The case of the disappearing angle units, or "the dangle of the angle"
08-14-2019, 04:08 PM
Post: #21
 Albert Chan Senior Member Posts: 2,516 Joined: Jul 2018
RE: The case of the disappearing angle units, or "the dangle of the angle"
Hi, ijabbott

I am reading Trigonometric Delights, by Eli Maor.
Chapter 4, Trigonometry becomes Analytic, page 53:

Quote:Euler and his Introductio to fully incorporate complex numbers into trigonometry: with him the subject becomes truly analytic.

These developments moved trigonometry ever farther from its original connection with a triangle.
The first to define the trigonometric functions as pure numbers, rather than ratios in a triangle ...
Today, of course, we go a step farther and define the independent variable itself as a real number rather than an angle.
08-16-2019, 01:38 AM (This post was last modified: 08-16-2019 11:55 AM by SlideRule.)
Post: #22
 SlideRule Senior Member Posts: 1,420 Joined: Dec 2013
RE: The case of the disappearing angle units, or "the dangle of the angle"
(07-31-2019 07:34 PM)ijabbott Wrote:  Question
Is it better to ignore angular units and just treat angles as plain old numbers (as long as an angle of 1 corresponds to 1 radian), or does the angular aspect have some cosmic significance that shouldn't be casually discarded? I guess this is more of a philosophical question

"The term circular statistics refers to a particular branch of the discipline of statistics that deals with data that can be represented as points on the circumference of the unit circle. Data of this type are themselves referred to as being circular, a term used to distinguish them from the usual linear data that we are more used to. More formally, we say that the support for circular data is the unit circle (as opposed to the real line which is the support for linear data).
… The requirements to specify the position of the origin and the direction taken to be positive do not arise for data on the real line; the origin is 0, values to the left of 0 are negative and those to the right are positive. For circular data, each angle defines a point on the circumference of the unit circle, just as each value of a linear variable defines a point on the real line … periodic nature of circular data that forces us to abandon standard statistical techniques designed for linear data in favour of those which respect the periodicity of circular data. emphasis mine.
As an illustration of what can go wrong if we treat circular data as being linear, suppose we measured the directions of flight, clockwise from north, of homing pigeons released at a certain location. If the angles measured for four birds were 10◦, 20◦, 340◦ and 350◦ then commonsense tells us that the birds generally fly approximately northwards. However, the arithmetic mean of these angles is 180◦—directly due south!" emphasis mine
Circular Statistics in R {Introduction pg-1}, Oxford University Press, © 2013.

A perspective in response to the aforementioned (hi-lite in red) question.

BEST!
SlideRule
08-16-2019, 12:02 PM
Post: #23
 ijabbott Senior Member Posts: 1,293 Joined: Jul 2015
RE: The case of the disappearing angle units, or "the dangle of the angle"
Thanks SlideRule, I should work out how to do mean directions sometime. It may come in useful!

— Ian Abbott
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