Rational curvature, winding and turning | Algebraic Topology | NJ Wildberger

This video introduces an important re-scaling of curvature, using the natural geometric unit rather than radians or degrees. We call this the turn-angle, or tangle, and use it to describe polygons, convex and otherwise. We also introduce winding numbers and the turning number of a planar curve.

This is the 11th lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J WIldberger at UNSW
Video Contents:
00:00 Introduction
01:20 Angle(Spread Rational Trigonometry- Overview
02:50 Q. How to describe the amount of turning?
06:40 Turn of the basics unit
10:00 Turn angles
15:54 Quadrilateral Turn angles
17:48 n-gon computation
19:48 Convex polygon
22:50 General n-gon
25:00 Alternate approach of the T angle
28:00 Total curvature of Convex Polygon
30:45 Winding numbers of a curve
34:48 The turning on different points of the curve
41:10 Turning number of a smooth curve

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[Myrslokstok]

The Prof is very easy to listen to and it gets better and better.

[hellenakinyi]

Video Content
00:00 Introduction
01:20 Angle(Spread Rational Trigonometry- Overview
02:50 Q. How to describe the amount of turning?
06:40 Turn of the basics unit
10:00 Turn angles
15:54 Quadrilateral Turn angles
17:48 n-gon computation
19:48 Convex polygon
22:50 General n-gon
25:00 Alternate approach of the T angle
28:00 Total curvature of Convex Polygon
30:45 Winding numbers of a curve
34:48 The turning on different points of the curve
41:10 Turning number of a smooth curve

[JasonQuackenbushonGoogle]

I'm so glad i found this video. it clarified a lot of the things id been trying to figure out about circular functions related to rational trig

[njwildberger]

No, in that case there is no contribution. You can see that by moving the curve just a little, so that north is no reached, or north is passed through in one direction and then immediately after that in the other direction.

[njwildberger]

If you must have a ''unit'', might it be ``turn''? So e.g. we go 1/4 turn when we go from horizontal to vertical etc.

But I think it good to have a discussion about whether a unit is actually required here, since the rational turn angle is a ratio of two similar things. Is there such a thing as a dimensionless unit??

[vice-sama]

Sure buddy. Sure. Very revolutionary of you to think in such a novel way. Can we get a fields medal over here. That aside, nice series of lectures

[njwildberger]

Admittedly the two concepts are pretty close, but often we want to think of a turn angle as a measurement, ie a number.

[njwildberger]

The usual way circular functions like sin x are ``defined'' is to treat x as a ``real number'', without any dimensions.

[ac-dpjk]

Sir, thank you very much for this great lecture. I have a question: Using tangles, do the complex numbers need re-scaling too ? For example, what happens to formulas like e^(i\pi)=-1 ?