1. Boolean Operations on Polygons Presented by Kevin Hardy

3. George Boole (1815-1864) Born Lincoln, England, on November 2, 1815. Boole's work in symbolic logic is collectively known as "Boolean algebra" Truth Tables/Logic Circuits Everyday usage- The use of "And, Or, and Not" when selecting the appropriate options for connecting search terms to find information in search engines

4. What is a Polygon A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others. The points at where these line segments join together are called vertices.

5. Picture of Polygons

6. The Boolean Operations with Polygons Union Intersection Difference

7. Union Definition If A is a Polygon and B is a Polygon then by the union of A with B we mean all of the the points contained in A and all of the points contained in B. The union of A with B will sometimes be denoted by A (the "cup" union sign) B and sometimes by A+B.

8. Union Picture A B A “UNION” B

9. Union Algorithm Build a list of vertices for each polygon A and B (must be in order) Find all the intersection points of the edges of A and B

10. Union Algorithm Rebuild lists including intersecting points in the order they arrive in the polygon Find a point on the boundary polygon of A that is outside of B

11. Union Algorithm Starting at that point, trace the boundary of A to the first intersection point with B (in the order of your list) At the intersection point, continue to trace the polygon according to list B

12. Union Algorithm Following this pattern, once you get to the next intersection point, continue to trace the polygon according to list A again The Algorithm is complete when you return to the initial starting point

13. Intersection Definiton If A is a Polygon and B is a Polygon then by the intersection of A with B we mean all of the points contained within Polygon A which are also contained within Polygon B. The intersection of A with B will sometimes be denoted by A (the "cap" intersection sign) B and sometimes by A*B.

14. Intersection Picture A B A “INTERSECT” B

15. Intersection Algorithm Build a list of vertices for each polygon A and B (must be in order) Find all the intersection points of the edges of A and B Rebuild lists including intersecting points in the order they arrive in the polygon Find a point on the boundary polygon A that is also on the boundary polygon of B (an intersecting point)

16. Intersection Algorithm Starting at that point, trace the boundary of A (according to list A) to the next intersection point with B At the intersection point, continue to trace the polygon according to list B The Algorithm is complete when you return to the initial starting point

17. Difference Definition If A is a Polygon and B is a Polygon then by the difference of A with B is meant all of the points contained within Polygon A without all of the points contained within Polygon B The of of A with B will sometimes be denoted by A-B

18. Difference Picture A - B B-A A B

19. Difference Algorithm Build a list of vertices for each polygon A and B (must be in order) Find all the intersection points of the edges of A and B Rebuild lists including intersecting points in the order they arrive in the polygon

20. Difference Algorithm Find a point on the boundary polygon of A that is outside of B Starting at point that point, trace the boundary of A (according to the list) to the next intersection point with B

21. Difference Algorithm At that intersection point, like the Union and Intersection algorithms, continue to trace according to list B, BUT in reverse order until the next intersection point with A At that Intersection point, continue to follow A, going forward through the list. Repeat this pattern. The algorithm is complete when you reach the original point.

22. Recapping Boolean Operations on Polygons Union- All of the points contained within Polygon A, and all of the points contained within Polygon B –Like logical operator OR Intersection- Takes all of the points within A that are also contained within B –Like logical operator AND Difference- Takes all of the points within A that are not also within B –Like logical operator NOT