point on the
circle and is given 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
for the velocity and using the chain rule and the definition of the
curvature and the normal vector to obtain the following important
To see why this equation is important, recall that is the
speed, so is the rate of change of the speed. In our
analogy of a car driving on a road, this term measures whether the car
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 with the symbol . The
component of the acceleration in the direction of the normal vector is
called the normal acceleration, denoted .
Computing these intrinsic properties of a curve 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 parametric curve , . Plot
the graph of this curve for
. Identify the points
on your graph that correspond to the values and and indicate the direction of motion on the graph as
increases. This is probably best done by hand on the printed copy.
- For each of the following parametric curves, plot the graph and
locate the point on your graph that corresponds to .
Explain why the graph is the same in each case, even though the
parametrizations are different.
. What is the
normal vector at ? At ? What about at ? What
- Show that the curve given by , is a parametrization of the ellipse
That is, show that the two functions satisfy the equation for the
ellipse for all values of .
- A hairpin turn on a roadway can be approximated as half of an
ellipse, including the half of the major (longer) axis. If the major
axis is 100
feet and the minor axis is 80 feet, what is the maximum (constant)
speed a car can go through the turn while keeping the centripetal
acceleration less than ?
Next: About this document ...
William W. Farr