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-rw-r--r--challenge-123/cheok-yin-fung/perl/ch-2.pl86
-rw-r--r--challenge-123/cheok-yin-fung/perl/ch-2a.pl59
-rw-r--r--challenge-123/cheok-yin-fung/perl/ch-2ax.pl86
3 files changed, 154 insertions, 77 deletions
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2.pl b/challenge-123/cheok-yin-fung/perl/ch-2.pl
index ed7142b2b9..7421f86fc7 100644
--- a/challenge-123/cheok-yin-fung/perl/ch-2.pl
+++ b/challenge-123/cheok-yin-fung/perl/ch-2.pl
@@ -8,7 +8,7 @@
use strict;
use warnings;
use v5.10.0;
-use Test::More tests => 11;
+use Test::More tests => 13;
my $D = $ARGV[0] || 2;
@@ -25,19 +25,24 @@ sub is_square {
my $v0 = vec_subtract($p0, $p1);
my $v1 = vec_subtract($p0, $p2);
my $v2 = vec_subtract($p0, $p3);
- # "cross product test";
- return 0 unless (vec_prod($v1, $v2) == 0) xor
- (vec_prod($v0, $v2) == 0) xor
- (vec_prod($v0, $v1) == 0);
+ # "dot product test";
+ # return 0 unless (vec_prod_f($v1, $v2) == 0) xor
+ # (vec_prod_f($v0, $v2) == 0) xor
+ # (vec_prod_f($v0, $v1) == 0);
my @n_vector = map { norm($_) } ($v0, $v1, $v2);
@n_vector = sort {$a<=>$b} @n_vector;
# "norm test"
#if ( $n_vector[0] == $n_vector[1] ) {
# the above conditional is fit for integter coordinates
-# the below is special arrangement starting from the 7th test case
+# the below is special arrangement starting from the 5th test case
# or floating point number in general
if ( sprintf("%f",$n_vector[0]) == sprintf("%f", $n_vector[1])
- && 2*sprintf("%f", $n_vector[0]) == sprintf("%f", $n_vector[2])
+ && 2*sprintf("%f", $n_vector[0]) == sprintf("%f", $n_vector[2])
+# the above line concerning length of diagonal is sqrt(2)
+# times the edge length would not be a necessary check
+# if we preserve the dot product test, because in
+# Euclidean space, if two vectors are orthogonal and in equal length,
+# we can apply the Pythagorean theorem.
) {
return 1;
}
@@ -53,6 +58,10 @@ sub vec_prod {
return $sum;
}
+sub vec_prod_f {
+ return sprintf("%f", vec_prod($_[0], $_[1]));
+}
+
sub norm {
my $p = $_[0];
my $sum = 0;
@@ -108,33 +117,66 @@ ok is_square( [5/sqrt(26), 1/sqrt(26)],
"inclined by arctan(1/5), centre at origin";
ok is_square(
- [cos(atan2(1,5)), sin(atan2(1,5))], [-sin(atan2(1,5)), cos(atan2(1,5))],
- [-cos(atan2(1,5)), -sin(atan2(1,5))],[sin(atan2(1,5)), -cos(atan2(1,5))]
+ [cos(atan2(1,5)), sin(atan2(1,5))],
+ [-sin(atan2(1,5)), cos(atan2(1,5))],
+ [-cos(atan2(1,5)), -sin(atan2(1,5))],
+ [sin(atan2(1,5)), -cos(atan2(1,5))]
)
== 1, "arctan(1/5) by atan2(), caught by the equalities";
ok is_square(
- [2.7*cos(atan2(1,5)), 2.7*sin(atan2(1,5))], [-2.7*sin(atan2(1,5)), 2.7*cos(atan2(1,5))],
- [-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],[2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
+ [2.7*cos(atan2(1,5)), 2.7*sin(atan2(1,5))],
+ [-2.7*sin(atan2(1,5)), 2.7*cos(atan2(1,5))],
+ [-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],
+ [2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.7), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.7), "
+ ."caught by the equalities";
ok is_square(
- [2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))], [-2.8*sin(atan2(1,5)), 2.8*cos(atan2(1,5))],
- [-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],[2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
+ [2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))],
+ [-2.8*sin(atan2(1,5)), 2.8*cos(atan2(1,5))],
+ [-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],
+ [2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.8), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.8), "
+ ."caught by the equalities";
ok is_square(
- [4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))], [-4.0*sin(atan2(1,5)), 4.0*cos(atan2(1,5))],
- [-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],[4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
+ [4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))],
+ [-4.0*sin(atan2(1,5)), 4.0*cos(atan2(1,5))],
+ [-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],
+ [4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 4.0), caught by the equalities";
-
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 4.0)"
+ .", caught by the equalities";
ok is_square(
- [0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))], [-0.0009*sin(atan2(1,5)), 0.0009*cos(atan2(1,5))],
- [-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],[0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
+ [0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))],
+ [-0.0009*sin(atan2(1,5)), 0.0009*cos(atan2(1,5))],
+ [-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],
+ [0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of a much smaller size (multiple by 0.0009), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of a much smaller size"
+ ." (multiple by 0.0009), caught by the equalities"
+ ." (expected fail)";
+;
+
+ok is_square(
+ [-2, 2*sqrt(3), -2*sqrt(3)-1 ],
+ [-2, 4+2*sqrt(3), -1+2*sqrt(3) ],
+ [ 6, 2*sqrt(3), -2*sqrt(3)-1],
+ [ 6, 4+2*sqrt(3), -1+2*sqrt(3)]
+ );
+
+ok is_square(
+ [ 6, -2*sqrt(3), 3-2*sqrt(3)],
+ [-2, -2*sqrt(3), 3-2*sqrt(3)],
+ [-2, 4-2*sqrt(3), 3+2*sqrt(3)],
+ [ 6, 4-2*sqrt(3), 3+2*sqrt(3)],
+ );
done_testing();
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2a.pl b/challenge-123/cheok-yin-fung/perl/ch-2a.pl
index 533bebf029..a7ea05eea8 100644
--- a/challenge-123/cheok-yin-fung/perl/ch-2a.pl
+++ b/challenge-123/cheok-yin-fung/perl/ch-2a.pl
@@ -1,10 +1,12 @@
#!/usr/bin/perl
# The Weekly Challenge 123
# Task 2 extension: Square/Cube/Hypercube Points
-# Usage: ch-2a.pl (optional $k) (optional)$D
+# Usage: $ ch-2a.pl (optional) $k (optional)$D
# $k: 2 or 3 or 4, stands for square or cube or hypercube, default is 3
# $D: 2 or above, cannot be smaller than $k, default is same as $k
+# Note: check out $ diff ch-2a.pl ch-2ax.pl
+
use strict;
use warnings;
use v5.10.0;
@@ -26,12 +28,18 @@ sub is_square {
my $v0 = vec_subtract($p0, $p1);
my $v1 = vec_subtract($p0, $p2);
my $v2 = vec_subtract($p0, $p3);
- return 0 unless (vec_prod($v1, $v2) == 0) xor
- (vec_prod($v0, $v2) == 0) xor
- (vec_prod($v0, $v1) == 0);
+ return 0 unless (vec_prod_f($v1, $v2) == 0) xor
+ (vec_prod_f($v0, $v2) == 0) xor
+ (vec_prod_f($v0, $v1) == 0);
+#========== BEGIN: diff of ch-2a.pl and ch-2ax.pl =========
return 0 unless vec_same($v0, vec_sum($v1, $v2) ) xor
vec_same($v1, vec_sum($v2, $v0) ) xor
vec_same($v2, vec_sum($v0, $v1) );
+#=========== END: diff of ch-2a.pl and ch-2ax.pl ==========
+# COMMENT:
+# this test is mathematically NOT necessary,
+# and if you check it against ch-2ax.pl,
+# you will find this section is the source of failed tests!
my @n_vector = map { norm_f($_) } ($v0, $v1, $v2);
@n_vector = sort {$a<=>$b} @n_vector;
if ( $n_vector[0] == $n_vector[1] ) {
@@ -46,11 +54,11 @@ sub is_cube {
my @p = @_;
my %v;
$v{$_} = vec_subtract($p[0], $p[$_]) for (1..7);
- my @ind = sort { norm($v{$a}) <=> norm($v{$b}) } keys %v;
+ my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
my ($N, $W, $U) = ($v{$ind[0]} , $v{$ind[1]} , $v{$ind[2]}) ;
return 0 unless norm_f($N) == norm_f($W) && norm_f($N) == norm_f($U);
- return 0 unless vec_prod($N,$W) == 0 && vec_prod($W,$U) == 0
- && vec_prod($U,$N) == 0;
+ return 0 unless vec_prod_f($N,$W) == 0 && vec_prod_f($W,$U) == 0
+ && vec_prod_f($U,$N) == 0;
my $NW = vec_sum($N, $W);
my $WU = vec_sum($W, $U);
my $UN = vec_sum($U, $N);
@@ -93,19 +101,19 @@ sub is_hypercube {
my @p = @_;
my %v;
$v{$_} = vec_subtract($p[0], $p[$_]) for (1..15);
- my @ind = sort { norm($v{$a}) <=> norm($v{$b}) } keys %v;
+ my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
my ($N, $W, $U, $A) = ( $v{$ind[0]}, $v{$ind[1]} ,
$v{$ind[2]}, $v{$ind[3]} );
return 0 unless
norm_f($N) == norm_f($W) && norm_f($W) == norm_f($U)
&& norm_f($U) == norm_f($A);
return 0 unless
- vec_prod($N, $W) == 0 &&
- vec_prod($N, $U) == 0 &&
- vec_prod($N, $A) == 0 &&
- vec_prod($A, $W) == 0 &&
- vec_prod($A, $U) == 0 &&
- vec_prod($W, $U) == 0 ;
+ vec_prod_f($N, $W) == 0 &&
+ vec_prod_f($N, $U) == 0 &&
+ vec_prod_f($N, $A) == 0 &&
+ vec_prod_f($A, $W) == 0 &&
+ vec_prod_f($A, $U) == 0 &&
+ vec_prod_f($W, $U) == 0 ;
my $AU = vec_sum($A, $U);
my $AW = vec_sum($A, $W);
@@ -159,6 +167,10 @@ sub vec_prod {
return $sum;
}
+sub vec_prod_f {
+ return sprintf("%f", vec_prod($_[0], $_[1]));
+}
+
sub norm {
my $p = $_[0];
my $sum = 0;
@@ -204,7 +216,7 @@ sub vec_subtract {
-# 4 tests
+# 9 tests
ok is_square( [1,0], [0,1], [-1,0],[0,-1]) == 1, "on x-axis and y-axis";
ok is_square( [5/sqrt(26), 1/sqrt(26)],
@@ -227,7 +239,9 @@ ok is_square(
[-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],
[2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.7), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.7),"
+ ."caught by the equalities";
ok is_square(
[2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))],
@@ -235,7 +249,9 @@ ok is_square(
[-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],
[2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.8), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.8),"
+ ."caught by the equalities";
ok is_square(
[4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))],
@@ -243,7 +259,9 @@ ok is_square(
[-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],
[4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 4.0), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 4.0),"
+ ." caught by the equalities";
ok is_square(
[0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))],
@@ -251,7 +269,10 @@ ok is_square(
[-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],
[0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of a much smaller size (multiple by 0.0009), caught by the equalities (_\"not ok\" is normal_)";
+ == 1,
+ "arctan(1/5) by atan2() of a much smaller size"
+ ."(multiple by 0.0009), caught by the equalities"
+ ."(_\"not ok\" is normal_)";
ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
ok is_square( [1, 1] , [-1, 1], [1,-1], [-1,-1] ) == 1, "centre at origin";
diff --git a/challenge-123/cheok-yin-fung/perl/ch-2ax.pl b/challenge-123/cheok-yin-fung/perl/ch-2ax.pl
index ac3ba42cce..cc5521e311 100644
--- a/challenge-123/cheok-yin-fung/perl/ch-2ax.pl
+++ b/challenge-123/cheok-yin-fung/perl/ch-2ax.pl
@@ -1,16 +1,17 @@
#!/usr/bin/perl
# The Weekly Challenge 123
# Task 2 extension: Square/Cube/Hypercube Points
-# Usage: ch-2a.pl (optional $k) (optional)$D
+# Usage: ch-2ax.pl (optional) $k (optional)$D
# $k: 2 or 3 or 4, stands for square or cube or hypercube, default is 3
# $D: 2 or above, cannot be smaller than $k, default is same as $k
+# Note: check out $ diff ch-2a.pl ch-2ax.pl
use strict;
use warnings;
use v5.10.0;
use Test::More tests => 14;
-use Algorithm::Combinatorics qw(permutations); #use for cube and hypercube
+use Algorithm::Combinatorics qw(permutations); #use for hypercube
my $k = $ARGV[0] || 3;
@@ -26,12 +27,12 @@ sub is_square {
my $v0 = vec_subtract($p0, $p1);
my $v1 = vec_subtract($p0, $p2);
my $v2 = vec_subtract($p0, $p3);
- return 0 unless (vec_prod($v1, $v2) == 0) xor
- (vec_prod($v0, $v2) == 0) xor
- (vec_prod($v0, $v1) == 0);
+ return 0 unless (vec_prod_f($v1, $v2) == 0) xor
+ (vec_prod_f($v0, $v2) == 0) xor
+ (vec_prod_f($v0, $v1) == 0);
my @n_vector = map { norm_f($_) } ($v0, $v1, $v2);
@n_vector = sort {$a<=>$b} @n_vector;
- if ( $n_vector[0] == $n_vector[1] && 2*$n_vector[0] == $n_vector[2] ) {
+ if ( $n_vector[0] == $n_vector[1] ) {
return 1;
}
else {
@@ -43,11 +44,11 @@ sub is_cube {
my @p = @_;
my %v;
$v{$_} = vec_subtract($p[0], $p[$_]) for (1..7);
- my @ind = sort { norm($v{$a}) <=> norm($v{$b}) } keys %v;
+ my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
my ($N, $W, $U) = ($v{$ind[0]} , $v{$ind[1]} , $v{$ind[2]}) ;
return 0 unless norm_f($N) == norm_f($W) && norm_f($N) == norm_f($U);
- return 0 unless vec_prod($N,$W) == 0 && vec_prod($W,$U) == 0
- && vec_prod($U,$N) == 0;
+ return 0 unless vec_prod_f($N,$W) == 0 && vec_prod_f($W,$U) == 0
+ && vec_prod_f($U,$N) == 0;
my $NW = vec_sum($N, $W);
my $WU = vec_sum($W, $U);
my $UN = vec_sum($U, $N);
@@ -81,10 +82,10 @@ sub is_cube {
if ( vec_same( $v{$ind[6]} , $NWU ) ) {
=pod
return 0 unless
- 2*norm($N) == norm($NW) &&
- norm($NW) == norm($WU) &&
- norm($WU) == norm($UN) &&
- 3*norm($N) == norm($NWU);
+ 2*norm_f($N) == norm_f($NW) &&
+ norm_f($NW) == norm_f($WU) &&
+ norm_f($WU) == norm_f($UN) &&
+ 3*norm_f($N) == norm_f($NWU);
=cut
return 1;
}
@@ -97,19 +98,19 @@ sub is_hypercube {
my @p = @_;
my %v;
$v{$_} = vec_subtract($p[0], $p[$_]) for (1..15);
- my @ind = sort { norm($v{$a}) <=> norm($v{$b}) } keys %v;
+ my @ind = sort { norm_f($v{$a}) <=> norm_f($v{$b}) } keys %v;
my ($N, $W, $U, $A) = ( $v{$ind[0]}, $v{$ind[1]} ,
$v{$ind[2]}, $v{$ind[3]} );
return 0 unless
norm_f($N) == norm_f($W) && norm_f($W) == norm_f($U)
&& norm_f($U) == norm_f($A);
return 0 unless
- vec_prod($N, $W) == 0 &&
- vec_prod($N, $U) == 0 &&
- vec_prod($N, $A) == 0 &&
- vec_prod($A, $W) == 0 &&
- vec_prod($A, $U) == 0 &&
- vec_prod($W, $U) == 0 ;
+ vec_prod_f($N, $W) == 0 &&
+ vec_prod_f($N, $U) == 0 &&
+ vec_prod_f($N, $A) == 0 &&
+ vec_prod_f($A, $W) == 0 &&
+ vec_prod_f($A, $U) == 0 &&
+ vec_prod_f($W, $U) == 0 ;
my $AU = vec_sum($A, $U);
my $AW = vec_sum($A, $W);
@@ -149,17 +150,17 @@ sub is_hypercube {
if ( vec_same($v{$ind[14]}, $AUNW) ) {
=pod
return 0 unless
- 2*norm($N) == norm($NW) &&
- norm($NW) == norm($AU) &&
- norm($NW) == norm($UN) &&
- norm($NW) == norm($WU) &&
- norm($NW) == norm($AW) &&
- norm($NW) == norm($AN) &&
- 3*norm($N) == norm($UNW) &&
- 3*norm($N) == norm($ANW) &&
- 3*norm($N) == norm($AWU) &&
- 3*norm($N) == norm($AUN) &&
- 4*norm($N) == norm($AUNW);
+ 2*norm_f($N) == norm_f($NW) &&
+ norm_f($NW) == norm_f($AU) &&
+ norm_f($NW) == norm_f($UN) &&
+ norm_f($NW) == norm_f($WU) &&
+ norm_f($NW) == norm_f($AW) &&
+ norm_f($NW) == norm_f($AN) &&
+ 3*norm_f($N) == norm_f($UNW) &&
+ 3*norm_f($N) == norm_f($ANW) &&
+ 3*norm_f($N) == norm_f($AWU) &&
+ 3*norm_f($N) == norm_f($AUN) &&
+ 4*norm_f($N) == norm_f($AUNW);
=cut
return 1;
}
@@ -177,6 +178,10 @@ sub vec_prod {
return $sum;
}
+sub vec_prod_f {
+ return sprintf("%f", vec_prod($_[0], $_[1]));
+}
+
sub norm {
my $p = $_[0];
my $sum = 0;
@@ -222,7 +227,7 @@ sub vec_subtract {
-# 4 tests
+# 9 tests
ok is_square( [1,0], [0,1], [-1,0],[0,-1]) == 1, "on x-axis and y-axis";
ok is_square( [5/sqrt(26), 1/sqrt(26)],
@@ -245,7 +250,9 @@ ok is_square(
[-2.7*cos(atan2(1,5)), -2.7*sin(atan2(1,5))],
[2.7*sin(atan2(1,5)), -2.7*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.7), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.7),"
+ ."caught by the equalities";
ok is_square(
[2.8*cos(atan2(1,5)), 2.8*sin(atan2(1,5))],
@@ -253,7 +260,9 @@ ok is_square(
[-2.8*cos(atan2(1,5)), -2.8*sin(atan2(1,5))],
[2.8*sin(atan2(1,5)), -2.8*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 2.8), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 2.8),"
+ ."caught by the equalities";
ok is_square(
[4.0*cos(atan2(1,5)), 4.0*sin(atan2(1,5))],
@@ -261,7 +270,9 @@ ok is_square(
[-4.0*cos(atan2(1,5)), -4.0*sin(atan2(1,5))],
[4.0*sin(atan2(1,5)), -4.0*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of larger size (multipled by 4.0), caught by the equalities";
+ == 1,
+ "arctan(1/5) by atan2() of larger size (multipled by 4.0),"
+ ." caught by the equalities";
ok is_square(
[0.0009*cos(atan2(1,5)), 0.0009*sin(atan2(1,5))],
@@ -269,7 +280,10 @@ ok is_square(
[-0.0009*cos(atan2(1,5)), -0.0009*sin(atan2(1,5))],
[0.0009*sin(atan2(1,5)), -0.0009*cos(atan2(1,5))]
)
- == 1, "arctan(1/5) by atan2() of a much smaller size (multiple by 0.0009), caught by the equalities (_\"not ok\" is normal_)";
+ == 1,
+ "arctan(1/5) by atan2() of a much smaller size"
+ ."(multiple by 0.0009), caught by the equalities"
+ ."(_\"not ok\" is normal_)";
ok is_square( [1, 2] , [4,3], [3,1], [2,4] ) == 1, "Knight's square";
ok is_square( [1, 1] , [-1, 1], [1,-1], [-1,-1] ) == 1, "centre at origin";
generated by cgit (git 2.34.1) at 2026-04-12 21:04:09 +0000