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In this section we want to briefly discuss the curvature of a smooth curve (recall
that for a smooth curve we require 
is continuous and ).
The curvature measures how fast a curve is changing direction at a given
point.
There are several formulas for determining the curvature for
a curve. The formal definition of
curvature is,
where is the unit tangent and s is the arc length. Recall
that we saw in a previous section
how to reparameterize a curve to get it into terms of the arc length.
In general the formal definition of the curvature is not
easy to use so there are two alternate formulas that we can use. Here they are.
These may not be particularly easy to deal with either, but
at least we don’t need to reparameterize the unit tangent.
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Example 1 Determine
the curvature for .
Solution
Back in the section when we
introduced the tangent vector we computed the tangent and unit tangent
vectors for this function. These were,
The derivative of the unit tangent is,
The magnitudes of the two vectors are,
The curvature is then,
In this case the curvature is constant. This means that the curve is changing
direction at the same rate at every point along it. Recalling that this curve is a helix this
result makes sense.
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Example 2 Determine
the curvature of .
Solution
In this case the second form of the curvature would
probably be easiest. Here are the first
couple of derivatives.
Next, we need the cross product.
The magnitudes are,
The curvature at any value of t is then,
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There is a special case that we can look at here as
well. Suppose that we have a curve given
by and we want to find its curvature.
As we saw when we first looked at vector functions we can write this
as follows,
If we then use the second formula for the curvature we will
arrive at the following formula for the curvature.