The math::geometry package is a collection of functions for computations and manipulations on two-dimensional geometrical objects, such as points, lines and polygons.

The geometrical objects are implemented as plain lists of coordinates. For instance a line is defined by a list of four numbers, the x- and y-coordinate of a first point and the x- and y-coordinates of a second point on the line.

The various types of object are recognised by the number of coordinate pairs and the context in which they are used: a list of four elements can be regarded as an infinite line, a finite line segment but also as a polyline of one segment and a point set of two points.

Currently the following types of objects are distinguished:

point - a list of two coordinates representing the x- and y-coordinates respectively.
line - a list of four coordinates, interpreted as the x- and y-coordinates of two distinct points on the line.
line segment - a list of four coordinates, interpreted as the x- and y-coordinates of the first and the last points on the line segment.
polyline - a list of an even number of coordinates, interpreted as the x- and y-coordinates of an ordered set of points.
polygon - like a polyline, but the implicit assumption is that the polyline is closed (if the first and last points do not coincide, the missing segment is automatically added).
point set - again a list of an even number of coordinates, but the points are regarded without any ordering.

PROCEDURES

The package defines the following public procedures:

::math::geometry::angle line

Calculate the angle from the positive x-axis to a given line (in two dimensions only).

list line: Coordinates of the line

::math::geometry::calculateDistanceToLine P line

Calculate the distance of point P to the (infinite) line and return the result

list P: List of two numbers, the coordinates of the point
list line: List of four numbers, the coordinates of two points on the line

::math::geometry::calculateDistanceToLineSegment P linesegment

Calculate the distance of point P to the (finite) line segment and return the result.

list P: List of two numbers, the coordinates of the point
list linesegment: List of four numbers, the coordinates of the first and last points of the line segment

::math::geometry::calculateDistanceToPolyline P polyline

Calculate the distance of point P to the polyline and return the result.

list P: List of two numbers, the coordinates of the point
list polyline: List of numbers, the coordinates of the vertices of the polyline

::math::geometry::findClosestPointOnLine P line

Return the point on a line which is closest to a given point.

list P: List of two numbers, the coordinates of the point
list line: List of four numbers, the coordinates of two points on the line

::math::geometry::findClosestPointOnLineSegment P linesegment

Return the point on a line segment which is closest to a given point.

list P: List of two numbers, the coordinates of the point
list linesegment: List of four numbers, the first and last points on the line segment

::math::geometry::findClosestPointOnPolyline P polyline

Return the point on a polyline which is closest to a given point.

list P: List of two numbers, the coordinates of the point
list polyline: List of numbers, the vertices of the polyline

::math::geometry::lengthOfPolyline polyline

Return the length of the polyline (note: it not regarded as a polygon)

list polyline: List of numbers, the vertices of the polyline

::math::geometry::movePointInDirection P direction dist

Move a point over a given distance in a given direction and return the new coordinates (in two dimensions only).

list P: Coordinates of the point to be moved
double direction: Direction (in degrees; 0 is to the right, 90 upwards)
list dist: Distance over which to move the point

::math::geometry::lineSegmentsIntersect linesegment1 linesegment2

Check if two line segments intersect or coincide. Returns 1 if that is the case, 0 otherwise (in two dimensions only).

list linesegment1: First line segment
list linesegment2: Second line segment

::math::geometry::findLineSegmentIntersection linesegment1 linesegment2

Find the intersection point of two line segments. Return the coordinates or the keywords "coincident" or "none" if the line segments coincide or have no points in common (in two dimensions only).

list linesegment1: First line segment
list linesegment2: Second line segment

::math::geometry::findLineIntersection line1 line2

Find the intersection point of two (infinite) lines. Return the coordinates or the keywords "coincident" or "none" if the lines coincide or have no points in common (in two dimensions only).

list line1: First line
list line2: Second line

::math::geometry::polylinesIntersect polyline1 polyline2

Check if two polylines intersect or not (in two dimensions only).

list polyline1: First polyline
list polyline2: Second polyline

::math::geometry::polylinesBoundingIntersect polyline1 polyline2 granularity

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.

list polyline1: First polyline
list polyline2: Second polyline
int granularity: Number of points in each part (<=1 means check every edge)

::math::geometry::intervalsOverlap y1 y2 y3 y4 strict