Arc Length and Curvature

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Arc Length and Curvature

• By Dr. Julia Arnold

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• Objectives
• Find the arc length of a space curve.
• Use the arc length parameter to describe a plane
  curve or space curve.
• Find the curvature of a curve at a point on the
  curve.
• Use a vector-valued function to find frictional
  force.

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Objective 1

• Find the arc length of a space curve.

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Given a smooth plane curve C that has parametric
equations x x(t) and y y(t) where
, the arc length s is given by
(See Section 10.3)
In vector form, where C is given by r(t)x(t)i
y(t)j, the above equation can be written as
We can extend this formula to space quite
naturally as follows
If C is a smooth curve given by r(t) x(t)i
y(t)j z(t)k on an interval a,b, then the arc
length C on the interval is
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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To help you visualize what is taking place look
at the curve and imagine taking steps from point
to point.
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Now let the points get closer and closer together
and sum them up.
Lets look a a few examples
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Example 1 Find the length of the space curve
over the given interval.
Set up the integral Find the derivative of the vector. Substitute into the formula. Simplify Integrate and evaluate.
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Objective 2

• 2. Use the arc length parameter to describe a
  plane curve or space curve.

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Curves can be represented by vector-valued
functions in different ways depending on the
choice of parameter. For example the following
two representations are equivalent. For
motion along a curve the most convenient
parameter is time t. However, for studying the
geometric properties of a curve, the convenient
parameter is often arc length s.
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If C is a smooth curve given by r(t) x(t)i
y(t)j z(t)k on an interval a,b, then the arc
length of C on the interval a,b , with alttltb is
C s(t) which is
The arc length s is called the arc length
parameter.
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Example 2 Consider the curve represented by the
vector-valued function
A. Write the length of the arc s as a function
of t by evaluating the integral
Solution
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Example 2 Consider the curve represented by the
vector-valued function
B. Solve for t in part A and substitute the
result into the original set of parametric
equations. This yields a parameterization of the
curve in terms of the arc length parameter s.
Solution
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Example 2 Consider the curve represented by the
vector-valued function
C. Find the coordinates of the point on the
curve for arc lengths
Solution
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Example 2 Consider the curve represented by the
vector-valued function
C. Find the coordinates of the point on the
curve for arc lengths
Solution
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Example 2 Consider the curve represented by the
vector-valued function
D. Verify that
Solution
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This brings us to a Theorem about the arc length
parameter, namely
If C is a smooth curve given by
Where s is the arc length parameter, then
Moreover, if t is any parameter for the
vector-valued function r such that Then t must
be the arc length parameter.
This theorem is stated without proof.
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Objective 3

• 3. Find the curvature of a curve at a point on
  the curve.

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Curvature An important use of the arc length
parameter is to find curvature.
Curvature is the measure of how sharply the curve
bends.
For example, in this helix we get more bend
here
Than here.
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We can calculate curvature by calculating the
magnitude of the rate of change of the unit
tangent vector T with respect to the arc length s.
T2
T3
T1
Definition of Curvature Let C be a smooth curve (
in the plane or in space) given by r(s), where s
is the arc length parameter. The curvature K at
s is given by
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Example 3 Find the curvature using s is the arc
length parameter, for
Solution This was the problem we did earlier
and found the arc length parameter to be
and the function to be Using
the formula for curvature K in terms of arc
length s, namely
we get
and knowing that
Since curvature K is
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Example 3 Find the curvature using s is the arc
length parameter, for
Solution
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c
Using winplot, this is the curve in question.
Since s
In terms of t the curvature would be
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We can see that the curvature of a circle is the
same everywhere and reason it to be a constant
which turns out to be 1/r where r is the radius
of the circle. See example 4 in your text.
Other formulas for curvature. Since the previous
definition depends on the arc length parameter,
it might be good to have some alternative
definitions which depend on an arbitrary
parameter t.
Two formulas for curvature Theorem 12.8 If C is a
smooth curve given by r(t), then the curvature K
of C at t is given by
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Example 4 using the alternative curvature
formula on the same vector-valued function
We can compare our answers. From Example 2A we
already know that
Next we need to find T(t).
Now we need T(t)
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Using the formula
Now we find
Which is what we got back on slide 37
Click on the purple crayon to get back to this
slide.
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Using the other formula
We have
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Solution to question on previous slide.
At (4,0) the curvature would be
Thus 25/2 would be the radius of the circle which
would be approximately 5.66
Using winplot I found the normal to the tangent
line at x 4 (blue line) and then found the
center to be approximately at (0,-4).
Figure 12.37
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Objective 4

• Use a vector-valued function to find
  frictional force.

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Example 6
Solution
Continued
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Example 6
Solution continued
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Figure 12.38
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Hint for HW Confusion of Mass and Weight A few
further comments should be added about the single
force which is a source of much confusion to
many students of physics - the force of gravity.
The force of gravity acting upon an object is
sometimes referred to as the weight of the
object. Many students of physics confuse weight
with mass. The mass of an object refers to the
amount of matter that is contained by the object
the weight of an object is the force of gravity
acting upon that object. Mass is related to how
much stuff is there and weight is related to the
pull of the Earth (or any other planet) upon that
stuff. The mass of an object (measured in kg)
will be the same no matter where in the universe
that object is located. Mass is never altered by
location, the pull of gravity, speed or even the
existence of other forces. For example, a 2-kg
object will have a mass of 2 kg whether it is
located on Earth, the moon, or Jupiter its mass
will be 2 kg whether it is moving or not (at
least for purposes of our study) and its mass
will be 2 kg whether it is being pushed upon or
not. On the other hand, the weight of an object
(measured in Newtons) will vary according to
where in the universe the object is. Weight
depends upon which planet is exerting the force
and the distance the object is from the planet.
Weight, being equivalent to the force of gravity,
is dependent upon the value of g.
On earth's surface g is 9.8 m/s2 (often
approximated as 10 m/s2) or 32 feet/s2. On the
moon's surface, g is 1.7 m/s2. Go to another
planet, and there will be another g value.
Furthermore, the g value is inversely
proportional to the distance from the center of
the planet. So if we were to measure g at a
distance of 400 km above the earth's surface,
then we would find the g value to be less than
9.8 m/s2.
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For comments on this presentation you may email
the author Dr. Julia Arnold at jarnold_at_tcc.edu