San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley U.S.A. |
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Theorems Concerning the Total Turning Angle of a Curve |
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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 (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
The exterior angles e_{i} of the polygon are
Therefore
But for every triangle
Hence
However Σa_{i} specifically is equal to 2π. Therefore
(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
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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