Boolean Operations on Polygons
Presented by Kevin Hardy
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
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.
Picture of Polygons
The Boolean Operations with
Polygons
• Union
• Intersection
• Difference
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.
Union Picture
A
B
A “UNION” B
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
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
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
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
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.
Intersection Picture
A
B
A “INTERSECT” B
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)
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
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
Difference Picture
A
B
A-B
B-A
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
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
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.
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