Turning Number
One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated on the right has a winding number of 4 (or −4), because the small loop is counted.
This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map.
This is called the turning number, and can be computed as the total curvature divided by 2π.
Read more about this topic: Winding Number
Famous quotes containing the words turning and/or number:
“Then turning to the disciples, Jesus said to them privately, “Blessed are the eyes that see what you see! For I tell you that many prophets and kings desired to see what you see, but did not see it, and to hear what you hear, but did not hear it.””
—Bible: New Testament, Luke 10:23,24.
“To finish the moment, to find the journey’s end in every step of the road, to live the greatest number of good hours, is wisdom. It is not the part of men, but of fanatics, or of mathematicians, if you will, to say, that, the shortness of life considered, it is not worth caring whether for so short a duration we were sprawling in want, or sitting high. Since our office is with moments, let us husband them.”
—Ralph Waldo Emerson (1803–1882)