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

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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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1 авг 2024

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Комментарии : 14

@Myrslokstok 12 лет назад

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

@JasonQuackenbushonGoogle 2 года назад

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

@brendawilliams8062 3 года назад

Very nice. Thx. Dr. Wildberger.

@ReeshaEducation 3 года назад

Thank you so much Sir.

@TheAbcdane1 12 лет назад

you're great.

@vice-sama3015 5 лет назад

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

@hellenakinyi7942 Год назад

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

@njwildberger 11 лет назад

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??

@ac-dp3jk 6 лет назад

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 ?

@millerfour2071 5 лет назад

Just replace \theta with 2\pi\tau. As mention by Albert Steiner below. In order to obtain the usual derivative relationships between cos and sin, hence to get their Taylor series and obtain Euler's formula, it is necessary to use the conventional definition so that differential arc length is rd\theta. So in that respect, the tangle is not very useful.

@njwildberger 12 лет назад

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 12 лет назад

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