🚀 go-pugleaf

Dr John Stockton <jrs@merlyn.demon.co.uk> wrote in message
news:zZIi7jAzW8c3EwDD@merlyn.demon.co.uk...
> >It will be interesting to see whether you've managed to avoid using a
> >value for Pi, implicitly or explicitly, in the work.

> Method 4 uses half-angle and angle-addition formulae to generate sines
> and cosines of evenly-spaced angles, starting with a "unit" of one
> right-angle

There was no angle-addition formula used in Method 4.

The half-angle recurrence formulas only involves a SQRT, not any
trig function.  The starting value is 0.0, i.e., cos(90 degrees).

The generated sequences of SIN and COS values from the
recurrence relationship were just algebraic expressions.
Is your objection to an "implicit" use of PI because SINs and
COSs were used in the explanation?  Sure seems like this could
have been expressed totally algebraically, e.g., SIN(angle) =
x / SQRT(x^2 + y^2) and if SQRT(x^2 + y^2) = 1, then SIN(angle) = x.
But expressing the relationships using the trig functions made
everything more understandable.  I don't see any dependence on
PI here.  I guess I haven't proven I can derive the half-angle formula
using only geometry without a dependence on PI.  Certainly
everything can be expressed just algebraically.

Once a sufficiently small angle was generated, namely 1/(2^N)-th of a
circle, angular rotation  [not angle-addition] was used to generate a
uniform sequence of points along a circular path.  While the rotation
matrix was expressed using SINs and COSs, the values were
calculated using only the algebraic expressions of the half-angle
recurrence formula (above).  So these points were all calculated using
nothing more complicated than a SQRT -- i.e., no trig functions.

What is the explicit dependence on PI in Method 4?
What is the implicit dependence on PI in Method 4?

>I guess that the true intention was to draw attention to the services
>you provide at efg's Computer Lab

My true intentions aren't any different than yours when you
answer math questions in the various newsgroups, or any
author publishing any paper.

efg
_________________________________
efg's Computer Lab:  http://www.efg2.com/lab
Delphi Books:  http://www.efg2.com/lab/TechBooks/Delphi.htm

Earl F. Glynn     E-Mail:  EarlGlynn@att.net
Overland Park, KS  USA