Previous: Plane Curves and Their Representation
2. Velocity Vectors and SpeedParametrizing a curve by t suggests considering the curve as the locus traced out by a point in motion. If the coordinate functions of the curve are differentiable, then at each point we have a well-defined velocity vector with coordinate functions x'(t), y'(t). (For a more precise definition of the derivative of a vector-valued function, see lab3.) We emphasize this in the next demonstration by including a "tape deck", giving an animation of the point moving along the curve, along with the velocity vector, both at the origin and at the point of the curve. | |||||
Demonstration 3: Position and Velocity Vectors of Plane Curves
This demo displays a plane curve, its position vectors emanating from the origin, and its velocity vectors emanating from the points on the curve with which they are associated. You can specify the curve, using a variable c . Note that the curve is described as a single vector. The default function is an ellipse with eccentricity controlled by c . Interval t determines the domain and resolution for your function. Running the tapedeck next to t0 in the control panel makes the parameter t0 run through its domain. For more information, see the tutorial. | |||||
For a circle, [cos(t),sin(t)] with 0 t < 2 | |||||
How does the direction of the velocity vector relate to the direction of the
position vector for the exponential spiral,
X(t)
=ect(cos(t)
,sin(t))? (This curve may be entered exp(c*t)*[cos(t),sin(t)]
) Explain what happens as c changes in
general and what happens when c equals zero. | |||||
How does the direction of the velocity vector relate to the direction of the
position vector for the arithmetic spiral, X(t)
=(tcos(t), tsin(t))
? | |||||
What happens to the velocity vector at the origin for the curve
X(t)
=(t3,t4)
? Why?
Consider the general polar coordinate curve, X(t)
=(r(t)cos(t), r(t)sin(t)).
Under what circumstances will the velocity vector and the position
vector be perpendicular? When will they be linearly independent?
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Demonstration 4: Effects of Parametrization This demo shows the tangent vector on a user-defined curve, X(t) =X(u(t )). There is another window which shows the unit tangent vector in yellow with its tail at the origin, so its head is always on the unit circle. The not-normalized tangent vector is also shown in red. As in the first demo, there is a tapedeck which traces out the curve. What makes this demo special is that the user can modify the way in which the curve depends on the tape-deck parameter. | |||||
This demonstration gives a good opportunity to display the effects of reparametrization of the domain of a curve. As an example, consider the circle X(t) =(Rcos(t),Rsin( t)), with X'(t) =(-Rsin(t),Rcos (t)). The velocity vector has constant length R and it is always perpendicular to the position vector. However if we consider Y'(t) =(Rcos(t2) ,Rsin(t2)), then Y'(t) =2tR(-Rsin(t2) ,Rcos(t2)) so the length of the velocity vector equals 2|t|R, a changing quantity; however Y'(t) is still perpendicular to Y(t) for all t. | |||||
Investigate the functions Y(t)
=X(t3+t) for the
curves discussed earlier. When you enter a reparametrization, you
want the graph to have the same endpoints as it did before. To
achieve this you will need to divide the suggested t(u) by some
contants and adjust the domain accordingyly.
More generally, investigate Y(t)
=X(u(t)) for a reparametrization u(t) such that u'(t) > 0 for all t.
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The speed of a parametrized curve at a point X(t) is defined to be the length of the velocity vector at the point. We can obtain this length by taking the square root of the dot product of the velocity vector with itself. | |||||
We denote the speed function by either
s'(t)
or
ds/dt. The speed may be thought of as the rate of change of the length of the curve
with respect to time. We can then integrate the speed from
t=a
to
t=b, the endpoints of the domain, to get the total length of the curve between
the endpoints
X(a)
and
X(b).
| |||||
More generally, the arclength function
s(t)
of a curve from a point
s(t0)
to a variable point
s(t)
is given by
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Arclength is one of the few properties that a curve has. It is a "natural parameter", and for a curve without singularities, it provides a way of "uniformizing" a curve. If you make a trip along a road, you can note the time at which you pass various landmarks, and then someone else could follow the same route and know when to look up to see a particular sight. But in order to do that, the second person would also have to know the velocity at each time. More uniform is the standard process of indicating noteworthy sights by telling how far along the route they are, for example, a rest stop at milestone 134 on the Ohio Turnpike, or a castle twenty kilometers south of Bonn on the road along the Rhein. Of course it is possible to use any unit of measurement, miles, kilometers, or what have you, but two travelers using the same units can be sure to be at the same place at the same time if they walk at one unit per hour. | |||||
As in the case of the velocity vectors, the definition of arclength is nearly identical in three dimensions. This subject is pursued in Arc Length in Lab 3.
Demonstration 5: Speed and Arclength Use the tapedeck to choose a current value tc of t . A pink cross marks X(tc) in the Curve window and the velocity vector at that point is shown emanating from the curve. A vertical pink line marks the current speed s'(tc) in the Speed Versus t window, leaving a series of vertical lines behind it as it moves across the graph. The area filled in by these vertical lines is equal to the distance s(tc) traveled along the curve, marked in the Distance along the Curve Versus t window. | |||||
This demo shows a plane curve, the graph of the speed s'(t) over the domain, and the graph of the distance s(t) along the curve over the domain.
Describe the distance functions and the speed functions for a function graph
of the form
X(t)
=(t,tm)
for various values of
m. | |||||
What does the distance function look like for a curve with a
cusp
, for example,
X(t)
=(t4,t5)
? When will the distance function have a horizontal tangent? | |||||
Calculate
s(t)
explicitly for a circle and for a straight line. Enter the equations for
a circle and for a straight line in the demo (for example, the circle
[cos(t),sin(t)]
, and the line [t,2*t])
and compare your expressions for the arclength with the graphs in
the demo. More generally you can look at a circle of radius R and a
line with slope m. | |||||
Describe the functions
s(t)
and
s'(t)
for the exponential spiral,
X(t)
=et(cos(t,sin(
t)))
. | |||||