In both the polygonal and polyhedral models, was a linear function. In the case of a semi-algebraic model for a 2D world, can be any polynomial with real-valued coefficients and variables and . For a 3D world, is a polynomial with variables , , and . The class of semi-algebraic models includes both polygonal and polyhedral models, which use first-degree polynomials. A point set determined by a single polynomial primitive is called an algebraic set; a point set that can be obtained by a finite number of unions and intersections of algebraic sets is called a semi-algebraic set.
Consider the case of a 2D world. A solid representation can be defined using algebraic primitives of the form
In the case of semi-algebraic models, the intersection of primitives does not necessarily result in a convex subset of . In general, however, it might be necessary to form by taking unions and intersections of algebraic primitives.
A logical predicate, , can once again be formed, and collision checking is still performed in time that is linear in the number of primitives. Note that it is still very efficient to evaluate every primitive; is just a polynomial that is evaluated on the point .
The semi-algebraic formulation generalizes easily to the case of a 3D world. This results in algebraic primitives of the form
Equations (3.10) and (3.13) are sufficient to express any model of interest. One may define many other primitives based on different relations, such as , , , , and ; however, most of them do not enhance the set of models that can be expressed. They might, however, be more convenient in certain contexts. To see that some primitives do not allow new models to be expressed, consider the primitive
Steven M LaValle 2012-04-20