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.