Note: for both a) and b) point p1 is inside the polygon as wn != 0, irrespective of added complexity of polygon in b). 1) wn++ for DE as it is upward crossing for which P1 is left of CD 2) wn- for BC as it is downward crossing for which P1 is right of BC As final winding number wn = 0 point P1 lies outside the polygon.Ī) wn = 1, as P1 is left of upward edge CD b) wn = 1, as P1 is left of upward edge FG c) wn = 0, as P1 is left of upward edge DE but right of downward edge BC. For c) part in the figure below: Initially wn = 0 Only edges DE and BC cut the line passing through P1 and parallel to y axis.
Although it should be noted that we are avoiding the added complexity of the polygon and hence make the algorithm more efficient. A similar case can be made for b) part of the figure. As final winding number wn = 1 which is not equal to zero point P1 lies inside the polygon. 1) wn++ for CD as it is an upward crossing for which P1 is left of CD 2) wn wont be changed as P1 is on left of AB (downward crossing). Example: For part a) in the figure below: Initially wn = 0 Only edges CD and AB cut the line passing through P1 and parallel to y axis. If the final winding number is non zero then the point lies inside the polygon. Increase the value of winding number (wn) by one if query point is on the left side of an upward crossing and decrease the wn by one if query point is on the right side of an downward crossing. For these edges check if the query point is on the left or right side of the edge when looking at all the edges in anticlockwise direction. Randolph Franklin is by simply observing the edges that actually wind around the query point whilst all the other edges can be ignored.Īlgorithm: For any polygon, find all the edges of the polygon that cut through the line passing through the query point and is parallel to y_axis. One way to reduce the complexity as described by W. The time complexity of this algorithm would be O(n) similar to ray casting algorithm, but it would involve repeated calculation of inverse trigonometric functions like atan2, to get the angle subtended by the sides of the polygon with the query point. If the summation of these angle add up to 2 π the point lies inside the polygon and if the sum is 0, the point lies outside. This is indicated by angles θ1, θ2, θ3 and θ4 for sides AB, BC, CD and DA respectively. One way to calculate the winding number would be to calculate the angle subtended by each side of the polygon with the query point. Winding number is defined by the number of times a curve travels counter clockwise around a point.
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C++ code for both these methods can be found here. I will then cover a simplified method for solving PIP for convex polygons. In this article I am going to explain the Winding Number Algorithm which solves PIP for any polygon.
The PIP problem is further simplified for convex polygons and we will discuss one such method to solve it. Also with the newer implementations it is also faster than ray casting algorithm. A point to note is that the winding number algorithm is more accurate then ray casting for points really close to the polygon. Some methods to solve the PIP problem is by using Ray Casting algorithmand Winding Number algorithm.
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