What is the value of the previous representation? As a simple example, we can define a logical predicate that serves as a collision detector. Recall from Section 2.4.1 that a predicate is a Boolean-valued function. Let be a predicate defined as TRUEFALSE, which returns TRUE for a point in that lies in , and FALSE otherwise. For a line given by , let denote a logical predicate that returns TRUE if , and FALSE otherwise.

A predicate that corresponds to a convex polygonal region is represented by a logical conjunction,

The predicate returns TRUE if the point lies in the convex polygonal region, and FALSE otherwise. An obstacle region that consists of convex polygons is represented by a logical disjunction of conjuncts,

Although more efficient methods exist, can check whether a point lies in in time , in which is the number of primitives that appear in the representation of (each primitive is evaluated in constant time).

Note the convenient connection between a logical predicate representation and a set-theoretic representation. Using the logical predicate, the unions and intersections of the set-theoretic representation are replaced by logical ORs and ANDs. It is well known from Boolean algebra that any complicated logical sentence can be reduced to a logical disjunction of conjunctions (this is often called ``sum of products'' in computer engineering). This is equivalent to our previous statement that can always be represented as a union of intersections of primitives.

Steven M LaValle 2020-08-14