Name
Math::Intersection::Circle::Line - Find the points at which circles and lines
intersect and the area of these intersections.
Synopsis
use Math::Intersection::Circle::Line q(:all);
use Test::More q(no_plan);
use utf8;
# Euler Line, see: https://en.wikipedia.org/wiki/Euler_line
if (1)
{my @t = (0, 0, 4, 0, 0, 3); # Corners of the triangle
&areaOfPolygon(sub {ok !$_[0]}, # Polygon formed by these points has zero area and so is a line or a point
&circumCircle (sub {@_[0,1]}, @t), # green
&ninePointCircle(sub {@_[0,1]}, @t), # red
&orthoCentre (sub {@_[0,1]}, @t), # blue
¢roid (sub {@_[0,1]}, @t)); # orange
}
# An isosceles tringle with an apex height of 3/4 of the radius of its
# circumcircle divides Euler's line into 6 equal pieces
if (1)
{my $r = 400; # Arbitrary but convenient radius
intersectionCircleLine # Find coordinates of equiangles of isoceles triangle
{my ($x, $y, $𝕩, $𝕪) = @_; # Coordinates of equiangles
my ($𝘅, $𝘆) = (0, $r); # Coordinates of apex
my ($nx, $ny, $nr) = ninePointCircle {@_} $x, $y, $𝘅, $𝘆, $𝕩, $𝕪; # Coordinates of centre and radius of nine point circle
my ($cx, $cy) = centroid {@_} $x, $y, $𝘅, $𝘆, $𝕩, $𝕪; # Coordinates of centroid
my ($ox, $oy) = orthoCentre {@_} $x, $y, $𝘅, $𝘆, $𝕩, $𝕪; # Coordinates of orthocentre
ok join(' ', $y, $cy, $y+$nr, $𝘆, $y+2*$nr, $oy) eq # Calculated points along Euler's line
join(' ', 100, 200, 300, 400, 500, 600); # The actual six equally spaced points along Euler's line
} 0, 0, $r, 0, $r/4, 1, $r/4; # Chord at 1/4 radius
}
# A line segment across a circle is never longer than the diameter
if (1) # Random circle and random line
{my ($x, $y, $r, $𝘅, $𝘆, $𝕩, $𝕪) = map {rand()} 1..7;
intersectionCircleLine # Find intersection of a circle and a line
{return ok 1 unless @_ == 4; # Ignore line unless it crosses circle
ok &vectorLength(@_) <= 2*$r; # Length if line segment is less than or equal to that of a diameter
} $x, $y, $r, $𝘅, $𝘆, $𝕩, $𝕪; # Circle and line to be intersected
}
# The length of a side of a hexagon is the radius of a circle inscribed through
# its vertices
if (1)
{my ($x, $y, $r) = map {rand()} 1..3; # Random circle
my @p = intersectionCircles {@_} $x, $y, $r, $x+$r, $y, $r; # First step of one radius
my @𝗽 = intersectionCircles {@_} $x, $y, $r, $p[0], $p[1], $r; # Second step of one radius
my @q = !&near($x+$r, $y, @𝗽[0,1]) ? @𝗽[0,1] : @𝗽[2,3]; # Away from start point
my @𝗾 = intersectionCircles {@_} $x, $y, $r, $q[0], $q[1], $r; # Third step of one radius
ok &near2(@𝗾[0,1], $x-$r, $y) or # Brings us to a point
&near2(@𝗾[2,3], $x-$r, $y); # opposite to the start point
}
# Circle through three points chosen at random has the same centre regardless of
# the pairing of the points
sub circleThrough3
{my ($x, $y, $𝘅, $𝘆, $𝕩, $𝕪) = @_; # Three points
&intersectionLines
(sub # Intersection of bisectors is the centre of the circle
{my @r =(&vectorLength(@_, $x, $y), # Radii from centre of circle to each point
&vectorLength(@_, $𝘅, $𝘆),
&vectorLength(@_, $𝕩, $𝕪));
ok &near(@r[0,1]); # Check radii are equal
ok &near(@r[1,2]);
@_ # Return centre
}, rotate90AroundMidPoint($x, $y, $𝘅, $𝘆), # Bisectors between pairs of points
rotate90AroundMidPoint($𝕩, $𝕪, $𝘅, $𝘆));
}
if (1)
{my (@points) = map {rand()} 1..6; # Three points chosen at random
ok &near2(circleThrough3(@points), circleThrough3(@points[2..5, 0..1])); # Circle has same centre regardless
ok &near2(circleThrough3(@points), circleThrough3(@points[4..5, 0..3])); # of the pairing of the points
}
Description
Find the points at which circles and lines intersect and the area of these
intersections.
Fast, fun and easy to use these functions are written in 100% Pure Perl.
areaOfTriangle sub triangle
Calls sub($a) where $a is the area of the specified triangle:
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
areaOfPolygon sub points...
Calls sub($a) where $a is the area of the polygon with vertices specified by
the point.
A point is specified by supplying a list of two numbers:
(𝘅, 𝘆)
centroid sub triangle
Calls sub($x,$y) where $x,$y are the coordinates of the centroid of the
specified triangle:
See: https://en.wikipedia.org/wiki/Centroid
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
circumCentre sub triangle
Calls sub($x,$y,$r) where $x,$y are the coordinates of the centre of the
circle drawn through the corners of the specified triangle and $r is its
radius:
See: https://en.wikipedia.org/wiki/Circumscribed_circle
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
circumCircle sub triangle
Calls sub($x,$y,$r) where $x,$y are the coordinates of the circum-centre of
the specified triangle and $r is its radius:
See: https://en.wikipedia.org/wiki/Circumscribed_circle
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
exCircles sub triangle
Calls sub([$x,$y,$r]...) where $x,$y are the coordinates of the centre of each
ex-circle and $r its radius for the specified triangle:
See: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
circleInscribedInTriangle sub triangle
Calls sub($x,$y,$r) where $x,$y are the coordinates of the centre of
a circle which touches each side of the triangle just once and $r is its radius:
See: https://en.wikipedia.org/wiki/Incircle_and_excircles_of_a_triangle#Incircle
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
intersectionCircles 𝘀𝘂𝗯 circle1, circle2
Find the points at which two circles intersect. Complains if the two circles
are identical.
𝘀𝘂𝗯 specifies a subroutine to be called with the coordinates of the
intersection points if there are any or an empty parameter list if there are
no points of intersection.
A circle is specified by supplying a list of three numbers:
(𝘅, 𝘆, 𝗿)
where (𝘅, 𝘆) are the coordinates of the centre of the circle and (𝗿) is its
radius.
Returns whatever is returned by 𝘀𝘂𝗯.
intersectionCirclesArea 𝘀𝘂𝗯 circle1, circle2
Find the area of overlap of two circles expressed as a fraction of the area of
the smallest circle. The fractional area is expressed as a number between 0
and 1.
𝘀𝘂𝗯 specifies a subroutine to be called with the fractional area.
A circle is specified by supplying a list of three numbers:
(𝘅, 𝘆, 𝗿)
where (𝘅, 𝘆) are the coordinates of the centre of the circle and (𝗿) is its
radius.
Returns whatever is returned by 𝘀𝘂𝗯.
intersectionCircleLine 𝘀𝘂𝗯 circle, line
Find the points at which a circle and a line intersect.
𝘀𝘂𝗯 specifies a subroutine to be called with the coordinates of the
intersection points if there are any or an empty parameter list if there are
no points of intersection.
A circle is specified by supplying a list of three numbers:
(𝘅, 𝘆, 𝗿)
where (𝘅, 𝘆) are the coordinates of the centre of the circle and (𝗿) is its
radius.
A line is specified by supplying a list of four numbers:
(x, y, 𝘅, 𝘆)
where (x, y) and (𝘅, 𝘆) are the coordinates of two points on the line.
Returns whatever is returned by 𝘀𝘂𝗯.
intersectionCircleLineArea 𝘀𝘂𝗯 circle, line
Find the fractional area of a circle occupied by a lune produced by an
intersecting line. The fractional area is expressed as a number
between 0 and 1.
𝘀𝘂𝗯 specifies a subroutine to be called with the fractional area.
A circle is specified by supplying a list of three numbers:
(𝘅, 𝘆, 𝗿)
where (𝘅, 𝘆) are the coordinates of the centre of the circle and (𝗿) is its
radius.
A line is specified by supplying a list of four numbers:
(x, y, 𝘅, 𝘆)
where (x, y) and (𝘅, 𝘆) are the coordinates of two points on the line.
Returns whatever is returned by 𝘀𝘂𝗯.
intersectionLines 𝘀𝘂𝗯 line1, line2
Finds the point at which two lines intersect.
𝘀𝘂𝗯 specifies a subroutine to be called with the coordinates of the
intersection point or an empty parameter list if the two lines do not
intersect.
Complains if the two lines are collinear.
A line is specified by supplying a list of four numbers:
(x, y, 𝘅, 𝘆)
where (x, y) and (𝘅, 𝘆) are the coordinates of two points on the line.
Returns whatever is returned by 𝘀𝘂𝗯.
intersectionLinePoint 𝘀𝘂𝗯 line, point
Find the point on a line closest to a specified point.
𝘀𝘂𝗯 specifies a subroutine to be called with the coordinates of the
intersection points if there are any.
A line is specified by supplying a list of four numbers:
(x, y, 𝘅, 𝘆)
where (x, y) and (𝘅, 𝘆) are the coordinates of two points on the line.
A point is specified by supplying a list of two numbers:
(𝘅, 𝘆)
where (𝘅, 𝘆) are the coordinates of the point.
Returns whatever is returned by 𝘀𝘂𝗯.
isEquilateralTriangle triangle
Return true if the specified triangle is close to being equilateral within the
definition of nearness.
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
isIsoscelesTriangle triangle
Return true if the specified triangle is close to being isosceles within the
definition of nearness.
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
isRightAngledTriangle triangle
Return true if the specified triangle is close to being right angled within
the definition of nearness.
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
ninePointCircle sub triangle
Calls sub($x,$y,$r) where $x,$y are the coordinates of the centre of the
circle drawn through the midpoints of each side of the specified triangle and
$r is its radius which guves the nine point circle:
See: https://en.wikipedia.org/wiki/Nine-point_circle
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
orthoCentre sub triangle
Calls sub($x,$y) where $x,$y are the coordinates of the ortho-centre of the
specified triangle:
See: https://en.wikipedia.org/wiki/Altitude_%28triangle%29
A triangle is specified by supplying a list of six numbers:
(x, y, 𝘅, 𝘆, 𝕩, 𝕪)
where (x, y), (𝘅, 𝘆) and (𝕩, 𝕪) are the coordinates of the vertices of the
triangle.
$Math::Intersection::Circle::Line::near
As a finite computer cannot represent an infinite plane of points it is
necessary to make the plane discrete by merging points closer than the
distance contained in this variable, which is set by default to 1e-6.
Export
The following functions are exported by default:
"areaOfPolygon()"
"areaOfTriangle()"
"centroid()"
"circumCentre()"
"circumCircle()"
"circleInscribedInTriangle()"
"circleThroughMidPointsOfTriangle()"
"exCircles()"
"intersectionCircleLine()"
"intersectionCircleLineArea()"
"intersectionCircles()"
"intersectionCircles()"
"intersectionCirclesArea()"
"intersectionLines()"
"intersectionLinePoint()"
"isEquilateralTriangle()"
"isIsoscelesTriangle()"
"isRightAngledTriangle()"
"orthoCentre()"
Optionally some useful helper functions can also be exported either by
specifying the tag :𝗮𝗹𝗹 or by naming the required functions individually:
"lengthsOfTheSidesOfAPolygon()"
"midPoint()"
"midPoint()"
"near()"
"near2()"
"near3()"
"near4()"
"rotate90CW()"
"rotate90CCW()"
"rotate90AroundMidPoint()"
"threeCollinearPoints()"
"vectorLength()"
"𝝿()"
Installation
Standard Module::Build process for building and installing modules:
perl Build.PL
./Build
./Build test
./Build install
Or, if you're on a platform (like DOS or Windows) that doesn't require
the "./" notation, you can do this:
perl Build.PL
Build
Build test
Build install
Author
Philip R Brenan at gmail dot com
http://www.appaapps.com
Copyright
Copyright (c) 2015 Philip R Brenan.
This module is free software. It may be used, redistributed and/or
modified under the same terms as Perl itself.