|San José State University|
& Tornado Alley
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.
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
The exterior angles ei of the polygon are
But for every triangle
However Σai specifically is equal to 2π. Therefore
(To be continued.)
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
This turn angle α may also be constructed by
The angle A after the completion of a circuit C can be represented as
If α(s) is differentiable then
otherwise Δα is the exterior angle of the curve at that point, the discontinuity.
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