🚀 go-pugleaf

JRS:  In article <7l165c$gqa$1@bgtnsc02.worldnet.att.net> of Fri, 25 Jun
1999 19:22:15 in news:comp.lang.pascal.delphi.misc, Earl F. Glynn
<EarlGlynn@att.net> wrote:
>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 rotation matrix is in effect a pair of addition formulae, surely?

>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?  

No; the full paragraph of mine reads "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 - these can then be
randomly chosen, though it would be better for the final Buffon value to
systematically use all of them (and to systematically evenly choose the
needle-centres).  But, since, for small x in radians, sin(x)=x, the
angle-halving is itself essentially capable of determining the radian as
a fraction of a right angle, which determines Pi."

You are using, as part of your emulation of Buffon's random-needle Pi
method, a systematic algorithm which itself essentially determines Pi;
you don't import Pi, you virtually make it!.



I can't imagine that it will make any significant difference to the
final result, as standard computer arithmetic is reasonably precise :
but, in principle, I think that one can improve (with a bit more head-
scratching while programming) on repeatedly bifurcating a right angle
and then continually rotating to get, in linear sequence, an equally-
spaced set of angles filling the right angle.  Starting with the zero-
degree point, rotate it by half the right angle to get a second point,
then repeatedly halve the angle and use rotation of each point by it to
double the set of points - rounding errors will build up even less.


If you have not already done so, it may be possible to make Method 3 a
little quicker by using FSINCOS in the FPU so as to generate both SIN &
COS in one move.


Rather than using two crossed needles on floorboards, one can equally
well, I imagine, use a single needle on a square-tiled floor.
 
-- 
 � John Stockton, Surrey, UK. jrs@merlyn.demon.co.uk / JR.Stockton@physics.org �
  Web <URL: http://www.merlyn.demon.co.uk/> - FAQish topics, acronyms, & links.
  Correct 4-line sig. separator is as above, a line precisely "-- " (SoRFC1036)
  Do not Mail News to me.    Before a reply, quote with ">" or "> " (SoRFC1036)