A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon

Problem for Complex Polygons

Michael Galetzka

1

and Patrick Glauner

2

1

Disy Informationssysteme GmbH, Ludwig-Erhard-Allee 6, 76131 Karlsruhe, Germany

2

Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg,

4 rue Alphonse Weicker, 2721 Luxembourg, Luxembourg

Keywords:

Complex Polygon, Even-Odd Algorithm, Point-in-Polygon.

Abstract:

Determining if a point is in a polygon or not is used by a lot of applications in computer graphics, computer

games and geoinformatics. Implementing this check is error-prone since there are many special cases to be

considered. This holds true in particular for complex polygons whose edges intersect each other creating

holes. In this paper we present a simple even-odd algorithm to solve this problem for complex polygons in

linear time and prove its correctness for all possible points and polygons. We furthermore provide examples

and implementation notes for this algorithm.

1 INTRODUCTION

At a ﬁrst glance, the point-in-polygon problem seems

to be a rather simple problem of geometry: given an

arbitrary point Q and a closed polygon P, the question

is whether the point lies inside or outside the poly-

gon. There exist different algorithms to solve this

problem, such as a cell-based algorithm (Zalik and

Kolingerova), the winding number algorithm (Hor-

mann and Agathos, 2001) or the even-odd algorithm

(Foley et al., 1990) that is used in this paper. The

problem is not as trivial as it seems to be if the edges

of the polygon can intersect other edges as seen in

Figure 1. This kind of polygon is also often called a

complex polygon since it can contain ”holes”.

Figure 1: A self-intersecting polygon.

A lot of special cases have to be considered (e.g.

if the point lies on an edge) and most of the exist-

ing even-odd algorithms either fail one or more of

these special cases or have to implement some kind

of workaround (Schirra, 2008). The rest of this paper

is organized as follows. In Section 2, we present an

even-odd algorithm and prove its correctness for all

possible points and polygons with no special cases to

be considered. In Section 3, we apply this algorithm

to an example complex polygon. We then provide im-

plementation notes in Section 4. Section 5 summa-

rizes this work.

2 THE EVEN-ODD ALGORITHM

The basic even-odd algorithm itself is fairly simple:

cast an inﬁnite ray from the point in question and

count how many edges of the polygon the ray inter-

sects (Foley et al., 1990).

Deﬁnition Let P be a polygon with n vertices

P

1

, ..., P

n−1

, P

n

where the vertices are sorted in such

a way that there exists an edge between P

i

and P

i+1

for every index 1 ≤ i < n and between P

1

and P

n

. The

algorithm computes if the arbitrary point Q lies either

inside or outside of P.

Galetzka M. and Glauner P.

A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon Problem for Complex Polygons.

DOI: 10.5220/0006040801750178

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 175-178

ISBN: 978-989-758-224-0

Copyright

c

2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

175

The Steps of the Algorithm. To ease the presenta-

tion of the steps of the algorithm, it deﬁnes its own co-

ordinate system with (0|0) = Q by moving the poly-

gon. This translation is only for illustration purposes.

The algorithm then uses the positive x-axis as ray to

calculate intersections with the edges of P. So the

start vertex of the ray is (0|0) and the end vertex is

(x

max

|0) where x

max

is an x value greater than that of

any of the vertices of P.

1. First it is determined if Q is equal to any of the

vertices of P or lies on any of the edges connecting

the vertices. If so the result is inside.

2. A vertex P

s

that does not lie on the x-axis is

searched in the set of vertices of P. If no such

vertex can be found the result is outside.

3. Set i to 1. Beginning from that vertex P

s

the fol-

lowing steps are repeated until all vertices of P

have been visited:

(a) The index s is increased to s + i until the next

vertex P

s+i

not lying on the x-axis is found. If

the index s + i > n then i is set to −s and the

search is continued.

(b) Depending on the course of step (a) one of the

following steps is taken:

i. No vertex has been skipped: the line segment

from P

s

to P

s+i

is intersected with the positive

x-axis.

ii. At least one vertex with a positive x-value has

been skipped: the line segment from P

s

to P

s+i

is intersected with the complete x-axis.

iii. At least one vertex with a negative x-value has

been skipped: nothing is done.

(c) P

s+i

is the starting vertex for the next iteration.

4. If the count of intersections with the x-axis is even

then the result is outside, if it is odd the result is

inside.

Proof of Correctness. To prove that the oven-odd

algorithm in general is correct one can generalize it

to the Jordan Curve Theorem and prove its correct-

ness (Tverberg, 1980). What has to be proven is that

the given algorithm is in fact a correct even-odd al-

gorithm, meaning that the count of edges is correct

under all circumstances.

A lot of challenges emerge when trying to inter-

sect two line segments and one of the segments has

one or two of its vertices lying on the other segment.

If the count of intersections with the x-axis are to be

counted correctly, then each edge of P must either

clearly intersect the x-axis or not intersect it at all.

There must be no case where the starting or end ver-

tex of a line segment lies on the line segment it should

be intersected with.

The x-axis is the ﬁrst line segment to look at, since

it is part of all the intersections. The start vertex of

the x-axis, namely Q, is guaranteed not to lie on any

edge or to be equal to any vertex of P. If this was the

case, then the ﬁrst step of the algorithm would have

already returned the correct result. The end vertex is

guaranteed to have an x value greater than any of the

vertices of P, so no vertex of P can be equal to it and

no edge of P can contain it.

Of course there still exists the challenge that one

or more vertices of P lie on the x-axis as seen in Fig-

ure 2.

Q(0|0)

P

1

P

2

P

3

P

4

P

5

Figure 2: One of the edges of P lies on the x-axis.

The algorithm deals with this kind of challenge by

ignoring the vertices lying on the x-axis and stepping

over them when trying to ﬁnd an edge to intersect. It

then creates a new auxiliary edge that it can intersect

safely. So starting for example at vertex P

1

it would

ignore P

2

and P

3

and then create a new edge between

P

1

and P

4

as shown in Figure 3.

Q(0|0)

P

1

P

2

P

3

P

4

P

5

Figure 3: A new auxiliary edge has been created.

At this point it is clear that none of the edges used

to calculate the count of intersections with the x-axis

has a vertex on the x-axis and is either clearly inter-

secting it or not. What has to be shown is that all

created auxiliary edges are correct substitutes to cal-

culate the intersections with the x-axis.

Indeed they are not a correct substitute to intersect

the positive x-axis, as can be seen in Figure 4. Here

GRAPP 2017 - International Conference on Computer Graphics Theory and Applications

176

the auxiliary edge would be the same as the edge be-

tween P

1

and P

3

and this edge does clearly not inter-

sect the positive x-axis. So the total count of inter-

sected edges of the polygon shown in Figure 4 would

be zero - which is obviously wrong.

The algorithm actually deals with this challenge

in step 3.b, by extending the ray in that special case

where a vertex lying on the positive x-axis has been

skipped. The new ray is then the complete x-axis and

not only the positive part of it. It can be seen that

this would create the desired result in the example of

Figure 4, because the total count of intersected edges

would then be one.

Q(0|0)

P

1

P

2

P

3

Figure 4: The auxiliary edge between P

1

and P

3

does not

intersect the positive x-axis.

The question remains if this method always yields

the correct result under all possible circumstances. If

the skipped vertex lies on the negative x-axis then

the auxiliary line will not be considered for intersec-

tion, since none of the original edges could have inter-

sected the positive x-axis. So all cases that have to be

looked at involve one or more vertices on the positive

x-axis. The skipped vertices can never change from

the positive to the negative x-axis since this would

have been caught in the ﬁrst step of the algorithm.

So, all auxiliary edges that intersect the x-axis are

created from the following order of vertices: P

u

which

does not lie on the x-axis, P

v

1

which does lie on the

positive x-axis and P

w

which does not lie on the x-

axis. Each of the vertices P

u

and P

w

can lie in one of

the four quadrants around the point Q and therefore

there exist 4 × 4 = 16 different versions of the auxil-

iary edge that have to be considered. This number can

be further reduced to 10 because six of these versions

are created by switching start and end vertex and do

not have to be considered as a separate case.

All possibilities of the locations of P

u

and P

w

are

shown in Table 2 as well as the desired intersection

1

Of course there could be more than just one vertex on

the x-axis but they are all skipped and the same auxiliary

edge is created no matter how many additional vertices are

on the x-axis.

count and the actual intersection count of the algo-

rithm. It is clear by looking at the table that the al-

gorithm satisﬁes the desired result for each possible

scenario. Therefore the auxiliary line is indeed a cor-

rect substitute for the original edges.

This leads to the conclusion that the algorithm

provided can correctly determine if there is an even

or an odd number of intersections with the polygon.

Time Complexity. All n vertices of the polygon

are visited once during the translation and the ﬁrst

three steps of the algorithm. All other computations

like step four can be completed within constant time.

Therefore the time complexity for this algorithm is

O(n).

3 EXAMPLE

The algorithm is applied to the sample complex poly-

gon P in Figure 5.

Q(0|0)

P

1

P

2

P

3

P

4

P

5

P

6

Figure 5: Sample complex polygon P.

1. Q lies neither on any vertex nor edge.

2. The start vertex P

s

is P

1

in this example.

3. All intersection and substitution steps can be

found in Table 1.

4. Since the count of intersections with the x-axis is

odd the result is inside.

Table 1: Intersection and substitution steps.

P

u

P

v

P

w

x-axis for intersection operation Inters.

P

1

− P

2

positive no

P

2

(P

3

, P

4

) P

5

complete no

P

5

P

6

P

1

complete yes

A Simple and Correct Even-Odd Algorithm for the Point-in-Polygon Problem for Complex Polygons

177

4 IMPLEMENTATION

The translation of P can be done together with the

ﬁrst step of the algorithm by moving the vertices

by the negative x- and y-values of Q. The geomet-

ric helper functions to intersect line segments can

be found in (Stueker, 1999) which are an improved

version of (Sedgewick, 1992). Our reference im-

plementation of the even-odd algorithm for complex

polygons presented in this paper is available as open

source: http://github.com/pglauner/point in polygon.

5 CONCLUSIONS

The implementation of point-in-polygon algorithms

for complex polygons is error-prone since there are

many special cases to be considered. We presented a

correct even-odd algorithm that prevents special cases

by substituting a sequence of edges with an auxil-

iary edge. The correctness of the algorithm has been

proven for all possible polygons, including complex

polygons. The algorithm is also time-efﬁcient since it

is O(n) for polygons with n vertices.

REFERENCES

Foley, J. D., van Dam, A., Feiner, S. K. and Hughes, J.

F. (1990). Computer Graphics: Principles and Prac-

tice. The Systems Programming Series. 2nd Edition.

Addison-Wesley, Reading.

Hormann, K. and Agathos, A. (2001). The point in poly-

gon problem for arbitrary polygons. Computational

Geometry, 20(3):131–144.

Schirra, S. (2008). How reliable are practical point-

in-polygon strategies? Algorithms - ESA 2008,

5193:744–755.

Sedgewick, R. (1992). Algorithms in C++. 1st Edition.

Addison-Wesley Professional.

Stueker, D. (1999). Line segment intersection and inclusion

in a polygon. Elementary geometric methods.

Tverberg, H. (1980). A proof of the jordan curve theo-

rem. Bulletin of the London Mathematical Society,

12(1):34–38.

Zalik, B. and Kolingerova, I. (2001). A cell-based point-

in-polygon algorithm suitable for large sets of points.

Computers and Geosciences, 27(10):1135 – 1145.

APPENDIX

Table 2: Possible cases when creating an auxiliary line

(blue). The values q1 to q4 correspond to the quadrants

of a cartesian coordinate system (q1 : x > 0 ∧ y > 0, q2 : x <

0 ∧ y > 0, ...). Rows that are just gained by switching the

vertices P

u

and P

w

are omitted due to symmetry.

P

u

P

w

Example Desired countInters.

q1q1

Q(0|0)

P

u

P

v

P

w

even no

q1q2

Q(0|0)

P

u

P

v

P

w

even no

q1q3

Q(0|0)

P

u

P

v

P

w

odd yes

q1q4

Q(0|0)

P

u

P

v

P

w

odd yes

q2q2

Q(0|0)

P

u

P

v

P

w

even no

q2q3

Q(0|0)

P

u

P

v

P

w

odd yes

q2q4

Q(0|0)

P

u

P

v

P

w

odd yes

q3q3

Q(0|0)

P

u

P

v

P

w

even no

q3q4

Q(0|0)

P

u

P

v

P

w

even no

q4q4

Q(0|0)

P

u

P

v

P

w

even no

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178