Abstract: The convex hull of two polygons having congruent corresponding angles with the same orientation can be determined by analyzing the relationship of each vertex of one of the polygons relative to its adjacent vertices. More particularly a line may be defined between a selected vertex on one of the polygons and its corresponding vertex on the other polygon. If the vertices adjacent to the selected vertex both fall on the same side of the line, then a line connecting the selected vertex with its corresponding vertex lies on the convex hull of the two polygons. If, on the other hand, the vertices adjacent to the selected vertex are located on different sides of the line, then the line connecting the selected vertex with its corresponding vertex does not lie on the convex hull.

Abstract: A locking arrangement for data structures is provided that prevents deadlocks, but still allows different threads to simultaneously obtain locks on different nodes of a data structure for both read and write operations. The locking system will fail a request to lock one or more resources in a data structure if access to those resources has already been restricted by a conflicting lock of an equal or higher priority. The locking system may also employ preemptable and non-preemptable locks such that, if a preemptable lock with a lower priority has restricted access to resources, then the locking system will preempt the lower priority lock in favor of a conflicting higher priority lock. Alternately, if a non-preemptable lock with a lower priority has restricted access to resources, then the locking system will wait until the lower priority lock is removed before implementing a requested conflicting higher priority lock.

Abstract: The convex hull of two polygons having congruent corresponding angles with the same orientation can be determined by analyzing the relationship of each vertex of one of the polygons relative to its adjacent vertices. More particularly a line may be defined between a selected vertex on one of the polygons and its corresponding vertex on the other polygon. If the vertices adjacent to the selected vertex both fall on the same side of the line, then a line connecting the selected vertex with its corresponding vertex lies on the convex hull of the two polygons. If, on the other hand, the vertices adjacent to the selected vertex are located on different sides of the line, then the line connecting the selected vertex with its corresponding vertex does not lie on the convex hull.

Abstract: The convex hull of two polygons having congruent corresponding angles with the same orientation can be determined by analyzing the relationship of each vertex of one of the polygons relative to its adjacent vertices. More particularly a line may be defined between a selected vertex on one of the polygons and its corresponding vertex on the other polygon. If the vertices adjacent to the selected vertex both fall on the same side of the line, then a line connecting the selected vertex with its corresponding vertex lies on the convex hull of the two polygons. If, on the other hand, the vertices adjacent to the selected vertex are located on different sides of the line, then the line connecting the selected vertex with its corresponding vertex does not lie on the convex hull.

Abstract: Systems and methods useful in communicating messages over a peer-to-peer network, optionally over transient connections to participants in a collaborative session (e.g., an “electronic whiteboard” session), are described. Examples of such systems and methods may include establishing a session for data transfer that enables participants to easily enter and leave the session without losing data or losing connections between the participants, even when participants leave the session or lose connection to the session, for example, because of the transient nature of their connection (e.g., wireless connections). These example systems and methods allow participants in a session to easily gain access to and catch up with the data previously transmitted in an ongoing session. Various disclosed data structures also are useful in various systems and methods for communicating messages. Such data structures may be used for transmitting message data to participants in a session.

Abstract: Various techniques are disclosed for easily and accurately estimating the amount of overlap between two or more polygons. With various examples of these techniques, a pattern of objects, such as points, is provided over the entire area of the first polygon. The amount of objects within the first polygon then is determined, as is the amount of objects within the first polygon that also are located within the second polygon. The ratio of the amount of objects located within both the first polygon and the second polygon to the total amount of objects located within the first polygon then provides an estimate of the amount of the first polygon overlapped by the second polygon.

Abstract: Methods for bridging subnet broadcasts across subnet boundaries include: (a) receiving a first broadcast message that originates in a first subnet; (b) sending the first broadcast message to a second subnet; and (c) broadcasting the first broadcast message on the second subnet, wherein the first broadcast message is not sent from the second subnet back to the first subnet and/or it is not rebroadcast on the first subnet. Other methods of processing broadcast messages include: (a) receiving a first broadcast message on a first subnet; (b) determining whether the first broadcast message originated in the first subnet; and (c) when the first broadcast message originated in the first subnet, sending the first broadcast message to a second subnet. Such methods may further include, at least in some examples, marking the first broadcast message to indicate that it originated from the first subnet.