Rational geometry experiments.

tests/rationalGeo.pl

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+#!/usr/bin/env perl
+
+# Preexperiments to rational geometry
+
+use strict;
+use warnings;
+
+use Math::Trig;
+
+# generate rational approximations to nth parts of an arc represented by sin and cos
+# approach leads to values in principal usable as first iteration, but has vastly to big numbers
+sub nthArc {
+  my $n = shift;
+  # basis triangle with half the angle 
+  my $prex = $n;
+  my $prey = 1;
+  dumpangles("pre", $prex, $prey, 45/$n);
+  # complex multiplication, double the angle
+  my $x = $prex*$prex - $prey*$prey;
+  my $y = 2*$prex*$prey;
+  dumpangles("square", $x, $y, 90/$n);
+  my $npowx = 1;
+  my $npowy = 0;
+  for(1..$n) {
+    my $nnpowx = $x*$npowx - $y*$npowy;
+    my $nnpowy = $x*$npowy + $y*$npowx;
+    $npowx = $nnpowx;
+    $npowy = $nnpowy;
+  }
+  dumpangles("n-powered", $npowx, $npowy, 90);
+}
+
+sub dumpangles {
+  my ($name, $x, $y, $compare) = @_;
+  my $goal = "";
+  $goal = " (goal: $compare)" if $compare;;
+  my $phi = atan2($y,$x)/pi*180;
+  print "[$name] $x $y -- $phi$goal\n";
+}
+
+# exp is
+# sum_n=0..inf  x^n/n!
+
+# 1/0! - x^2/2! + x^4/4! - x^6/6!
+sub preCos {
+  my ($steps, $x) = @_;
+  my $mxsq = -$x*$x;
+  my $sum = 0;
+  my $fa = 1;
+  for(1..$steps) {
+    my $step2 = $_ * 2;
+    $sum += $fa;
+    $fa *= $mxsq/($step2*($step2-1))
+  }
+  return $sum
+}
+
+# x/1! - x^3/3! + x^5/5! - x^7/7!
+sub preSin {
+  my ($steps, $x) = @_;
+  my $mxsq = -$x*$x;
+  my $sum = 0;
+  my $fa = $x;
+  for(1..$steps) {
+    my $step2 = $_ * 2;
+    $sum += $fa;
+    $fa *= $mxsq/($step2*($step2+1))
+  }
+  return $sum
+}
+
+
+while(<>) {
+#  if(m#^([1-9][0-9]*)$#) {
+#    nthArc($1)
+#  } else {
+#    print "Bad parameter\n";
+#  }
+  if(m#^([1-9][0-9]*)\s+([0-9\.]+)$#) {
+    my ($st, $x) = ($1, $2);
+    my $c = cos($x);
+    my $cappr = preCos($st, $x);
+    print "$x -- c:$c  cappr:$cappr\n";
+    my $s = sin($x);
+    my $sappr = preSin($st, $x);
+    print "$x -- s:$s  sappr:$sappr\n";
+  } elsif(m#pi\s+([0-9]+)\s+([0-9]+)#) {
+    my ($bigit, $smallit) = ($1, $2);
+    my $pi = 22/7;
+    for(1..$bigit) {
+      $pi = $pi + preSin($smallit, $pi);
+    }
+    print "piappr: $pi  -- truepu: ", pi, "\n"
+  } else {
+    print "Bad parameters\n";
+  }
+}
+
+