By Shahabuddin Amerudin
Introduction
Point in polygon analysis is a spatial analysis technique used in Geographic Information Systems (GIS) to determine whether a point is inside or outside of a polygon. It is a common operation used in many applications, such as environmental monitoring, land-use planning, and market analysis.
The process involves comparing the geographic coordinates of a point with the boundaries of a polygon. If the point is located within the polygon’s boundaries, it is said to be inside the polygon; otherwise, it is outside. The analysis can be performed using various methods, including ray casting, crossing number, and winding number algorithms.
The point in polygon analysis is useful for a wide range of applications, such as determining if a house is located within a flood zone, identifying which census block a particular address belongs to, or analyzing the spatial distribution of crime incidents within a city.
Algorithms
The algorithms for point in polygon analysis have been developed by various researchers in the field of computational geometry and spatial analysis. The earliest algorithm was the crossing number algorithm, which was first proposed by Paul Bourke in 1984. The ray casting algorithm was later developed by Franklin Antonio in 1987, and the winding number algorithm was proposed by Jack Ritter in 1989.
Since then, many researchers have proposed various modifications and improvements to these algorithms to make them more efficient and accurate. For example, the point in polygon algorithms have been extended to handle more complex polygon shapes, such as polygons with holes or self-intersecting polygons. Some researchers have also proposed hybrid algorithms that combine the strengths of multiple algorithms to improve performance.
There are many algorithms that have been developed for point in polygon analysis. The most commonly used algorithms are the ray casting algorithm, crossing number algorithm, and winding number algorithm.
The ray casting algorithm involves casting a ray from the point in question to the right, and counting the number of times it intersects with the polygon boundary. If the number of intersections is odd, the point is inside the polygon; if it is even, the point is outside the polygon.
The crossing number algorithm involves counting the number of times the polygon boundary crosses a horizontal line passing through the point. If the number of crossings is odd, the point is inside the polygon; if it is even, the point is outside the polygon.
The winding number algorithm involves calculating the sum of the angles made by the polygon edges at the point. If the sum is 0, the point is outside the polygon; if it is not 0, the point is inside the polygon.
These algorithms have been implemented in many GIS software and programming languages, including Python, R, and MATLAB.
Common Algorithm Used in GIS Software
The ray casting algorithm is considered one of the most common algorithms used in GIS software for point in polygon analysis. This is because it is relatively simple to implement, has good performance, and works well for most polygon shapes.
Many popular GIS software packages, such as ArcGIS, QGIS, and MapInfo, use the ray casting algorithm to perform point in polygon analysis. The algorithm is also implemented in many programming languages, including Python, Java, and C++, and is widely used in various applications that involve spatial analysis.
While other algorithms, such as the crossing number and winding number algorithms, are also used in GIS software, the ray casting algorithm is generally preferred because of its simplicity and efficiency. However, the choice of algorithm may depend on the specific requirements of the application and the characteristics of the polygon data being analyzed.
A high-level Pseudocode implementation of the ray casting algorithm:
// Input: point (x,y), polygon vertices [(x1,y1), (x2,y2), ..., (xn,yn)]
// Output: True if point is inside polygon, False otherwise
num_intersections = 0
for i in range(num_vertices):
v1 = vertices[i]
v2 = vertices[(i+1) % num_vertices] // wrap around to first vertex for last edge
if ((v1.y > point.y) != (v2.y > point.y)) and (point.x < (v2.x - v1.x) * (point.y - v1.y) / (v2.y - v1.y) + v1.x):
num_intersections += 1
return num_intersections % 2 == 1
In this algorithm, we iterate over each edge of the polygon and check whether a horizontal ray from the point intersects that edge. We use the fact that the y-coordinate of the point lies between the y-coordinates of the two vertices defining an edge to determine whether the ray intersects the edge. If an odd number of intersections are found, the point is inside the polygon; if an even number of intersections are found, the point is outside the polygon.
Note that this algorithm assumes that the polygon vertices are given in counterclockwise order. If the vertices are given in clockwise order, the result will be the opposite of what we expect. To handle polygons with holes, we can use the winding number algorithm or other advanced techniques.
An implementation of the ray casting algorithm in VB.NET:
' Input: point (x,y), polygon vertices [(x1,y1), (x2,y2), ..., (xn,yn)]
' Output: True if point is inside polygon, False otherwise
Function IsPointInPolygon(ByVal point As Point, ByVal vertices() As Point) As Boolean
Dim num_intersections As Integer = 0
Dim num_vertices As Integer = vertices.Length
For i As Integer = 0 To num_vertices - 1
Dim v1 As Point = vertices(i)
Dim v2 As Point = vertices((i + 1) Mod num_vertices) ' wrap around to first vertex for last edge
If ((v1.Y > point.Y) <> (v2.Y > point.Y)) AndAlso (point.X < (v2.X - v1.X) * (point.Y - v1.Y) / (v2.Y - v1.Y) + v1.X) Then
num_intersections += 1
End If
Next
Return num_intersections Mod 2 = 1
End Function
This VB.NET implementation is very similar to the pseudocode implementation I provided earlier. We use a For loop to iterate over each edge of the polygon and check whether a horizontal ray from the point intersects that edge. The Mod operator is used to calculate the remainder of the division of num_intersections by 2 to determine whether the point is inside or outside the polygon.
An implementation of the ray casting algorithm in Python:
# Input: point (x,y), polygon vertices [(x1,y1), (x2,y2), ..., (xn,yn)]
# Output: True if point is inside polygon, False otherwise
def is_point_in_polygon(point, vertices):
num_intersections = 0
num_vertices = len(vertices)
for i in range(num_vertices):
v1 = vertices[i]
v2 = vertices[(i + 1) % num_vertices] # wrap around to first vertex for last edge
if ((v1[1] > point[1]) != (v2[1] > point[1])) and \
(point[0] < (v2[0] - v1[0]) * (point[1] - v1[1]) / (v2[1] - v1[1]) + v1[0]):
num_intersections += 1
return num_intersections % 2 == 1
The Python implementation is very similar to the other implementations I provided earlier. We use a for loop to iterate over each edge of the polygon and check whether a horizontal ray from the point intersects that edge. The modulo operator (%) is used to calculate the remainder of the division of num_intersections by 2 to determine whether the point is inside or outside the polygon.
Algorithm Speed
The time complexity of the ray casting algorithm for point-in-polygon testing is O(n), where n is the number of vertices of the polygon. This is because we need to iterate over each edge of the polygon and perform a simple arithmetic calculation for each edge.
In practice, the algorithm is generally very fast and efficient, even for polygons with a large number of vertices. For most practical applications, the performance of the algorithm is more than sufficient. However, there are more advanced algorithms for point-in-polygon testing that can achieve faster performance in certain cases, such as the Sweep Line Algorithm or the Weiler–Atherton clipping algorithm. These algorithms can achieve logarithmic or sublinear time complexity, but they are more complex to implement and may not be necessary for most applications.
Conclusion
In conclusion, the ray casting algorithm is a widely-used algorithm for determining whether a point is inside a polygon or not. It is a simple and efficient algorithm with a time complexity of O(n), where n is the number of vertices in the polygon. The algorithm works by casting a horizontal ray from the point and counting the number of times it intersects with the edges of the polygon. If the number of intersections is odd, then the point is inside the polygon; otherwise, it is outside.
The ray casting algorithm is commonly used in GIS software and other applications that involve spatial analysis. It is relatively easy to implement and can handle polygons of arbitrary shape and complexity. However, for some specialized applications, more advanced algorithms such as the Sweep Line Algorithm or the Weiler–Atherton clipping algorithm may be more appropriate. Overall, the ray casting algorithm is a valuable tool in spatial analysis and is widely used in many different fields, including geography, architecture, and engineering.
Suggestion for Citation:
Amerudin, S. (2023). Understanding the Ray Casting Algorithm for Point-in-Polygon Analysis. [Online] Available at: https://people.utm.my/shahabuddin/?p=6277 (Accessed: 8 April 2023).
