We have learned that given
a smooth curve 
with non-vanishing derivative 
for the parameterization 
, then there is a property of the points in the curve itself which is independent
of the parameterization. That property is the curvature at the point, or viewed
another way, the radius of the osculating circle.
We say that a mapping
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is an isometry if it preserves distances. That is if 
are any points in the plane 
then
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Orientation preserving isometries
Orientation-preserving isometries
(ones that would map the letter F into something recognizable as the letter
F and not its mirror image) are compositions of translations and rotations about
points. All isometries can be obtained as compositions of reflections across
lines. An even number of reflections preserves orientation. An odd number of
reflections does not preserve orientation. All isometries are 1-1 mappings of
the plane onto itself.
Curvature as (geo)metric property of a curve
Now if 
is a curve, then for any isometry 
is another curve. It is not difficult to see from our velocity frame interpretation
of curvature that the curvature of 
is the same as the curvature of 
.
The image of the curve
has the same size and shape as the image of the curve 
.
Curvature is therefore called a metric property of points on a curve.
In fact, curvature is invariant under all transformations
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where 
is an isometry and 
is a smooth "change of parameter" a map with non-vanishing derivative.
In this section, we will give a final, beautiful interpretation of plane curvature in the spirit of Gauss' construction for surfaces. The interpretation is easiest to describe for closed curves, that is, smooth mappings
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where
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and 
has a smooth extension to a slightly larger open interval so that all the derivatives
of 
at a are equal to the corresponding derivatives of 
at b. All of the examples in the explorations of this section are closed
curves. These curves may be considered in more advanced settings to be maps
from the unit circle to the plane. To see where we are going, recall the velocity
frame for curves with non-vanishing velocity. We constructed 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 as we traverse the curve
from a to b, (or really back to a) watch the progress of
the vector 
. You see that 
describes a curve that remains entirely in the unit circle. The mapping 
will be called the Gauss map for the curve 
.
For each t, the vector 
, if translated up to the curve at the point 
would be perpendicular to the curve, or, as we sometimes say, normal
to the curve. If we leave the tail of 
at the origin, so that its head is in the unit circle, we see that, on one circuit
of the curve, 
returns to its starting point: 
.
Next, imagine that 
carries along with it a string that it lets out as it moves. It may happen that
when it returns to the starting point, the string wrapped once around the circle.
Or perhaps it wrapped twice or three times. It might not be wrapped at all (0
times), or it may wrap in the opposite sense from the sense of traversal of
the curve.
In each case, an integer
is determined: the number of times the "string" of
wrapped around the unit circle. This integer is another invariant of the curve
and is intimately related to the curvature as we will see. In fact, if the curve
does not cross itself, the integer is 
depending on the sense in which the curve was traversed. If it crosses itself
once the integer is 0 or 
as in the pictures below.
Winding number 0:
Winding number 2:
As
the curve 
is traversed, at each point, it is traversed with a certain positive speed:
as we have observed. We may consider for a small interval of "time"
starting at t, that a length of curve is traversed, 
,
that is approximately equal to 
. This is because, as we showed, 
.
Now suppose that the velocity
of 
along the unit circle (which is a real number that may be positive, negative,
or zero) is denoted 
. We will explain how to calculate it below. Then 
is the approximate extent of arc on the unit circle traversed in the interval
.
We will show below that the
ratio of these numbers: 
is a number whose absolute value is the curvature at the point 
. It is the signed curvature with positive sign if the circle
is traversed in the counter clockwise sense, and negative sign if it is traversed
in the clockwise sense.
For example, if we revisit
the winding number 0 case illustrated above, we see easily that as the curve
is traversed in the counterclockwise sense (following the red arrow) the normal
vector is turning in the clockwise sense in the region indicated below:
In this region, the signed curvature is negative.
Now
suppose we want to approximate the amount of arc on the unit circle through
which 
"turns" as t varies in the interval 
. One way to do that would be to choose a partition
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of the parameter interval, with all 
small. Then call 
the length of arc on the curve 
between 
.
Then
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is an approximation to the (signed) arc on the unit circle that 
turns through in the interval 
.
Therefore we may suspect
that if 
is the curvature at the point 
then
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will be equal to the change in arc on the unit circle as t varies in
the interval 
.
This reasoning depends on
two separate things: First, we must justify the claim that 
is the curvature 
and interpret the signed curvature 
, and next, we must show that 
is really an approximating function for the change in arc. Once we do that,
we will formulate things in invariant form.
Digression: Let us take a short detour to see how this idea fits into
"modern" mathematics (circa 1820). What is really being asserted is
that if the curve does not intersect itself, then if the image of the curve
is denoted 
then we may consider the curvature 
to be defined on that image! This idea is a starting point for many of the ideas
of modern algebraic and differential topology.
Integral curvature and manifolds
Gauss realized that 
is an object in its own right, now called a manifold. It is (except
for shape) essentially a circle. It is possible to integrate functions over
such objects. We thus may speak of calculating
And Gauss saw that when this
metric concept was extended to 2-dimensional surfaces, another invariant
number would emerge, and this invariant is called Gaussian curvature. Riemann
later extended this construction to surfaces of all dimensions, and still later,
Einstein saw in this generalization (Riemannian Geometry) the basis for his
4-dimensional General Theory of Relativity, which is, as we said, the modern
theory of gravity. The basic fruitful idea is that the curvature is "really"
an intrinsic property of the points in 
(or the surface or manifold) itself and does not depend on how we describe 
to ourselves (the parameterization). We say that spacetime with its distribution
of matter and energy has an intrinsic curvature. And one manifestation of that
curvature is gravity.
We say that two manifolds
(like curves or surfaces) are topologically equivalent if one can be
deformed continuously and smoothly into the other, allowing stretching and turning,
but not tearing. It happens that 2-dimensional closed surfaces come in a much
greater variety of topologically distinct shapes than do simple 1-dimensional
curves. The latter are all topologically equivalent to circles! But in 3-dimensions,
we can have such inequivalent surfaces as:
The
sphere has constant Gaussian curvature, just as the circle has constant plane
curvature. The Gaussian curvature of the torus varies from point to point. But
Gauss observed (in this more advanced context of surfaces) that the integral
of the Gaussian curvature over the surface is independent of the shape
and size of the surface. It is a multiple of 2p and two surfaces are topologically equivalent
if and only if these integrals are equal! The content of the Gauss-Bonnet
theorem is that this integral is 
where 
is an integer associated with any surface called its "Euler Characteristic."
For the sphere, 
and for the torus, 
. These surfaces are different!
Returning
to plane curves, we will see that the integral of the plane curvature 
over 
is independent of the shape of 
.
It is always equal to 2p, because
that is the arc that 
winds through when the curve does not cross itself. Of course we have not, and
will not, prove that last fact. We will be content simply to interpret 
as the arc that 
winds through.
Notice that we have made
a subtle shift of viewpoint. We are speaking now of the integral of a function
defined directly on 
: 
.
Above, we spoke of calculating the integral of a function defined on 
:
.
In order to justify this way of thinking, we will have to show that the latter
integral 
is, like the curvature itself, independent of the parameterization
. This brings us to the "change of variable" formula for integration
which is the door to the modern understanding of the concepts discussed above
in the digression.
Let us return to the picture
of integration discussed in the previous page. In that view, we imagine that
there is a parameter or variable that varies from one fixed
number to another, and determines as it increases, the growing and shrinking
amounts of the quantity to be measured. This simply means that the quantity
to be measured, call it Q, is a function of the parameter, call it t.
So Q depends on t and it is a continuous function of t.
We imagine that at some starting value of the parameter: to
we know the value Q(to) and then that we would like to determine
the value of Q at a "later" value of t: t1
> to, that we will call 
.
Given a quantity Q
as above, a continuous function defined on a parameter interval 
, with parameter t. Then a continuous function
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was called a general approximating function of order k for Q if there is a continuous function
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with
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and such that if 
is small enough, and 
,
then there are 
such that
.
Then the Fundamental Theorem
of Calculus asserts that Q is differentiable, and moreover
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The trick as we saw, is to discover a general approximating function f that really does give Riemann Sum approximations to the quantity you want to measure. Then, if you can antidifferentiate f, that is, find a function algebraically that satisfies:
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then the quantity 
is
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for that antiderivative 
.
Now it often happens that
there will be other ways to parametrize the quantity Q. In practice this
means that there is an increasing differentiable function 
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with
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Then we may think of s as a new parameter
for 
where 
.
We may find it convenient to measure
using this new parameter. If we do, then we have made a "change of variable".
Now suppose that 
is an approximating function for Q with parameter t, so that
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If we have an antiderivative for f, a function P such that 
, then of course the chain rule tells us that
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So in that case, 
is an antiderivative for 
and so
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Change of variable strategy in integration
This formula allows us to
measure 
with respect to a new parameter s if we can antidifferentiate the
approximating function 
.
So it is not really useful, except as a heuristic to help us remember what to
do in the important case where cannot find an antiderivative for
algebraically. In that case, it may still be possible to change variable in
such a way that we can find an antiderivative algebraically for the new function
.
We will show that in any
case, 
is an approximating function for 
and so,
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and so if we can find an antiderivative 
for 
, we will have measured 
.
It will be 
. This is neatly summarized in the "change of variable" equation:
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Theorem 1: The function
is an approximating function for 
if 
is an approximating function for 
.
Proof: Suppose that continuous function
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is a general approximating function of order k for Q. Thus, there is a continuous function
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with
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and such that if 
is small enough, and 
,
then there are 
such that
.
Now let the corresponding
partition for 
be denoted
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and let 
. Then
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and there is a 
such
that 
by the Mean Value Theorem.
Therefore
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So if we let
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by
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then the function
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is an approximating function for 
.
Change of integration variable formula
This fact justifies
the change if integration variable formula:
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For example, if we wanted to calculate the integral
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Then we may observe that
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has positive derivative on 
and that it is not difficult at all to calculate
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by finding an antiderivative of 
. This is an example of trigonometric substitution.
We need the change
of integration variable formula to justify the idea that, for a given (simple,
closed) curve 
the image set 
is a geometric object in its own right, independent of the parameterization
( 
)
that we used to present it. We will know that the (signed) curvature 
at a point 
depends only on P, and not on the parameterization. But we would now
like to speak about the integral of the function 
over 
. We want to give meaning to the symbol:
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To that end, imagine
that we have divided 
into a sequence of adjacent arcs as we did in the previous exploration. Let
be one of those arcs. At each point 
the signed curvature 
will be defined as sketched above by a procedure that we will explain later.
For now, let us assume it is defined. In fact, for the purpose of this discussion,
could
be any function defined on J.
Now the idea is to
define 
using a particular oriented parameterization, and then to show that we will
get the same number if we calculate this interval using another oriented parameterization.
"Oriented" means that both parameterizations go in the same direction.
Thus, let J
be the arc 
(oriented by starting at A and ending at B) and choose an interval
along with a parameterization of the arc 
where 
never vanishes. As before, call
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Definition: Let the oriented integral 
be defined to be the number 
.
To say the integral is oriented simply means that the parameterization satisfies
. We will now show that this number does not depend on the parameterization
.
Theorem 2: If 
is a parameterization of 
satisfying 
never vanishes
and if 
is a second parameterization of 
satisfying 
never vanishes
and if 
and 
,
then
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Proof: We define the "change of variable" 
defined by the rule
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It is a straightforward if technical matter to show that the maps 
and 
are 1-1 and onto (bijective) and have differentiable inverses (see the Inverse
Function Theorem). We will not enter into the details of that plausible assertion.
Now apply the change of integration variable formula to write
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Unpacking this with the observation 
we have
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Finally, we will show from the chain rule that
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and that will finish the proof.
Write 
Then differentiate both sides with respect to s,
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Now for the velocity
frame (which is the same for both parameterizations at 
since both parameterizations have the same orientation) let the unit radial
vector be 
. Then,
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and
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So substituting, we see that
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From this, it follows
that
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so
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We now know what it
means to integrate the curvature 
over 
.
And we define
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for any parameterization of 
going once around.
Gauss' theorem and total winding
This brings us finally
to Gauss' beautiful insights. That is that
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as the simple (non self intersecting) curve is traversed once. And in general
for closed curves that may be traversed several times by some parameterization:
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We will prove both statements at once.
Recall the construction
in the Radius of Curvature exploration of the previous Chapter: Conservation
of Energy.
For a given parameterization of the curve: 
, we constructed a "velocity moving frame" at 
,
with unit vectors 
where 
is a vector of length 1 tangent to c at 
.
In particular,
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where 
is the speed, that is 
and 
and 
is obtained by rotating 
by 90 degrees.
Then we let 
be the acceleration vector
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We then expressed the vector 
in terms of the velocity frame.
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Finally, we learned that the number 
,
the 
coordinate of the vector 
is independent of the parameterization. It what we have called above the signed
curvature 
.
The absolute value of this number is the reciprocal of the radius of curvature
at 
and we have called it the plane curvature 
at 
.
Now, you will see
in the exploration that there is a function of the parameter t that determines
the angle 
that the frame vector 
makes with the positive x-axis (measured in radians and in the counter-clockwise
sense.) We can thus write
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and since 
is rotated 
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This angle varies
with t of course. Let us also write the unit vector 
in coordinates as 
,
so
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Question 1: Write the acceleration at parameter t as 
.
Then show that the projection 
of 
on 
is
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We
use this fact to derive another characterization of 
. Since
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it is clear that
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This just means that
the normal vector is always perpendicular to the velocity vector. We will explain
the algebra of "dot products" in a later section, since we do not
need the details here.
It follows that
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When we do the differentiation we see that
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Now
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as you saw in Question 1.
Therefore,
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Now since 
it follows that
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and since 
,
we have finally that
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The signed curvature
is therefore the ratio of the rate at which 
sweeps out angle as it moves in the unit circle, and the rate at which
sweeps out arc length as it moves along the curve. 
may change in the positive or the negative sense -- or not at all. The speed
is always positive.
So 
We see therefore that
for general closed curves: 
.
This total winding is sometimes called the "integral curvature."
Now the exploration
for this page gives an interactive illustration of Gauss' construction in 1-dimension.
We first construct, as on previous pages, a smooth mapping
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where
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and 
has a smooth extension to a slightly larger open interval so that all the derivatives
of 
at 0 are equal to the corresponding derivatives of 
at 
. These curves may be considered in more advanced settings to be maps from the
unit circle to the plane.
Begin by entering
the component functions for your curve
Then use the control buttons 
to draw your curve.
If you use the default
curve you will see
where the point 
is marked with the small red sprite, and the unit normal vector 
of the velocity frame is marked by a dark red arrow sprite.
You will also see
in the lower left corner, now translated to the origin and determining a point
on the unit circle.
Now the aim of the exercise
is to illustrate the idea that the integral of the signed curvature over an
arc 
is the change in angle of 
. And the integral of the signed curvature over the entire arc 
is an integer multiple of 
. That integer is sometimes called the "winding number" of 
.
The idea behind the Gauss
map construction that maps the curve 
to the unit circle 
is that it "wraps" the curve around the circle an integer number of
times, and that integer provides a topological description of the curve itself.
In modern topology, such an integer is characterized as an integral cohomology
class of 
.
You may do the experiment
by determining T in the arc 
.
Simply move the slider
and the vector 
will move along the curve, and you may watch it "wind" in the smaller
graph2d on the left. The current value of T is reported both in radians
below the slider and as a multiple of 
to the right of it.
For example, if we move t
through 
radians, we see
which indicates that the normal vector has turned almost 
radians in the counter clockwise sense. We can see this by following the arrow
as we move the slider, but we get better information if we calculate the integral
of the curvature on the arc corresponding to parameter interval 
. Do this by pressing the Integrate Curvature button.
The length of arc along the
curve 
as we vary t from 0 to 
radians is reported in radians. Here it is roughly 11.7867891 radians. And the
signed angle through which 
wound as we vary t from 0 to 
radians is reported as a multiple of 
radians. Here it is 
radians, or almost 
radians.
Now calcultate the total
integral curvature for this curve by taking the slider to the right end ( 
radians ).
The sprite returns to its
original position
but press the Integrate Curvature button and you see:
The total winding (integral of the curvature) is 0, but the arc traversed
along the curve 
(the length of the arc) is roughly 47.1471564 radians
Exercise: Create the circle
is traversed twice as t varies in 
. Now calculate the total winding on 
and on 
. What is the radius of this circle? What is the constant curvature?
Exercise: Try to construct a curve that intersects itself at precisely one point for which the integral of the curvature is not 0 as it was in our figure 8 example. Do you see what you have to do?
