# What’s the fastest way of checking if a point is inside a polygon in python

Each Answer to this Q is separated by one/two green lines.

I found two main methods to look if a point belongs inside a polygon. One is using the ray tracing method used here, which is the most recommended answer, the other is using matplotlib path.contains_points (which seems a bit obscure to me). I will have to check lots of points continuously. Does anybody know if any of these two is more recommendable than the other or if there are even better third options?

UPDATE:

I checked the two methods and matplotlib looks much faster.

from time import time
import numpy as np
import matplotlib.path as mpltPath

# regular polygon for testing
lenpoly = 100
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in np.linspace(0,2*np.pi,lenpoly)[:-1]]

# random points set of points to test
N = 10000
points = np.random.rand(N,2)

# Ray tracing
def ray_tracing_method(x,y,poly):

n = len(poly)
inside = False

p1x,p1y = poly[0]
for i in range(n+1):
p2x,p2y = poly[i % n]
if y > min(p1y,p2y):
if y <= max(p1y,p2y):
if x <= max(p1x,p2x):
if p1y != p2y:
xints = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x or x <= xints:
inside = not inside
p1x,p1y = p2x,p2y

return inside

start_time = time()
inside1 = [ray_tracing_method(point[0], point[1], polygon) for point in points]
print("Ray Tracing Elapsed time: " + str(time()-start_time))

# Matplotlib mplPath
start_time = time()
path = mpltPath.Path(polygon)
inside2 = path.contains_points(points)
print("Matplotlib contains_points Elapsed time: " + str(time()-start_time))

which gives,

Ray Tracing Elapsed time: 0.441395998001
Matplotlib contains_points Elapsed time: 0.00994491577148

Same relative difference was obtained one using a triangle instead of the 100 sides polygon. I will also check shapely since it looks a package just devoted to these kind of problems

You can consider shapely:

from shapely.geometry import Point
from shapely.geometry.polygon import Polygon

point = Point(0.5, 0.5)
polygon = Polygon([(0, 0), (0, 1), (1, 1), (1, 0)])
print(polygon.contains(point))

From the methods you’ve mentioned I’ve only used the second, path.contains_points, and it works fine. In any case depending on the precision you need for your test I would suggest creating a numpy bool grid with all nodes inside the polygon to be True (False if not). If you are going to make a test for a lot of points this might be faster (although notice this relies you are making a test within a “pixel” tolerance):

from matplotlib import path
import matplotlib.pyplot as plt
import numpy as np

first = -3
size  = (3-first)/100
xv,yv = np.meshgrid(np.linspace(-3,3,100),np.linspace(-3,3,100))
p = path.Path([(0,0), (0, 1), (1, 1), (1, 0)])  # square with legs length 1 and bottom left corner at the origin
flags = p.contains_points(np.hstack((xv.flatten()[:,np.newaxis],yv.flatten()[:,np.newaxis])))
grid = np.zeros((101,101),dtype="bool")
grid[((xv.flatten()-first)/size).astype('int'),((yv.flatten()-first)/size).astype('int')] = flags

xi,yi = np.random.randint(-300,300,100)/100,np.random.randint(-300,300,100)/100
vflag = grid[((xi-first)/size).astype('int'),((yi-first)/size).astype('int')]
plt.imshow(grid.T,origin='lower',interpolation='nearest',cmap='binary')
plt.scatter(((xi-first)/size).astype('int'),((yi-first)/size).astype('int'),c=vflag,cmap='Greens',s=90)
plt.show()

, the results is this:

If speed is what you need and extra dependencies are not a problem, you maybe find numba quite useful (now it is pretty easy to install, on any platform). The classic ray_tracing approach you proposed can be easily ported to numba by using numba @jit decorator and casting the polygon to a numpy array. The code should look like:

@jit(nopython=True)
def ray_tracing(x,y,poly):
n = len(poly)
inside = False
p2x = 0.0
p2y = 0.0
xints = 0.0
p1x,p1y = poly[0]
for i in range(n+1):
p2x,p2y = poly[i % n]
if y > min(p1y,p2y):
if y <= max(p1y,p2y):
if x <= max(p1x,p2x):
if p1y != p2y:
xints = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x or x <= xints:
inside = not inside
p1x,p1y = p2x,p2y

return inside

The first execution will take a little longer than any subsequent call:

%%time
polygon=np.array(polygon)
inside1 = [numba_ray_tracing_method(point[0], point[1], polygon) for
point in points]

CPU times: user 129 ms, sys: 4.08 ms, total: 133 ms
Wall time: 132 ms

Which, after compilation will decrease to:

CPU times: user 18.7 ms, sys: 320 µs, total: 19.1 ms
Wall time: 18.4 ms

If you need speed at the first call of the function you can then pre-compile the code in a module using pycc. Store the function in a src.py like:

from numba import jit
from numba.pycc import CC
cc = CC('nbspatial')

@cc.export('ray_tracing',  'b1(f8, f8, f8[:,:])')
@jit(nopython=True)
def ray_tracing(x,y,poly):
n = len(poly)
inside = False
p2x = 0.0
p2y = 0.0
xints = 0.0
p1x,p1y = poly[0]
for i in range(n+1):
p2x,p2y = poly[i % n]
if y > min(p1y,p2y):
if y <= max(p1y,p2y):
if x <= max(p1x,p2x):
if p1y != p2y:
xints = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x or x <= xints:
inside = not inside
p1x,p1y = p2x,p2y

return inside

if __name__ == "__main__":
cc.compile()

Build it with python src.py and run:

import nbspatial

import numpy as np
lenpoly = 100
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in
np.linspace(0,2*np.pi,lenpoly)[:-1]]

# random points set of points to test
N = 10000
# making a list instead of a generator to help debug
points = zip(np.random.random(N),np.random.random(N))

polygon = np.array(polygon)

%%time
result = [nbspatial.ray_tracing(point[0], point[1], polygon) for point in points]

CPU times: user 20.7 ms, sys: 64 µs, total: 20.8 ms
Wall time: 19.9 ms

In the numba code I used:
‘b1(f8, f8, f8[:,:])’

In order to compile with nopython=True, each var needs to be declared before the for loop.

In the prebuild src code the line:

@cc.export('ray_tracing' , 'b1(f8, f8, f8[:,:])')

Is used to declare the function name and its I/O var types, a boolean output b1 and two floats f8 and a two-dimensional array of floats f8[:,:] as input.

## Edit Jan/4/2021

For my use case, I need to check if multiple points are inside a single polygon – In such a context, it is useful to take advantage of numba parallel capabilities to loop over a series of points. The example above can be changed to:

from numba import jit, njit
import numba
import numpy as np

@jit(nopython=True)
def pointinpolygon(x,y,poly):
n = len(poly)
inside = False
p2x = 0.0
p2y = 0.0
xints = 0.0
p1x,p1y = poly[0]
for i in numba.prange(n+1):
p2x,p2y = poly[i % n]
if y > min(p1y,p2y):
if y <= max(p1y,p2y):
if x <= max(p1x,p2x):
if p1y != p2y:
xints = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x or x <= xints:
inside = not inside
p1x,p1y = p2x,p2y

return inside

@njit(parallel=True)
def parallelpointinpolygon(points, polygon):
D = np.empty(len(points), dtype=numba.boolean)
for i in numba.prange(0, len(D)):
D[i] = pointinpolygon(points[i,0], points[i,1], polygon)
return D

Note: pre-compiling the above code will not enable the parallel capabilities of numba (parallel CPU target is not supported by pycc/AOT compilation) see: https://github.com/numba/numba/issues/3336

Test:

import numpy as np
lenpoly = 100
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in np.linspace(0,2*np.pi,lenpoly)[:-1]]
polygon = np.array(polygon)
N = 10000
points = np.random.uniform(-1.5, 1.5, size=(N, 2))

For N=10000 on a 72 core machine, returns:

%%timeit
parallelpointinpolygon(points, polygon)
# 480 µs ± 8.19 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

## Edit 17 Feb ’21:

• fixing loop to start from 0 instead of 1 (thanks @mehdi):

for i in numba.prange(0, len(D))

## Edit 20 Feb ’21:

Follow-up on the comparison made by @mehdi, I am adding a GPU-based method below. It uses the point_in_polygon method, from the cuspatial library:

import numpy as np
import cudf
import cuspatial

N = 100000002
lenpoly = 1000
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in
np.linspace(0,2*np.pi,lenpoly)]
polygon = np.array(polygon)
points = np.random.uniform(-1.5, 1.5, size=(N, 2))

x_pnt = points[:,0]
y_pnt = points[:,1]
x_poly =polygon[:,0]
y_poly = polygon[:,1]
result = cuspatial.point_in_polygon(
x_pnt,
y_pnt,
cudf.Series([0], index=['geom']),
cudf.Series([0], name="r_pos", dtype="int32"),
x_poly,
y_poly,
)

Following @Mehdi comparison. For N=100000002 and lenpoly=1000 – I got the following results:

time_parallelpointinpolygon:         161.54760098457336
time_mpltPath:                       307.1664695739746
time_ray_tracing_numpy_numba:        353.07356882095337
time_is_inside_sm_parallel:          37.45389246940613
time_is_inside_postgis_parallel:     127.13793849945068
time_is_inside_rapids:               4.246025562286377

hardware specs:

• CPU Intel xeon E1240
• GPU Nvidia GTX 1070

Notes:

• The cuspatial.point_in_poligon method, is quite robust and powerful, it offers the ability to work with multiple and complex polygons (I guess at the expense of performance)

• The numba methods can also be ‘ported’ on the GPU – it will be interesting to see a comparison which includes a porting to cuda of fastest method mentioned by @Mehdi (is_inside_sm).

Your test is good, but it measures only some specific situation:
we have one polygon with many vertices, and long array of points to check them within polygon.

Moreover, I suppose that you’re measuring not
matplotlib-inside-polygon-method vs ray-method,
but
matplotlib-somehow-optimized-iteration vs simple-list-iteration

Let’s make N independent comparisons (N pairs of point and polygon)?

lenpoly = 100
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in np.linspace(0,2*np.pi,lenpoly)[:-1]]

M = 10000
start_time = time()
# Ray tracing
for i in range(M):
x,y = np.random.random(), np.random.random()
inside1 = ray_tracing_method(x,y, polygon)
print "Ray Tracing Elapsed time: " + str(time()-start_time)

# Matplotlib mplPath
start_time = time()
for i in range(M):
x,y = np.random.random(), np.random.random()
inside2 = path.contains_points([[x,y]])
print "Matplotlib contains_points Elapsed time: " + str(time()-start_time)

Result:

Ray Tracing Elapsed time: 0.548588991165
Matplotlib contains_points Elapsed time: 0.103765010834

Matplotlib is still much better, but not 100 times better.
Now let’s try much simpler polygon…

lenpoly = 5
# ... same code

result:

Ray Tracing Elapsed time: 0.0727779865265
Matplotlib contains_points Elapsed time: 0.105288982391

I will just leave it here, just rewrote the code above using numpy, maybe somebody finds it useful:

def ray_tracing_numpy(x,y,poly):
n = len(poly)
inside = np.zeros(len(x),np.bool_)
p2x = 0.0
p2y = 0.0
xints = 0.0
p1x,p1y = poly[0]
for i in range(n+1):
p2x,p2y = poly[i % n]
idx = np.nonzero((y > min(p1y,p2y)) & (y <= max(p1y,p2y)) & (x <= max(p1x,p2x)))[0]
if p1y != p2y:
xints = (y[idx]-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x:
inside[idx] = ~inside[idx]
else:
idxx = idx[x[idx] <= xints]
inside[idxx] = ~inside[idxx]

p1x,p1y = p2x,p2y
return inside

Wrapped ray_tracing into

def ray_tracing_mult(x,y,poly):
return [ray_tracing(xi, yi, poly[:-1,:]) for xi,yi in zip(x,y)]

Tested on 100000 points, results:

ray_tracing_mult 0:00:00.850656
ray_tracing_numpy 0:00:00.003769

## Comparison of different methods

I found other methods to check if a point is inside a polygon (here). I tested two of them only (is_inside_sm and is_inside_postgis) and the results were the same as the other methods.

Thanks to @epifanio, I parallelized the codes and compared them with @epifanio and @user3274748 (ray_tracing_numpy) methods. Note that both methods had a bug so I fixed them as shown in their codes below.

One more thing that I found is that the code provided for creating a polygon does not generate a closed path np.linspace(0,2*np.pi,lenpoly)[:-1]. As a result, the codes provided in above GitHub repository may not work properly. So It’s better to create a closed path (first and last points should be the same).

Codes

Method 1: parallelpointinpolygon

from numba import jit, njit
import numba
import numpy as np

@jit(nopython=True)
def pointinpolygon(x,y,poly):
n = len(poly)
inside = False
p2x = 0.0
p2y = 0.0
xints = 0.0
p1x,p1y = poly[0]
for i in numba.prange(n+1):
p2x,p2y = poly[i % n]
if y > min(p1y,p2y):
if y <= max(p1y,p2y):
if x <= max(p1x,p2x):
if p1y != p2y:
xints = (y-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x or x <= xints:
inside = not inside
p1x,p1y = p2x,p2y

return inside

@njit(parallel=True)
def parallelpointinpolygon(points, polygon):
D = np.empty(len(points), dtype=numba.boolean)
for i in numba.prange(0, len(D)):   #<-- Fixed here, must start from zero
D[i] = pointinpolygon(points[i,0], points[i,1], polygon)
return D

Method 2: ray_tracing_numpy_numba

@jit(nopython=True)
def ray_tracing_numpy_numba(points,poly):
x,y = points[:,0], points[:,1]
n = len(poly)
inside = np.zeros(len(x),np.bool_)
p2x = 0.0
p2y = 0.0
p1x,p1y = poly[0]
for i in range(n+1):
p2x,p2y = poly[i % n]
idx = np.nonzero((y > min(p1y,p2y)) & (y <= max(p1y,p2y)) & (x <= max(p1x,p2x)))[0]
if len(idx):    # <-- Fixed here. If idx is null skip comparisons below.
if p1y != p2y:
xints = (y[idx]-p1y)*(p2x-p1x)/(p2y-p1y)+p1x
if p1x == p2x:
inside[idx] = ~inside[idx]
else:
idxx = idx[x[idx] <= xints]
inside[idxx] = ~inside[idxx]

p1x,p1y = p2x,p2y
return inside

Method 3: Matplotlib contains_points

path = mpltPath.Path(polygon,closed=True)  # <-- Very important to mention that the path
#     is closed (default is false)

Method 4: is_inside_sm (got it from here)

@jit(nopython=True)
def is_inside_sm(polygon, point):
length = len(polygon)-1
dy2 = point[1] - polygon[0][1]
intersections = 0
ii = 0
jj = 1

while ii<length:
dy  = dy2
dy2 = point[1] - polygon[jj][1]

# consider only lines which are not completely above/bellow/right from the point
if dy*dy2 <= 0.0 and (point[0] >= polygon[ii][0] or point[0] >= polygon[jj][0]):

# non-horizontal line
if dy<0 or dy2<0:
F = dy*(polygon[jj][0] - polygon[ii][0])/(dy-dy2) + polygon[ii][0]

if point[0] > F: # if line is left from the point - the ray moving towards left, will intersect it
intersections += 1
elif point[0] == F: # point on line
return 2

# point on upper peak (dy2=dx2=0) or horizontal line (dy=dy2=0 and dx*dx2<=0)
elif dy2==0 and (point[0]==polygon[jj][0] or (dy==0 and (point[0]-polygon[ii][0])*(point[0]-polygon[jj][0])<=0)):
return 2

ii = jj
jj += 1

#print 'intersections=", intersections
return intersections & 1

@njit(parallel=True)
def is_inside_sm_parallel(points, polygon):
ln = len(points)
D = np.empty(ln, dtype=numba.boolean)
for i in numba.prange(ln):
D[i] = is_inside_sm(polygon,points[i])
return D

Method 5: is_inside_postgis (got it from here)

@jit(nopython=True)
def is_inside_postgis(polygon, point):
length = len(polygon)
intersections = 0

dx2 = point[0] - polygon[0][0]
dy2 = point[1] - polygon[0][1]
ii = 0
jj = 1

while jj<length:
dx  = dx2
dy  = dy2
dx2 = point[0] - polygon[jj][0]
dy2 = point[1] - polygon[jj][1]

F =(dx-dx2)*dy - dx*(dy-dy2);
if 0.0==F and dx*dx2<=0 and dy*dy2<=0:
return 2;

if (dy>=0 and dy2<0) or (dy2>=0 and dy<0):
if F > 0:
intersections += 1
elif F < 0:
intersections -= 1

ii = jj
jj += 1

#print "intersections=", intersections
return intersections != 0

@njit(parallel=True)
def is_inside_postgis_parallel(points, polygon):
ln = len(points)
D = np.empty(ln, dtype=numba.boolean)
for i in numba.prange(ln):
D[i] = is_inside_postgis(polygon,points[i])
return D

## Benchmark

Timing for 10 million points:

parallelpointinpolygon Elapsed time:      4.0122294425964355
Matplotlib contains_points Elapsed time: 14.117807388305664
ray_tracing_numpy_numba Elapsed time:     7.908452272415161
sm_parallel Elapsed time:                 0.7710440158843994
is_inside_postgis_parallel Elapsed time:  2.131121873855591

Here is the code.

import matplotlib.pyplot as plt
import matplotlib.path as mpltPath
from time import time
import numpy as np

np.random.seed(2)

time_parallelpointinpolygon=[]
time_mpltPath=[]
time_ray_tracing_numpy_numba=[]
time_is_inside_sm_parallel=[]
time_is_inside_postgis_parallel=[]
n_points=[]

for i in range(1, 10000002, 1000000):
n_points.append(i)

lenpoly = 100
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in np.linspace(0,2*np.pi,lenpoly)]
polygon = np.array(polygon)
N = i
points = np.random.uniform(-1.5, 1.5, size=(N, 2))

#Method 1
start_time = time()
inside1=parallelpointinpolygon(points, polygon)
time_parallelpointinpolygon.append(time()-start_time)

# Method 2
start_time = time()
path = mpltPath.Path(polygon,closed=True)
inside2 = path.contains_points(points)
time_mpltPath.append(time()-start_time)

# Method 3
start_time = time()
inside3=ray_tracing_numpy_numba(points,polygon)
time_ray_tracing_numpy_numba.append(time()-start_time)

# Method 4
start_time = time()
inside4=is_inside_sm_parallel(points,polygon)
time_is_inside_sm_parallel.append(time()-start_time)

# Method 5
start_time = time()
inside5=is_inside_postgis_parallel(points,polygon)
time_is_inside_postgis_parallel.append(time()-start_time)

plt.plot(n_points,time_parallelpointinpolygon,label="parallelpointinpolygon')
plt.plot(n_points,time_mpltPath,label="mpltPath")
plt.plot(n_points,time_ray_tracing_numpy_numba,label="ray_tracing_numpy_numba")
plt.plot(n_points,time_is_inside_sm_parallel,label="is_inside_sm_parallel")
plt.plot(n_points,time_is_inside_postgis_parallel,label="is_inside_postgis_parallel")
plt.xlabel("N points")
plt.ylabel("time (sec)")
plt.legend(loc="best")
plt.show()

CONCLUSION

The fastest algorithms are:

1- is_inside_sm_parallel

2- is_inside_postgis_parallel

3- parallelpointinpolygon (@epifanio)

pure numpy vectorized implementation of the Even-odd rule

The other answers are either a slow python loop or requires external dependancies or cython treatment.

import numpy as np

def points_in_polygon(polygon, pts):
pts = np.asarray(pts,dtype="float32")
polygon = np.asarray(polygon,dtype="float32")
contour2 = np.vstack((polygon[1:], polygon[:1]))
test_diff = contour2-polygon
m1 = (polygon[:,1] > pts[:,None,1]) != (contour2[:,1] > pts[:,None,1])
slope = ((pts[:,None,0]-polygon[:,0])*test_diff[:,1])-(test_diff[:,0]*(pts[:,None,1]-polygon[:,1]))
m2 = slope == 0
m3 = (slope < 0) != (contour2[:,1] < polygon[:,1])
m4 = m1 & m3
count = np.count_nonzero(m4,axis=-1)

N = 1000000
lenpoly = 1000
polygon = [[np.sin(x)+0.5,np.cos(x)+0.5] for x in np.linspace(0,2*np.pi,lenpoly)]
polygon = np.array(polygon,dtype="float32")
points = np.random.uniform(-1.5, 1.5, size=(N, 2)).astype('float32')