San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

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 (ai, bi, ci) with ai being the angle impinging upon P.

The interior angles di of the polygon are of the form

di = ci-1 + bi
except for
d1 = cn + b1

The exterior angles ei of the polygon are

ei = π − di

Therefore

Σei = nπ − Σdi = nπ − (Σci + Σbi)

But for every triangle

ai + bi + ci = π

Hence

Σai + Σbi + Σci = nπ

However Σai specifically is equal to 2π. Therefore

Σbi + Σci = (n−2)π
and hence
Σei = nπ − (n−2)π = 2π

(To be continued.)


Appendix:

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) = Tx(t) = cos(α)
and hence
α = cos-1(Tx(t))
where
Tx(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 = ∫Cdα(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.

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