tcllib 1.19-dfsg-2 / usr / share / tcltk / tcllib1.19 / math / geometry.tcl

# geometry.tcl --
#
#	Collection of geometry functions.
#
# Copyright (c) 2001 by Ideogramic ApS and other parties.
# Copyright (c) 2004 Arjen Markus
# Copyright (c) 2010 Andreas Kupries
# Copyright (c) 2010 Kevin Kenny
#
# See the file "license.terms" for information on usage and redistribution
# of this file, and for a DISCLAIMER OF ALL WARRANTIES.
#
# RCS: @(#) $Id: geometry.tcl,v 1.12 2010/05/24 21:44:16 andreas_kupries Exp $

namespace eval ::math::geometry {}

package require Tcl 8.5
package require math

###
#
# POINTS
#
#    A point P consists of an x-coordinate, Px, and a y-coordinate, Py,
#    and both coordinates are floating point values.
#
#    Points are usually denoted by A, B, C, P, or Q.
#
###
#
# LINES
#
#    There are basically three types of lines:
#         line           A line is defined by two points A and B as the
#                        _infinite_ line going through these two points.
#                        Often a line is given as a list of 4 coordinates
#                        instead of 2 points.
#         line segment   A line segment is defined by two points A and B
#                        as the _finite_ that starts in A and ends in B.
#                        Often a line segment is given as a list of 4
#                        coordinates instead of 2 points.
#         polyline       A polyline is a sequence of connected line segments.
#
#    Please note that given a point P, the closest point on a line is given
#    by the projection of P onto the line. The closest point on a line segment
#    may be the projection, but it may also be one of the end points of the
#    line segment.
#
###
#
# DISTANCES
#
#    The distances in this package are all floating point values.
#
###

# Point constructor
proc ::math::geometry::p {x y} {
    return [list $x $y]
}

# Vector addition
proc ::math::geometry::+ {pa pb} {
    lassign $pa ax ay; lassign $pb bx by
    return [list [expr {$ax + $bx}] [expr {$ay + $by}]]
}

# Vector difference
proc ::math::geometry::- {pa pb} {
    lassign $pa ax ay; lassign $pb bx by
    return [list [expr {$ax - $bx}] [expr {$ay - $by}]]
}

# Distance between 2 points
proc ::math::geometry::distance {pa pb} {
    lassign $pa ax ay; lassign $pb bx by
    return [expr {hypot($bx-$ax,$by-$ay)}]
}

# Length of a vector
proc ::math::geometry::length {v} {
    lassign $v x y
    return [expr {hypot($x,$y)}]
}

# Scaling a vector by a factor
proc ::math::geometry::s* {factor p} {
    lassign $p x y
    return [list [expr {$x * $factor}] [expr {$y * $factor}]]
}

# Unit vector into specific direction given by angle (degrees)
proc ::math::geometry::direction {angle} {
    variable torad
    set x [expr {  cos($angle * $torad)}]
    set y [expr {- sin($angle * $torad)}]
    return [list $x $y]
}

# Vertical vector of specified length.
proc ::math::geometry::v {h} {
    return [list 0 $h]
}

# Horizontal vector of specified length.
proc ::math::geometry::h {w} {
    return [list $w 0]
}

# Find point on a line between 2 points at a distance
# distance 0 => a, distance 1 => b
proc ::math::geometry::between {pa pb s} {
    return [+ $pa [s* $s [- $pb $pa]]]
}

# Find direction octant the point (vector) lies in.
proc ::math::geometry::octant {p} {
    variable todeg
    lassign $p x y

    set a [expr {(atan2(-$y,$x)*$todeg)}]
    while {$a >  360} {set a [expr {$a - 360}]}
    while {$a < -360} {set a [expr {$a + 360}]}
    if {$a < 0} {set a [expr {360 + $a}]}

    #puts "p ($x, $y) @ angle $a | [expr {atan2($y,$x)}] | [expr {atan2($y,$x)*$todeg}]"
    # XXX : Add outer conditions to make a log2 tree of checks.

    if {$a <= 157.5} {
	if {$a <= 67.5} {
	    if {$a <= 22.5} { return east }
	    return northeast
	}
	if {$a <=  112.5} { return north }
	return northwest
    } else {
	if {$a <=  247.5} {
	    if {$a <=  202.5} { return west }
	    return southwest
	}
	if {$a <=  337.5} {
	    if {$a <=  292.5} { return south }
	    return southeast
	}
	return east ; # a <= 360.0
    }
}

# Return the NW and SE corners of the rectangle.
proc ::math::geometry::nwse {rect} {
    lassign $rect xnw ynw xse yse
    return [list [p $xnw $ynw] [p $xse $yse]]
}

# Construct rectangle from NW and SE corners.
proc ::math::geometry::rect {pa pb} {
    lassign $pa ax ay; lassign $pb bx by
    return [list $ax $ay $bx $by]
}

proc ::math::geometry::conjx {p} {
    lassign $p x y
    return [list [expr {- $x}] $y]
}

proc ::math::geometry::conjy {p} {
    lassign $p x y
    return [list $x [expr {- $y}]]
}

proc ::math::geometry::x {p} {
    return [lindex $p 0]
}

proc ::math::geometry::y {p} {
    return [lindex $p 1]
}

# ::math::geometry::calculateDistanceToLine
#
#       Calculate the distance between a point and a line.
#
# Arguments:
#       P             a point
#       line          a line
#
# Results:
#       dist          the smallest distance between P and the line
#
# Examples:
#     - calculateDistanceToLine {5 10} {0 0 10 10}
#       Result: 3.53553390593
#     - calculateDistanceToLine {-10 0} {0 0 10 10}
#       Result: 7.07106781187
#
proc ::math::geometry::calculateDistanceToLine {P line} {
    # solution based on FAQ 1.02 on comp.graphics.algorithms
    # L = hypot( Bx-Ax, By-Ay )
    #     (Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay)
    # s = -----------------------------
    #                 L^2
    # dist = |s|*L
    #
    # =>
    #
    #        | (Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay) |
    # dist = ---------------------------------
    #                       L
    set Ax [lindex $line 0]
    set Ay [lindex $line 1]
    set Bx [lindex $line 2]
    set By [lindex $line 3]
    set Cx [lindex $P 0]
    set Cy [lindex $P 1]
    if {$Ax==$Bx && $Ay==$By} {
	return [lengthOfPolyline [concat $P [lrange $line 0 1]]]
    } else {
	set L [expr {hypot($Bx-$Ax,$By-$Ay)}]
	return [expr {abs(($Ay-$Cy)*($Bx-$Ax)-($Ax-$Cx)*($By-$Ay)) / $L}]
    }
}

# ::math::geometry::findClosestPointOnLine
#
#       Return the point on a line which is closest to a given point.
#
# Arguments:
#       P             a point
#       line          a line
#
# Results:
#       Q             the point on the line that has the smallest
#                     distance to P
#
# Examples:
#     - findClosestPointOnLine {5 10} {0 0 10 10}
#       Result: 7.5 7.5
#     - findClosestPointOnLine {-10 0} {0 0 10 10}
#       Result: -5.0 -5.0
#
proc ::math::geometry::findClosestPointOnLine {P line} {
    return [lindex [findClosestPointOnLineImpl $P $line] 0]
}

# ::math::geometry::findClosestPointOnLineImpl
#
#       PRIVATE FUNCTION USED BY OTHER FUNCTIONS.
#       Find the point on a line that is closest to a given point.
#
# Arguments:
#       P             a point
#       line          a line defined by points A and B
#
# Results:
#       Q             the point on the line that has the smallest
#                     distance to P
#       r             r has the following meaning:
#                        r=0      P = A
#                        r=1      P = B
#                        r<0      P is on the backward extension of AB
#                        r>1      P is on the forward extension of AB
#                        0<r<1    P is interior to AB
#
proc ::math::geometry::findClosestPointOnLineImpl {P line} {
    # solution based on FAQ 1.02 on comp.graphics.algorithms - but avoid the
    # chain of pow( sqrt(...) ,2) for better precision (& performance).
    #   L^2 = (Bx-Ax)^2 + (By-Ay)^2
    #        (Cx-Ax)(Bx-Ax) + (Cy-Ay)(By-Ay)
    #   r = -------------------------------
    #                     L^2
    #   Px = Ax + r(Bx-Ax)
    #   Py = Ay + r(By-Ay)
    set Ax [lindex $line 0]
    set Ay [lindex $line 1]
    set Bx [lindex $line 2]
    set By [lindex $line 3]
    set Cx [lindex $P 0]
    set Cy [lindex $P 1]
    if {$Ax==$Bx && $Ay==$By} {
	return [list [list $Ax $Ay] 0]
    } else {
	set Lsquared [expr {pow($Bx-$Ax,2) + pow($By-$Ay,2)}]
	set r [expr {(($Cx-$Ax)*($Bx-$Ax) + ($Cy-$Ay)*($By-$Ay))/$Lsquared}]
	set Px [expr {$Ax + $r*($Bx-$Ax)}]
	set Py [expr {$Ay + $r*($By-$Ay)}]
	return [list [list $Px $Py] $r]
    }
}

# ::math::geometry::calculateDistanceToLineSegment
#
#       Calculate the distance between a point and a line segment.
#
# Arguments:
#       P             a point
#       linesegment   a line segment
#
# Results:
#       dist          the smallest distance between P and any point
#                     on the line segment
#
# Examples:
#     - calculateDistanceToLineSegment {5 10} {0 0 10 10}
#       Result: 3.53553390593
#     - calculateDistanceToLineSegment {-10 0} {0 0 10 10}
#       Result: 10.0
#
proc ::math::geometry::calculateDistanceToLineSegment {P linesegment} {
    set result [calculateDistanceToLineSegmentImpl $P $linesegment]
    set distToLine [lindex $result 0]
    set r [lindex $result 1]
    if {$r<0} {
	return [lengthOfPolyline [concat $P [lrange $linesegment 0 1]]]
    } elseif {$r>1} {
	return [lengthOfPolyline [concat $P [lrange $linesegment 2 3]]]
    } else {
	return $distToLine
    }
}

# ::math::geometry::calculateDistanceToLineSegmentImpl
#
#       PRIVATE FUNCTION USED BY OTHER FUNCTIONS.
#       Find the distance between a point and a line.
#
# Arguments:
#       P             a point
#       linesegment   a line segment A->B
#
# Results:
#       dist          the smallest distance between P and the line
#       r             r has the following meaning:
#                        r=0      P = A
#                        r=1      P = B
#                        r<0      P is on the backward extension of AB
#                        r>1      P is on the forward extension of AB
#                        0<r<1    P is interior to AB
#
proc ::math::geometry::calculateDistanceToLineSegmentImpl {P linesegment} {
    # solution based on FAQ 1.02 on comp.graphics.algorithms
    # L = hypot( Bx-Ax , By-Ay )
    #     (Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay)
    # s = -----------------------------
    #                 L^2
    #      (Cx-Ax)(Bx-Ax) + (Cy-Ay)(By-Ay)
    # r = -------------------------------
    #                   L^2
    # dist = |s|*L
    #
    # =>
    #
    #        | (Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay) |
    # dist = ---------------------------------
    #                       L
    set Ax [lindex $linesegment 0]
    set Ay [lindex $linesegment 1]
    set Bx [lindex $linesegment 2]
    set By [lindex $linesegment 3]
    set Cx [lindex $P 0]
    set Cy [lindex $P 1]
    if {$Ax==$Bx && $Ay==$By} {
	return [list [lengthOfPolyline [concat $P [lrange $linesegment 0 1]]] 0]
    } else {
	set L [expr {hypot($Bx-$Ax,$By-$Ay)}]
	set r [expr {(($Cx-$Ax)*($Bx-$Ax) + ($Cy-$Ay)*($By-$Ay))/pow($L,2)}]
	return [list [expr {abs(($Ay-$Cy)*($Bx-$Ax)-($Ax-$Cx)*($By-$Ay)) / $L}] $r]
    }
}

# ::math::geometry::findClosestPointOnLineSegment
#
#       Return the point on a line segment which is closest to a given point.
#
# Arguments:
#       P             a point
#       linesegment   a line segment
#
# Results:
#       Q             the point on the line segment that has the
#                     smallest distance to P
#
# Examples:
#     - findClosestPointOnLineSegment {5 10} {0 0 10 10}
#       Result: 7.5 7.5
#     - findClosestPointOnLineSegment {-10 0} {0 0 10 10}
#       Result: 0 0
#
proc ::math::geometry::findClosestPointOnLineSegment {P linesegment} {
    set result [findClosestPointOnLineImpl $P $linesegment]
    set Q [lindex $result 0]
    set r [lindex $result 1]
    if {$r<0} {
	return [lrange $linesegment 0 1]
    } elseif {$r>1} {
	return [lrange $linesegment 2 3]
    } else {
	return $Q
    }

}

# ::math::geometry::calculateDistanceToPolyline
#
#       Calculate the distance between a point and a polyline.
#
# Arguments:
#       P           a point
#       polyline    a polyline
#
# Results:
#       dist        the smallest distance between P and any point
#                   on the polyline
#
# Examples:
#     - calculateDistanceToPolyline {10 10} {0 0 10 5 20 0}
#       Result: 5.0
#     - calculateDistanceToPolyline {5 10} {0 0 10 5 20 0}
#       Result: 6.7082039325
#
proc ::math::geometry::calculateDistanceToPolyline {P polyline} {
    set minDist "Inf"
    foreach {Bx By} [lassign $polyline Ax Ay] {
	set dist [calculateDistanceToLineSegment $P [list $Ax $Ay $Bx $By]]
	if {$dist < $minDist} {
	    set minDist $dist
	}
	set Ax $Bx; set Ay $By
    }
    return $minDist
}

# ::math::geometry::calculateDistanceToPolygon
#
#       Calculate the distance between a point and a polygon.
#
# Arguments:
#       P           a point
#       polygon     a polygon
#
# Results:
#       dist        the smallest distance between P and any point
#                   on the polygon
#
# Note:
#       The polygon does not need to be closed - this is taken
#       care of in the procedure.
#
proc ::math::geometry::calculateDistanceToPolygon {P polygon} {
    return [::math::geometry::calculateDistanceToPolyline $P [ClosedPolygon $polygon]]
}

# ::math::geometry::findClosestPointOnPolyline
#
#       Return the point on a polyline which is closest to a given point.
#
# Arguments:
#       P           a point
#       polyline    a polyline
#
# Results:
#       Q           the point on the polyline that has the smallest
#                   distance to P
#
# Examples:
#     - findClosestPointOnPolyline {10 10} {0 0 10 5 20 0}
#       Result: 10 5
#     - findClosestPointOnPolyline {5 10} {0 0 10 5 20 0}
#       Result: 8.0 4.0
#
proc ::math::geometry::findClosestPointOnPolyline {P polyline} {
    set closestPoint "none"; set closestDistance "Inf"
    foreach {Bx By} [lassign $polyline Ax Ay] {
	set Q [findClosestPointOnLineSegment $P [list $Ax $Ay $Bx $By]]
	set dist [lengthOfPolyline [concat $P $Q]]
	if {$dist<$closestDistance} {
	    set closestPoint $Q
	    set closestDistance $dist
	}
	set Ax $Bx; set Ay $By
    }
    return $closestPoint
}






# ::math::geometry::lengthOfPolyline
#
#       Find the length of a polyline, i.e., the sum of the
#       lengths of the individual line segments.
#
# Arguments:
#       polyline      a polyline
#
# Results:
#       length        the length of the polyline
#
# Examples:
#     - lengthOfPolyline {0 0 5 0 5 10}
#       Result: 15.0
#
proc ::math::geometry::lengthOfPolyline {polyline} {
    set length 0
    foreach {x2 y2} [lassign $polyline x1 y1] {
	set length [expr {$length + hypot($x1-$x2,$y1-$y2)}]
	set x1 $x2; set y1 $y2
    }
    return $length
}




# ::math::geometry::movePointInDirection
#
#       Move a point in a given direction.
#
# Arguments:
#       P             the starting point
#       direction     the direction from P
#                     The direction is in 360-degrees going counter-clockwise,
#                     with "straight right" being 0 degrees
#       dist          the distance from P
#
# Results:
#       Q             the point which is found by starting in P and going
#                     in the given direction, until the distance between
#                     P and Q is dist
#
# Examples:
#     - movePointInDirection {0 0} 45.0 10
#       Result: 7.07106781187 7.07106781187
#
proc ::math::geometry::movePointInDirection {P direction dist} {
    set x [lindex $P 0]
    set y [lindex $P 1]
    set pi [expr {4*atan(1)}]
    set xt [expr {$x + $dist*cos(($direction*$pi)/180)}]
    set yt [expr {$y + $dist*sin(($direction*$pi)/180)}]
    return [list $xt $yt]
}


# ::math::geometry::angle
#
#       Calculates angle from the horizon (0,0)->(1,0) to a line.
#
# Arguments:
#       line          a line defined by two points A and B
#
# Results:
#       angle         the angle between the line (0,0)->(1,0) and (Ax,Ay)->(Bx,By).
#                     Angle is in 360-degrees going counter-clockwise
#
# Examples:
#     - angle {10 10 15 13}
#       Result: 30.9637565321
#
proc ::math::geometry::angle {line} {
    set x1 [lindex $line 0]
    set y1 [lindex $line 1]
    set x2 [lindex $line 2]
    set y2 [lindex $line 3]
    # - handle vertical lines
    if {$x1==$x2} {if {$y1<$y2} {return 90} else {return 270}}
    # - handle other lines
    set a [expr {atan(abs((1.0*$y1-$y2)/(1.0*$x1-$x2)))}] ; # a is between 0 and pi/2
    set pi [expr {4*atan(1)}]
    if {$y1<=$y2} {
	# line is going upwards
	if {$x1<$x2} {set b $a} else {set b [expr {$pi-$a}]}
    } else {
	# line is going downwards
	if {$x1<$x2} {set b [expr {2*$pi-$a}]} else {set b [expr {$pi+$a}]}
    }
    return [expr {$b/$pi*180}] ; # convert b to degrees
}




###
#
# Intersection procedures
#
###

# ::math::geometry::lineSegmentsIntersect
#
#       Checks whether two line segments intersect.
#
# Arguments:
#       linesegment1  the first line segment
#       linesegment2  the second line segment
#
# Results:
#       dointersect   a boolean saying whether the line segments intersect
#                     (i.e., have any points in common)
#
# Examples:
#     - lineSegmentsIntersect {0 0 10 10} {0 10 10 0}
#       Result: 1
#     - lineSegmentsIntersect {0 0 10 10} {20 20 20 30}
#       Result: 0
#     - lineSegmentsIntersect {0 0 10 10} {10 10 15 15}
#       Result: 1
#
proc ::math::geometry::lineSegmentsIntersect {linesegment1 linesegment2} {
    # Algorithm based on Sedgewick.
    set l1x1 [lindex $linesegment1 0]
    set l1y1 [lindex $linesegment1 1]
    set l1x2 [lindex $linesegment1 2]
    set l1y2 [lindex $linesegment1 3]
    set l2x1 [lindex $linesegment2 0]
    set l2y1 [lindex $linesegment2 1]
    set l2x2 [lindex $linesegment2 2]
    set l2y2 [lindex $linesegment2 3]

    #
    # First check the distance between the endpoints
    #
    set margin 1.0e-7
    if { [calculateDistanceToLineSegment [lrange $linesegment1 0 1] $linesegment2] < $margin } {
        return 1
    }
    if { [calculateDistanceToLineSegment [lrange $linesegment1 2 3] $linesegment2] < $margin } {
        return 1
    }
    if { [calculateDistanceToLineSegment [lrange $linesegment2 0 1] $linesegment1] < $margin } {
        return 1
    }
    if { [calculateDistanceToLineSegment [lrange $linesegment2 2 3] $linesegment1] < $margin } {
        return 1
    }

    return [expr {([ccw [list $l1x1 $l1y1] [list $l1x2 $l1y2] [list $l2x1 $l2y1]]\
	    *[ccw [list $l1x1 $l1y1] [list $l1x2 $l1y2] [list $l2x2 $l2y2]] <= 0) \
	    && ([ccw [list $l2x1 $l2y1] [list $l2x2 $l2y2] [list $l1x1 $l1y1]]\
	    *[ccw [list $l2x1 $l2y1] [list $l2x2 $l2y2] [list $l1x2 $l1y2]] <= 0)}]
}

# ::math::geometry::findLineSegmentIntersection
#
#       Returns the intersection point of two line segments.
#       Note: may also return "coincident" and "none".
#
# Arguments:
#       linesegment1  the first line segment
#       linesegment2  the second line segment
#
# Results:
#       P             the intersection point of linesegment1 and linesegment2.
#                     If linesegment1 and linesegment2 have an infinite number
#                     of points in common, the procedure returns "coincident".
#                     If there are no intersection points, the procedure
#                     returns "none".
#
# Examples:
#     - findLineSegmentIntersection {0 0 10 10} {0 10 10 0}
#       Result: 5.0 5.0
#     - findLineSegmentIntersection {0 0 10 10} {20 20 20 30}
#       Result: none
#     - findLineSegmentIntersection {0 0 10 10} {10 10 15 15}
#       Result: 10.0 10.0
#     - findLineSegmentIntersection {0 0 10 10} {5 5 15 15}
#       Result: coincident
#
proc ::math::geometry::findLineSegmentIntersection {linesegment1 linesegment2} {
    if {[lineSegmentsIntersect $linesegment1 $linesegment2]} {
	set lineintersect [findLineIntersection $linesegment1 $linesegment2]
	switch -- $lineintersect {

	    "coincident" {
		# lines are coincident
		set l1x1 [lindex $linesegment1 0]
		set l1y1 [lindex $linesegment1 1]
		set l1x2 [lindex $linesegment1 2]
		set l1y2 [lindex $linesegment1 3]
		set l2x1 [lindex $linesegment2 0]
		set l2y1 [lindex $linesegment2 1]
		set l2x2 [lindex $linesegment2 2]
		set l2y2 [lindex $linesegment2 3]
		# check if the line SEGMENTS overlap
		# (NOT enough to check if the x-intervals overlap (vertical lines!))
		set overlapx [intervalsOverlap $l1x1 $l1x2 $l2x1 $l2x2 0]
		set overlapy [intervalsOverlap $l1y1 $l1y2 $l2y1 $l2y2 0]
		if {$overlapx && $overlapy} {
		    return "coincident"
		} else {
		    return "none"
		}
	    }

	    "none" {
		# should never happen, because we call "lineSegmentsIntersect" first
		puts stderr "::math::geometry::findLineSegmentIntersection: suddenly no intersection?"
		return "none"
	    }

	    default {
		# lineintersect = the intersection point
		return $lineintersect
	    }
	}
    } else {
	return "none"
    }
}

# ::math::geometry::findLineIntersection {line1 line2}
#
#       Returns the intersection point of two lines.
#       Note: may also return "coincident" and "none".
#
# Arguments:
#       line1         the first line
#       line2         the second line
#
# Results:
#       P             the intersection point of line1 and line2.
#                     If line1 and line2 have an infinite number of points
#                     in common, the procedure returns "coincident".
#                     If there are no intersection points, the procedure
#                     returns "none".
#
# Examples:
#     - findLineIntersection {0 0 10 10} {0 10 10 0}
#       Result: 5.0 5.0
#     - findLineIntersection {0 0 10 10} {20 20 20 30}
#       Result: 20.0 20.0
#     - findLineIntersection {0 0 10 10} {10 10 15 15}
#       Result: coincident
#     - findLineIntersection {0 0 10 10} {5 5 15 15}
#       Result: coincident
#     - findLineIntersection {0 0 10 10} {0 1 10 11}
#       Result: none
#
proc ::math::geometry::findLineIntersection {line1 line2} {

    # References:
    # http://wiki.tcl.tk/12070 (Kevin Kenny)
    # http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/

    set l1x1 [lindex $line1 0]
    set l1y1 [lindex $line1 1]
    set l1x2 [lindex $line1 2]
    set l1y2 [lindex $line1 3]

    set l2x1 [lindex $line2 0]
    set l2y1 [lindex $line2 1]
    set l2x2 [lindex $line2 2]
    set l2y2 [lindex $line2 3]

    set d [expr {($l2y2 - $l2y1) * ($l1x2 - $l1x1) -
		 ($l2x2 - $l2x1) * ($l1y2 - $l1y1)}]
    set na [expr {($l2x2 - $l2x1) * ($l1y1 - $l2y1) -
		  ($l2y2 - $l2y1) * ($l1x1 - $l2x1)}]

    # http://local.wasp.uwa.edu.au/~pbourke/geometry/lineline2d/
    if {$d == 0} {
	if {$na == 0} {
	    return "coincident"
	} else {
	    return "none"
	}
    }
    set r [list \
               [expr {$l1x1 + $na * ($l1x2 - $l1x1) / $d}] \
               [expr {$l1y1 + $na * ($l1y2 - $l1y1) / $d}]]
    return $r
}


# ::math::geometry::polylinesIntersect
#
#       Checks whether two polylines intersect.
#
# Arguments;
#       polyline1     the first polyline
#       polyline2     the second polyline
#
# Results:
#       dointersect   a boolean saying whether the polylines intersect
#
# Examples:
#     - polylinesIntersect {0 0 10 10 10 20} {0 10 10 0}
#       Result: 1
#     - polylinesIntersect {0 0 10 10 10 20} {5 4 10 4}
#       Result: 0
#
proc ::math::geometry::polylinesIntersect {polyline1 polyline2} {
    return [polylinesBoundingIntersect $polyline1 $polyline2 0]
}

# ::math::geometry::polylinesBoundingIntersect
#
#       Check whether two polylines intersect, but reduce
#       the correctness of the result to the given granularity.
#       Use this for faster, but weaker, intersection checking.
#
#       How it works:
#          Each polyline is split into a number of smaller polylines,
#          consisting of granularity points each. If a pair of those smaller
#          lines' bounding boxes intersect, then this procedure returns 1,
#          otherwise it returns 0.
#
# Arguments:
#       polyline1     the first polyline
#       polyline2     the second polyline
#       granularity   the number of points in each part-polyline
#                     granularity<=1 means full correctness
#
# Results:
#       dointersect   a boolean saying whether the polylines intersect
#
# Examples:
#     - polylinesBoundingIntersect {0 0 10 10 10 20} {0 10 10 0} 2
#       Result: 1
#     - polylinesBoundingIntersect {0 0 10 10 10 20} {5 4 10 4} 2
#       Result: 1
#
proc ::math::geometry::polylinesBoundingIntersect {polyline1 polyline2 granularity} {
    if {$granularity<=1} {
	# Use perfect intersect
	# => first pin down where an intersection point may be, and then
	#    call MultilinesIntersectPerfect on those parts
	set granularity 10 ; # optimal search granularity?
	set perfectmatch 1
    } else {
	set perfectmatch 0
    }

    # split the lines into parts consisting of $granularity points
    set polyline1parts {}
    for {set i 0} {$i<[llength $polyline1]} {incr i [expr {2*$granularity-2}]} {
	lappend polyline1parts [lrange $polyline1 $i [expr {$i+2*$granularity-1}]]
    }
    set polyline2parts {}
    for {set i 0} {$i<[llength $polyline2]} {incr i [expr {2*$granularity-2}]} {
	lappend polyline2parts [lrange $polyline2 $i [expr {$i+2*$granularity-1}]]
    }

    # do any of the parts overlap?
    foreach part1 $polyline1parts {
	foreach part2 $polyline2parts {
	    set part1bbox [bbox $part1]
	    set part2bbox [bbox $part2]
	    if {[rectanglesOverlap [lrange $part1bbox 0 1] [lrange $part1bbox 2 3] \
		    [lrange $part2bbox 0 1] [lrange $part2bbox 2 3] 0]} {
		# the lines' bounding boxes intersect
		if {$perfectmatch} {
		    foreach {l1x2 l1y2} [lassign $part1 l1x1 l1y1] {
			foreach {l2x2 l2y2} [lassign $part2 l2x1 l2y1] {
			    if {[lineSegmentsIntersect [list $l1x1 $l1y1 $l1x2 $l1y2] \
				    [list $l2x1 $l2y1 $l2x2 $l2y2]]} {
				# two line segments overlap
				return 1
			    }
			    set l2x1 $l2x2; set l2y1 $l2y2
			}
			set l1x1 $l1x2; set l1y1 $l1y2
		    }
		    return 0
		} else {
		    return 1
		}
	    }
	}
    }
    return 0
}

# ::math::geometry::ccw
#
#       PRIVATE FUNCTION USED BY OTHER FUNCTIONS.
#       Returns whether traversing from A to B to C is CounterClockWise
#       Algorithm by Sedgewick.
#
# Arguments:
#       A             first point
#       B             second point
#       C             third point
#
# Reeults:
#       ccw           a boolean saying whether traversing from A to B to C
#                     is CounterClockWise
#
proc ::math::geometry::ccw {A B C} {
    set Ax [lindex $A 0]
    set Ay [lindex $A 1]
    set Bx [lindex $B 0]
    set By [lindex $B 1]
    set Cx [lindex $C 0]
    set Cy [lindex $C 1]
    set dx1 [expr {$Bx - $Ax}]
    set dy1 [expr {$By - $Ay}]
    set dx2 [expr {$Cx - $Ax}]
    set dy2 [expr {$Cy - $Ay}]
    if {$dx1*$dy2 > $dy1*$dx2} {return 1}
    if {$dx1*$dy2 < $dy1*$dx2} {return -1}
    if {($dx1*$dx2 < 0) || ($dy1*$dy2 < 0)} {return -1}
    if {($dx1*$dx1 + $dy1*$dy1) < ($dx2*$dx2+$dy2*$dy2)} {return 1}
    return 0
}







###
#
# Overlap procedures
#
###

# ::math::geometry::intervalsOverlap
#
#       Check whether two intervals overlap.
#       Examples:
#         - (2,4) and (5,3) overlap with strict=0 and strict=1
#         - (2,4) and (1,2) overlap with strict=0 but not with strict=1
#
# Arguments:
#       y1,y2         the first interval
#       y3,y4         the second interval
#       strict        choosing strict or non-strict interpretation
#
# Results:
#       dooverlap     a boolean saying whether the intervals overlap
#
# Examples:
#     - intervalsOverlap 2 4 4 6 1
#       Result: 0
#     - intervalsOverlap 2 4 4 6 0
#       Result: 1
#     - intervalsOverlap 4 2 3 5 0
#       Result: 1
#
proc ::math::geometry::intervalsOverlap {y1 y2 y3 y4 strict} {
    if {$y1>$y2} {
	set temp $y1
	set y1 $y2
	set y2 $temp
    }
    if {$y3>$y4} {
	set temp $y3
	set y3 $y4
	set y4 $temp
    }
    if {$strict} {
	return [expr {$y2>$y3 && $y4>$y1}]
    } else {
	return [expr {$y2>=$y3 && $y4>=$y1}]
    }
}

# ::math::geometry::rectanglesOverlap
#
#       Check whether two rectangles overlap (see also intervalsOverlap).
#
# Arguments:
#       P1            upper-left corner of the first rectangle
#       P2            lower-right corner of the first rectangle
#       Q1            upper-left corner of the second rectangle
#       Q2            lower-right corner of the second rectangle
#       strict        choosing strict or non-strict interpretation
#
# Results:
#       dooverlap     a boolean saying whether the rectangles overlap
#
# Examples:
#     - rectanglesOverlap {0 10} {10 0} {10 10} {20 0} 1
#       Result: 0
#     - rectanglesOverlap {0 10} {10 0} {10 10} {20 0} 0
#       Result: 1
#
proc ::math::geometry::rectanglesOverlap {P1 P2 Q1 Q2 strict} {
    set b1x1 [lindex $P1 0]
    set b1y1 [lindex $P1 1]
    set b1x2 [lindex $P2 0]
    set b1y2 [lindex $P2 1]
    set b2x1 [lindex $Q1 0]
    set b2y1 [lindex $Q1 1]
    set b2x2 [lindex $Q2 0]
    set b2y2 [lindex $Q2 1]
    # ensure b1x1<=b1x2 etc.
    if {$b1x1 > $b1x2} {
	set temp $b1x1
	set b1x1 $b1x2
	set b1x2 $temp
    }
    if {$b1y1 > $b1y2} {
	set temp $b1y1
	set b1y1 $b1y2
	set b1y2 $temp
    }
    if {$b2x1 > $b2x2} {
	set temp $b2x1
	set b2x1 $b2x2
	set b2x2 $temp
    }
    if {$b2y1 > $b2y2} {
	set temp $b2y1
	set b2y1 $b2y2
	set b2y2 $temp
    }
    # Check if the boxes intersect
    # (From: Cormen, Leiserson, and Rivests' "Algorithms", page 889)
    if {$strict} {
	return [expr {($b1x2>$b2x1) && ($b2x2>$b1x1) \
		&& ($b1y2>$b2y1) && ($b2y2>$b1y1)}]
    } else {
	return [expr {($b1x2>=$b2x1) && ($b2x2>=$b1x1) \
		&& ($b1y2>=$b2y1) && ($b2y2>=$b1y1)}]
    }
}



# ::math::geometry::bbox
#
#       Calculate the bounding box of a polyline.
#
# Arguments:
#       polyline      a polyline
#
# Results:
#       x1,y1,x2,y2   four coordinates where (x1,y1) is the upper-left corner
#                     of the bounding box, and (x2,y2) is the lower-right corner
#
# Examples:
#     - bbox {0 10 4 1 6 23 -12 5}
#       Result: -12 1 6 23
#
proc ::math::geometry::bbox {polyline} {
    set minX [lindex $polyline 0]
    set maxX $minX
    set minY [lindex $polyline 1]
    set maxY $minY
    foreach {x y} $polyline {
	if {$x < $minX} {set minX $x}
	if {$x > $maxX} {set maxX $x}
	if {$y < $minY} {set minY $y}
	if {$y > $maxY} {set maxY $y}
    }
    return [list $minX $minY $maxX $maxY]
}

# ::math::geometry::ClosedPolygon
#
#       Return a closed polygon - used internally
#
# Arguments:
#       polygon       a polygon
#
# Results:
#       closedpolygon a polygon whose first and last vertices
#                     coincide
#
proc ::math::geometry::ClosedPolygon {polygon} {

    lassign $polygon x y
    if { $x != [lindex $polygon end-1] ||
         $y != [lindex $polygon end]     } {

        lappend polygon $x $y

    }
    return $polygon
}


# ::math::geometry::pointInsidePolygon
#
#       Determine if a point is completely inside a polygon. If the point
#       touches the polygon, then the point is not complete inside the
#       polygon.
#
# Arguments:
#       P             a point
#       polygon       a polygon
#
# Results:
#       isinside      a boolean saying whether the point is
#                     completely inside the polygon or not
#
# Examples:
#     - pointInsidePolygon {5 5} {4 4 4 6 6 6 6 4}
#       Result: 1
#     - pointInsidePolygon {5 5} {6 6 6 7 7 7}
#       Result: 0
#
proc ::math::geometry::pointInsidePolygon {P polygon} {
    # check if P is on one of the polygon's sides (if so, P is not
    # inside the polygon)
    set closedPolygon [ClosedPolygon $polygon]

    foreach {x2 y2} [lassign $closedPolygon x1 y1] {
	if {[calculateDistanceToLineSegment $P [list $x1 $y1 $x2 $y2]]<0.0000001} {
	    return 0
	}
	set x1 $x2; set y1 $y2
    }

    # Algorithm
    #
    # Consider a straight line going from P to a point far away from both
    # P and the polygon (in particular outside the polygon).
    #   - If the line intersects with 0 of the polygon's sides, then
    #     P must be outside the polygon.
    #   - If the line intersects with 1 of the polygon's sides, then
    #     P must be inside the polygon (since the other end of the line
    #     is outside the polygon).
    #   - If the line intersects with 2 of the polygon's sides, then
    #     the line must pass into one polygon area and out of it again,
    #     and hence P is outside the polygon.
    #   - In general: if the line intersects with the polygon's sides an odd
    #     number of times, then P is inside the polygon. Note: we also have
    #     to check whether the line crosses one of the polygon's
    #     bend points for the same reason.

    # get point far away and define the line
    set polygonBbox [bbox $polygon]

    set pointFarAway [list \
        [expr {[lindex $polygonBbox 0]-[lindex $polygonBbox 2]}] \
        [expr {[lindex $polygonBbox 1]-0.1*[lindex $polygonBbox 3]}]]

    set infinityLine [concat $pointFarAway $P]

    # calculate number of intersections
    set noOfIntersections 0
    #   1. count intersections between the line and the polygon's sides
    foreach {x2 y2} [lassign $closedPolygon x1 y1] {
	if {[lineSegmentsIntersect $infinityLine [list $x1 $y1 $x2 $y2]]} {
	    incr noOfIntersections
	}
	set x1 $x2; set y1 $y2
    }
    #   2. count intersections between the line and the polygon's points
    foreach {x1 y1} $closedPolygon {
	if {[calculateDistanceToLineSegment [list $x1 $y1] $infinityLine]<0.0000001} {
	    incr noOfIntersections
	}
    }
    return [expr {$noOfIntersections % 2}]
}

# See ticket [dc49af96c2]
# Original code found at: https://www.ecse.rpi.edu/~wrf/Research/Short_Notes/pnpoly.html
# Thanks to Christian Gollwitzer, Peter Lewerin and Eduard Zozuly
# Replaced by:
proc ::math::geometry::pointInsidePolygon {point polygon} {
    lassign $point testx testy
    foreach {x y} $polygon {
        lappend vertx $x
        lappend verty $y
    }
    set c 0
    set nvert [llength $vertx]
    for {set i 0 ; set j [expr {$nvert-1}]} {$i < $nvert} {set j $i ; incr i} {
        if {
            (([lindex $verty $i]>$testy) != ([lindex $verty $j]>$testy)) &&
            ($testx < ([lindex $vertx $j] - [lindex $vertx $i]) *
            ($testy - [lindex $verty $i]) /
            ([lindex $verty $j] - [lindex $verty $i]) + [lindex $vertx $i])
        } {
            set c [expr {!$c}]
        }
    }
    return $c
}

# ::math::geometry::pointInsidePolygonAlt
#
#       Determine if a point is completely inside a polygon. If the point
#       touches the polygon, then the point is not complete inside the
#       polygon.
#       This alternative algorithm works with complex (self-intersecting)
#       polygons in a "natural" way. It uses the winding number instead
#       of the number of crossings.
#
#       See: http://geomalgorithms.com/a03-_inclusion.html
#
# Arguments:
#       P             a point
#       polygon       a polygon
#
# Results:
#       isinside      a boolean saying whether the point is
#                     completely inside the polygon or not
#

# Auxiliary procedure:
#     > 0 if point 2 left of line through points 0 and 1
#     < 0 if point 2 right of the line
#     = 0 if point on the line
#
proc ::math::geometry::LeftOfEdge {x0 y0 x1 y1 x2 y2} {
    expr {($x1 - $x0) * ($y2 - $y0) - ($x2 - $x0) * ($y1 - $y0)}
}

proc ::math::geometry::pointInsidePolygonAlt {point polygon} {
    lassign $point testx testy
    foreach {x y} $polygon {
        lappend vertx $x
        lappend verty $y
    }
    set w 0
    set nvert [llength $vertx]
    for {set i 0} {$i < $nvert} {incr i} {
        set j [expr {$i+1}]
        if { $j == $nvert } {
            set j 0
        }
        if { [lindex $verty $i] <= $testy } {
            if { [lindex $verty $j] > $testy } {
                if { [LeftOfEdge [lindex $vertx $i] [lindex $verty $i] [lindex $vertx $j] [lindex $verty $j] $testx $testy] > 0.0 } {
                    incr w
                }
            }
        } else {
            if { [lindex $verty $j] <= $testy } {
                if { [LeftOfEdge [lindex $vertx $i] [lindex $verty $i] [lindex $vertx $j] [lindex $verty $j] $testx $testy] < 0.0 } {
                    incr w -1
                }
            }
        }
    }
    return [expr {$w != 0}]
}

# ::math::geometry::rectangleInsidePolygon
#
#       Determine if a rectangle is completely inside a polygon. If polygon
#       touches the rectangle, then the rectangle is not complete inside the
#       polygon.
#
# Arguments:
#       P1            upper-left corner of the rectangle
#       P2            lower-right corner of the rectangle
#       polygon       a polygon
#
# Results:
#       isinside      a boolean saying whether the rectangle is
#                     completely inside the polygon or not
#
# Examples:
#     - rectangleInsidePolygon {0 10} {10 0} {-10 -10 0 11 11 11 11 0}
#       Result: 1
#     - rectangleInsidePolygon {0 0} {0 0} {-16 14 5 -16 -16 -25 -21 16 -19 24}
#       Result: 1
#     - rectangleInsidePolygon {0 0} {0 0} {2 2 2 4 4 4 4 2}
#       Result: 0
#
proc ::math::geometry::rectangleInsidePolygon {P1 P2 polygon} {
    # get coordinates of rectangle
    set bx1 [lindex $P1 0]
    set by1 [lindex $P1 1]
    set bx2 [lindex $P2 0]
    set by2 [lindex $P2 1]

    # if rectangle does not overlap with the bbox of polygon, then the
    # rectangle cannot be inside the polygon (this is a quick way to
    # get an answer in many cases)
    set polygonBbox [bbox $polygon]
    set polygonP1x [lindex $polygonBbox 0]
    set polygonP1y [lindex $polygonBbox 1]
    set polygonP2x [lindex $polygonBbox 2]
    set polygonP2y [lindex $polygonBbox 3]
    if {![rectanglesOverlap [list $bx1 $by1] [list $bx2 $by2] \
	    [list $polygonP1x $polygonP1y] [list $polygonP2x $polygonP2y] 0]} {
	return 0
    }

    # 1. if one of the points of the polygon is inside the rectangle,
    # then the rectangle cannot be inside the polygon
    foreach {x y} $polygon {
	if {$bx1<$x && $x<$bx2 && $by1<$y && $y<$by2} {
	    return 0
	}
    }

    # 2. if one of the line segments of the polygon intersect with the
    # rectangle, then the rectangle cannot be inside the polygon
    set rectanglePolyline [list $bx1 $by1 $bx2 $by1 $bx2 $by2 $bx1 $by2 $bx1 $by1]
    set closedPolygon [ClosedPolygon $polygon]
    if {[polylinesIntersect $closedPolygon $rectanglePolyline]} {
	return 0
    }

    # at this point we know that:
    #  1. the polygon has no points inside the rectangle
    #  2. the polygon's sides don't intersect with the rectangle
    # therefore:
    #  either the rectangle is (completely) inside the polygon, or
    #  the rectangle is (completely) outside the polygon

    # final test: if one of the points on the rectangle is inside the
    # polygon, then the whole rectangle must be inside the rectangle
    return [pointInsidePolygon [list $bx1 $by1] $polygon]
}


# ::math::geometry::areaPolygon
#
#       Determine the area enclosed by a (non-complex) polygon
#
# Arguments:
#       polygon       a polygon
#
# Results:
#       area          the area enclosed by the polygon
#
# Examples:
#     - areaPolygon {-10 -10 10 -10 10 10 -10 10}
#       Result: 400
#
proc ::math::geometry::areaPolygon {polygon} {

    # get last pair of the polygon for start:
    set b1 [lindex $polygon end-1]; set b2 [lindex $polygon end]

    set area 0.0
    foreach {c1 c2} $polygon {
        set area [expr {$area + ($b1*$c2 - $b2*$c1)}]
        set b1   $c1
        set b2   $c2
    }
    expr {0.5*abs($area)}
}

# ::math::geometry::inproduct
#
#       Determine the inproduct of two vectors
#
# Arguments:
#       vector1       first vector
#       vector2       second vector
#
# Results:
#       inproduct     the inproduct
#
proc ::math::geometry::inproduct {vector1 vector2} {

    set inproduct 0.0
    foreach v1 $vector1 v2 $vector2 {
        set inproduct [expr {$inproduct + $v1 * $v2}]
    }

    return $inproduct
}

# ::math::geometry::angleBetween
#
#       Determine the angle between two vectors (degrees)
#
# Arguments:
#       vector1       first vector
#       vector2       second vector
#
# Results:
#       angle         the angle in degrees
#
proc ::math::geometry::angleBetween {vector1 vector2} {
    variable todeg

    set inproduct 0.0
    set length1   0.0
    set length2   0.0
    foreach v1 $vector1 v2 $vector2 {
        set inproduct [expr {$inproduct + $v1 * $v2}]
        set length1   [expr {$length1   + $v1 * $v1}]
        set length2   [expr {$length2   + $v2 * $v2}]
    }
    set angle [expr {acos($inproduct/sqrt($length1 * $length2)) * $todeg}]

    return $angle
}

# ::math::geometry::areaParallellogram
#
#       Determine the area of the parallellogram spanned by two vectors
#
# Arguments:
#       vector1       first vector
#       vector2       second vector
#
# Results:
#       area          the area of the parallellogram
#
proc ::math::geometry::areaParallellogram {vector1 vector2} {

    lassign $vector1 x1 y1; lassign $vector2 x2 y2

    set area [expr {abs($x2 * $y1 - $x1 * $y2}]

    return $area
}

# ::math::geometry::translate
#
#       Translate a polyline over a given vector
#
# Arguments:
#       vector        Translation vector
#       polyline      Polyline (or any list of coordinate pairs)
#
# Results:
#       newPolyline   Translated poyline
#
proc ::math::geometry::translate {vector polyline} {

    set newPolyline $polyline

    lassign $vector xt yt

    set idx 0
    foreach {x y} $polyline {
        lset newPolyline $idx [expr {$x + $xt}]
        incr idx
        lset newPolyline $idx [expr {$y + $yt}]
        incr idx
    }

    return $newPolyline
}

# ::math::geometry::rotate
#
#       Rotate a polyline over a given angle (degrees) around the origin
#
# Arguments:
#       angle         rotation angle (degrees)
#       polyline      polyline (or any list of coordinate pairs)
#
# Results:
#       newPolyline   rotated polyline
#
# Note:
#       rotation is counterclockwise
#
proc ::math::geometry::rotate {angle polyline} {
    variable torad

    set angle [expr {$torad * $angle}]
    set cosa  [expr {cos($angle)}]
    set sina  [expr {sin($angle)}]

    set newPolyline $polyline

    set idx 0
    foreach {x y} $polyline {
        set newx [expr {$cosa * $x - $sina *$y}]
        set newy [expr {$sina * $x + $cosa *$y}]

        lset newPolyline $idx $newx
        incr idx
        lset newPolyline $idx $newy
        incr idx
    }

    return $newPolyline
}

# ::math::geometry::reflect
#
#       Reflect a polyline in a line through the origin at a given angle to the x-axis
#
# Arguments:
#       angle         angle of the line of reflection (degrees)
#       polyline      polyline (or any list of coordinate pairs)
#
# Results:
#       newPolyline   reflected polyline
#
# Note:
#       the angle is used counterclockwise
#
proc ::math::geometry::reflect {angle polyline} {
    variable torad

    set angle [expr {2.0 * $torad * $angle}]
    set cosa  [expr {cos($angle)}]
    set sina  [expr {sin($angle)}]

    set newPolyline $polyline

    set idx 0
    foreach {x y} $polyline {
        set newx [expr {$cosa * $x + $sina *$y}]
        set newy [expr {$sina * $x - $cosa *$y}]

        lset newPolyline $idx $newx
        incr idx
        lset newPolyline $idx $newy
        incr idx
    }

    return $newPolyline
}

# ::math::geometry::degToRad
#
#       Convert from degrees to radians
#
# Arguments:
#       angle         angle (degrees)
#
# Results:
#       angle         angle in radians
#
proc ::math::geometry::degToRad {angle} {
    variable torad

    return [expr {$angle * $torad}]
}

# ::math::geometry::radToDeg
#
#       Convert from radians to degrees
#
# Arguments:
#       angle         angle (radians)
#
# Results:
#       angle         angle in degrees
#
proc ::math::geometry::radToDeg {angle} {
    variable todeg

    return [expr {$angle * $todeg}]
}

# # ## ### ##### #############

namespace eval ::math::geometry {
    variable pi    [expr { 4 * atan(1) }]
    variable torad [expr { (4 * atan(1)) / 180.0 }]
    variable todeg [expr { 180.0 / (4 * atan(1)) }]

    namespace export \
	+ - s* direction v h p between distance length \
	nwse rect octant findLineSegmentIntersection \
	findLineIntersection bbox x y conjx conjy \
	calculateDistanceToLine findClosestPointOnLine \
	calculateDistanceToLineSegment findClosestPointOnLineSegment \
	calculateDistanceToPolylineSegment findClosestPointOnPolyline lengthOfPolyline \
	movePointInDirection lineSegmentsIntersect findLineSegmentIntersection findLineIntersection \
	polylinesIntersect polylinesBoundingIntersect intervalsOverlap rectanglesOverlap pointInsidePolygon pointInsidePolygonAlt \
	rectangleInsidePolygon areaPolygon translate rotate reflect degToRad radToDeg
}

package provide math::geometry 1.2.3