Re: Sine and Cosine Accuracy

[Marc Espie <espie@nerim.net>]

On Sun, May 29, 2005 at 08:59:00PM +0200, Georg Bauhaus wrote:
> Marc Espie wrote:
> >Sorry for chiming in after all this time, but I can't let this pass.
> >
> >Scott, where on earth did you pick up your trig books ?
> 
> Sorry, too, but why one earth do modern time mathematics scholars
> think that sine and cosine are bound to have to do with an equally
> modern notion of real numbers that clearly exceed what a circle
> has to offer? What is a plain unit circle of a circumference that
> exceeds 2???
> How can a real mathematical circle of the normal kind have
> more than 360 non-fractional sections?
> By "real circle" I mean a thing that is not obfuscated by the useful
> but strange ways in which things are redefined by mathematicians;
> cf. Halmos for some humor.

Err, because it all makes sense ? Because there is no reason to do stuff
from 0 to 360 instead of -180 to 180 ?

> And yes, I know that all the other stuff mentioned in this thread
> explains very well that there exist useful definitions of sine for real
> numbers outside "(co)sine related ranges", and that these definitions
> are frequently used. Still, at what longitude does your your trip around
> the world start in Paris, at 2°20' or at 362°20', if you tell the story
> to a seaman? Cutting a pizza at 2.0^90. Huh?!

At 0.0. Did you know that, before Greenwhich, the meridian for the
origin of longitude was going through Paris ? Your idea would make some
sense if you talked about a latitude (well, even though the notion of
north pole is not THAT easy to define, and neither is the earth round).

Heck, I can plot trajectories on a sphere that do not follow great circles,
and that extend over 360 degrees in longitude.  I don't see why I should be
restricted from doing that.

> Have a look into e.g. "Mathematics for the Million" by Lancelot
> Hogben for an impression of how astounding works of architecture
> have been done without those weird ways of extending angle related
> computations into arbitrarily inflated numbers of which no one knows
> how to distinguish one from the other in sine (what you have dared to call
> "obvious", when it is just one useful convention. Apparently some
> applications derive from different conventions if I understand Scott's
> remarks correctly).

There are some arbitrary convenient definitions in modern mathematics.
The angle units have been chosen so that derivation of sine/cosine is 
obvious.  The definition of sine/cosine extends naturally to the whole
real axis which gives a sense to mechanics, rotation speeds, complex functions
and everything that's been done in mathematics over the last four centuries
or so.

You can decide to restrict this stuff to plain old 2D geometry, and this would
be fine for teaching in elementary school, but this makes absolutely 
no sense with respect to any kind of modern mathematics.

Maybe playing with modern mathematical notions for years has obfuscated
my mind ? or maybe I just find those definitions to be really obvious and
intuitive.   Actually, I would find arbitrary boundaries to be unintuitive.

There is absolutely nothing magical wrt trigonometric functions, if I
compare them to any other kind of floating point arithmetic: as soon as
you try to map `real' numbers into approximations, you have to be VERY wary
if you don't want to lose all precision.  There's nothing special, nor
conventional about sine and cosine.

Again, if you want ARBITRARY conventions, then look at reverse trig functions,
or at logarithms. There you will find arbitrary discontinuities 
that can't be avoided.