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Theorems Concerning the Total Turning Angle of a Curve

When a curve such as the following one is traversed the direction of travel completes a full circle of 2π radians.

In the following it is assumed that a curve is transversed in a counterclockwise direction; i.e., movement along the curve
is with the interior of the curve on the left.

Let A be the accumulated change in the direction angle for the curve at the beginning and end of its traversing.

If the curve is simple; i.e., non-self-intersecting then A is equal to 2π. The starting point on the curve for the traversing does not matter.

For a figure 8 curve A is equal to zero.

For the curve below A is equal to 2(2π)=4π.

The Net Turning Angle for a Polygon

Consider a polygon with n sides (and corners). For now its interior will be assumed to be convex and later
the consequences of any nonconvexity will be examined.

Choose a point P in the interior of the polygon and draw straight lines from P to each of the corner points.

This
creates n triangles. Let these triangles be labeled sequentially from 1 to n and their angles designated (a_{i}, b_{i}, c_{i})
with a_{i} being the angle impinging upon P.

The interior angles d_{i} of the polygon are of the form

Let X(t) be the vector of the coordinates x(t) and y(t) of a plane curve. The functions x(t) and y(t) are continuous and at least one of the right and left derivatives X'(t_{-}
and X'(t_{+}, exist at every point. These derivatives define unit tangent vectors, T(t_{-} and T(t_{+}, at each point, where possibly the right and left unit tangents may be
different. LIkewise there are unit normal vectors, N(t_{-} and N(t_{+}, defined at each point.

The angle α between a unit tangent vector and the unit vector in the x direction is given by

T(t)·(1, 0) = T_{x}(t) = cos(α)
and hence
α = cos^{-1}(T_{x}(t))
where
T_{x}(t) = x'(t)/[(x'(t))²+(y'(t))²]^{½}

This turn angle α may also be constructed by

dα/dt = k(t)
if T(t_{-})=T(t_{+})
and otherwise
Δα(t) = cos^{-1}(T(t_{-})·T(t_{+}))

The angle A after the completion of a circuit C can be represented as

A = ∫_{C}dα(s)

If α(s) is differentiable then

dα/ds = k, the curvature

otherwise Δα is the exterior angle of the curve at that point, the discontinuity.