RAY CASTING ALGORITHM

Fig1 :- A Polygon

One simple way of finding whether the point is inside or outside a simple polygon is to test how many times a ray starting from the point intersects the edges of the polygon. If the point in question is not on the boundary of the polygon, the number of intersections is an even number if the point is outside, and it is odd if inside. This algorithm is sometimes also known as the crossing number algorithm or the even-odd rule algorithm. Now why the name even-odd rule algorithm?? If we see the fig1 carefully then there are two points in and out. Our first task is to draw a straight line to the right of the point till it crosses the boundary of the polygon possibly several times. Now if the point under consideration is outside the polygon then the first it will go inside the polygon and then again out and ultimately outside the polygon which signifies that cn is even where cn counts the number of times a ray starting from the point P crosses the polygon boundary edges. Similarly when a point is inside the polygon then its next phase will be outside the polygon and several inside and outside and ultimately outside which signifies that cn is odd since it started with outside and ended with outside so it is quite obvious that cn will be odd. Due to the even and odd calculation this algorithm is known as Even-Odd Rule Algorithm.

There are some rules which has to be followed during the design of the algorithm. These are:1. An upward edge includes its starting endpoint, and excludes its final endpoint.2. A downward edge excludes its starting endpoint, and includes its final endpoint.3. Horizontal edges are excluded.4. The edge-ray intersection point must be strictly right of the point P.

int point_in_poly(int N, int tgtx, int tgty) { int counter = 0; int i; double xintrsct; int a,b,c,d; a = x[0]; b = y[0]; for (i=1; i <=N; i++) { c = x[i % N]; d = y[i % N]; if (tgty > (b < d ? b : d)) { // Downward Crossing. if (tgty <= (b > d ? b : d)) { // Upward Crossing. if (tgtx <= (a > c ? a : c)) { // Line Extended To The Right. if (b != d) { // compute the actual edge-ray intersect x-coordinate xintrsct = (tgty-b) * (c-a) / (d-b) +a; if (tgtx <= xintrsct) counter++; } } }}

a=c;b=d;

} if (counter % 2 == 0) return(0);// Outside The Polygon. else return(1);// Inside The Polygon.}

In this algorithm tgtx and tgty is the x and y coordinate of the point under consideration. The variables a, b, c, d hold the coordinates of the polygon. Now after that all the Edge Crossing Rules are verified then we form the equation of the line. Our aim is to draw a single straight line parallel to x-axis. Now when it is parallel to x-axis so its y-axis is same as the y coordinate of the point under consideration. Now our next task is to find the point of

intersection i.e. to find the x coordinate of the point of intersection. Now we have to check that whether this x coordinate is having a value greater than the point under consideration, if it is then it is discarded otherwise accepted and the variable counter is incremented by 1. Now when all the points are traversed then we check whether count is even or odd. If it is even then it is outside the polygon otherwise it is inside. Since our aim is to check that whether a point is inside or outside a polygon so we return 1 if it is inside the polygon and 0 if outside the polygon.