If we have a plane curve 
where 
are smooth (infinitely differentiable) functions of their parameter t,
then if the geometry of the plane is fixed once and for all, so that distance
has a meaning, we see that the shape of the curve should not depend on the parameter.
If it is a circle for my parameter, it should be a circle for yours. A different
parameterization of 
,
say
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would yield the same circle. So we will seek geometric properties of plane curves that do not depend on the parameterization.
The property that we are after
in this section is how the curve "bends". As we saw in the previous
Chapter with the radius of curvature, this is a positive radius that we can
associate with each point on the curve. When it is infinite, the curve is locally
flat. The smaller it is, the more the curve "bends". We will call
the reciprocal of this number the curvature of the curve at each
point. Our approach will seem roundabout -- all first approaches to this idea
are -- but it is closely related to the polar coordinate approach that has served
us well so far in the analysis of satellite orbits.
The idea is that each smooth
curve 
defines another curve 
and
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We call 
the velocity curve. In all that follows in this section, we will assume
that
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That is, the velocity never vanishes.
We may thus follow the strategy
of the previous Chapter. If we connect the point 
to the origin 
by an imaginary line, that line intersects the unit circle, the
set of points at distance 1 from the origin, in a unique point 
. So the curve 
is the projection of the curve 
onto the unit circle. Represent the velocity curve
Let
us construct a "velocity moving frame" at 
, with unit vectors 
where 
is a vector of length 1 tangent to 
at 
and 
is obtained by rotating 
by 90 degrees counterclockwise.
Now, let 
be the acceleration vector
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Then
we may represent this acceleration vector in the moving frame 
as
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where of course 
are smooth real valued functions. We may think of 
as the "radial" component of acceleration 
and 
as the "angular" component of the acceleration.
Recalling that 
and that 
is never 0, we will see that 
is an important invariant of the curve, independent of its parameterization.
We will interpret it in the next section in terms of arc-length parameterization,
and again in the following section in terms of the 'Gauss map'.
Measure of how a curve bends: parameterization independence
Theorem: Suppose that 
we define the curve 
and if 
then 
. This defines a new curve 
with parameter s. Now 
is independent of the parameterization in the sense that if we write 
then at 
we have the same velocity frame 
.
And if we write the acceleration
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in the form
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and if
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then
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Proof: The proof of this fact is an exercise in the chain rule. Since
it follows that
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This is not 
. Thus the unit vector
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So the velocity frame at 
is 
.
Now using the chain rule again, we see that
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If we write 
we see that
and since 
Now
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so
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and we see that
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Now you are probably asking
yourself: So, what does this mean? First, notice that 
is what we called the angular component of acceleration. There is no
such invariant quantity for 
, the radial component of the acceleration. The ratio of this angular
component with the squared speed (think of kinetic energy) measures in an intrinsic
sense how the curve "bends" at a point.
We will have in the next
section a more vivid interpretation of the invariant 
. For now, we call it the curvature of the curve c at the point 
.
Exploration: Change of parameter
The exploration for this
page will help you visualize this invariance. As we did in the Radius of Curvature
section of the previous Chapter, we let you define a curve. This time, the domain
is restricted to 
so you may want to use trigonometric functions to define the curve.
Begin by defining your curve
by giving its component functions
The smallest and largest values for t are now 0 and 
Next,
Click Graph to
draw the curve. 
You will then be able to select a point on your curve by clicking in the window at the bottom of the screen and dragging the pointer to the desired position. Right click to select it. As you do, a sprite will move along the curve.
At the same time, the current values for the point will be continually displayed at the top of the window.
Finally, press the 
button to draw the osculating circle. You will see something like:
The light red vector is 
.
The projection of this vector on the dark red one gives the curvature (here,
roughly 0.5574). This projection when multiplied by 
(the squared speed) is the acceleration a particle would feel if it moved uniformly
about the circle with speed. 
.
The light red vector when multiplied by 
is the actual acceleration at that point:
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The small graph2d on the
left shows a magnified version of the velocity frame with 
in it:
The absolute value of the vertical component of the light red vector in the frame is the invariant that we call the curvature.
Notice that the horizontal
component corresponds under a change of parameterization (as above) to
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and so is not invariant, although the change introduced by the term 
can be quite small.
Now to see the invariance,
you may change the parameter for this curve with the slider in the upper right:
The 4 choices, say n, give new parameterization with 
These nonlinear changes of
parameter can help you see the invariance. Of course, if 
you have the original parameterization of the curve.
Once you set the slider to your choice, press 
A slightly different picture
will be drawn in the graph2d
The other picture will not be changed at all. The original coordinates of the tip of the light red arrow in the frame were: [0.2022674,-0.5574626]
The new coordinates with 
are reported just below the button as [0.2216276,-0.5574626]. You see
that the first coordinate has changed slightly -- very slightly, but the second
component remains the same. For 
there is further change in the first coordinate, [0.2313077,-0.5574626]
but the second coordinate remains the same.
Experiment!
