The problem of exactly representing the offset to curves (in the plane) or surfaces (in the space) will be investigated by developing techniques allowing to manipulate offsets by using their implicit equation, properly represented in order to avoid the huge size when obtained in the monomial basis. For low degree cases, the rationality will also be analyzed in order to develop a toolbox to deal exactly with offsets of conics, quadrics and cubics in a very efficient and robust way.

Among other things this will necessitate to adapt several symbolic-numeric algorithms in Numerical Linear Algebra to the study of generalized eigenvalue problems on polynomial matrices derived from resultant-like matrices and to understand the Real Algebraic Geometry behind the geometric extraneous components appearing when computing the offset’s implicit equation.

The proper understanding of this way of representing offsets “exactly” will be used to deal with these geometric objects in practical situations: detecting self-intersections, sweeping objects, etc.

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