How to determine left and right side of crossing lines? [closed]

Question

Given two lines of which we know the starting XY points and intersecting point, I need to find which line is left and which line is right. The lines are always crossing and can be turned in any direction. Here are 4 possible scenarios as an image where A is always on the left line:

Below are the same four scenarios as data, where every scenario has two crossing lines. Each line has a start point (x, y) and an end (x, y) point. In the scenarios below, the starting point of A is always on the left of the starting point B and therefore lineA is on the left of lineB:

line_A1 = ((1, 1), (0, 0))
line_B1 = ((0, 1), (1, 0))

line_A2 = ((0, 1), (1, 0))
line_B2 = ((0, 0), (1, 1))

line_A3 = ((0, 0), (1, 1))
line_B3 = ((1, 0), (0, 1))

line_A4 = ((1, 0), (0, 1))
line_B4 = ((1, 1), (0, 0))

middlepoint_C = (0.5, 0.5) #The middle point in all 4 scenarios is (0.5, 0.5)

How can we determine left and right side of the crossing lines in the four scenarios above?

Preference for solution would be with built in Python modules or shapely, but all solutions are welcome since we are kinda stuck.

Answer 1

Consider using the cross product approach as discussed here. Pure mathematical approach that does not rely on any third party module.

I would take the start point and test against the line. I think this code assumes the line is a simple line composed of 2 vertices and not a polyline.

Answer 2

Here is an approach using Postgis, it shouldn't be difficult to replicate in Python, code finds the azimuths from the intersection to the starting points, if it geom A's azimuth is greater than geom B's and its difference (azimuth A - azimuth B) is not greater than 1pi (180 degrees turns the direction to the opposite, to take into account cases where startpoints fall both sides close to the north), left will be A.

WITH 
  tbla AS (select st_geomfromtext('LINESTRING(1 1, 0 0)',4326) as A, st_geomfromtext('LINESTRING(0 1, 1 0)',4326) as B), 
  tblb as (select st_astext(a) as a,st_astext(b) as b, st_astext(st_intersection(a,b)) as aib, st_azimuth(st_intersection(a,b), st_startpoint(a)) as za, 
  st_azimuth(st_intersection(a,b), st_startpoint(b)) as zb from tbla) 
select a, b, aib, za, zb, 
case when za > zb and za-zb < 3.14 then 'left is a' 
     when zb > za and zb-za < 3.14 then 'left is b' 
     when za > zb and za-zb > 3.14 then 'left is b' 
     else 'left is a'  end direction from tblb;
          a          |          b          |      aib       |        za         |        zb        | direction 
---------------------+---------------------+----------------+-------------------+------------------+-----------
 LINESTRING(1 1,0 0) | LINESTRING(0 1,1 0) | POINT(0.5 0.5) | 0.785398163397448 | 5.49778714378214 | left is a

Answer 3

Here is working code example in Python based on great answer of Elio in Postgis.

from shapely.ops import LineString, Point
import math

def calc_azimuth(point1, point2):
  """Calculate azimuth in radians between 2 shapely points."""
  angle = math.atan2(point2.x - point1.x, point2.y - point1.y)
  return angle

A4 = Point(2, 0)
B4 = Point(2, 1)
C4 = Point(1, 0.5)
geomA = LineString([A4, C4])
geomB = LineString([B4, C4])
azimuthA = calc_azimuth(Point(geomA.coords[0]), geomC)
azimuthB = calc_azimuth(Point(geomB.coords[0]), geomC)
if (azimuthA > azimuthB) and ((azimuthA - azimuthB) < math.pi):
  print('1: Left is A')
elif (azimuthB > azimuthA) and ((azimuthB - azimuthA) < math.pi):
  print('2: Left is B')
elif (azimuthA > azimuthB) and ((azimuthA - azimuthB) > math.pi):
  print('3: Left is B')
else:
  print('4: Left is A')