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Techniques from algebraic geometry and commutative algebra are adopted to establish sufficient polynomial conditions for the validity of implicitization by the method of moving quadrics both for rectangular tensor product surfaces of bi-degree (m, n) and for triangular surfaces of total degree n in the absence of base points.

A function F (x, y, t) that assigns to each parameter t an algebraic curve F (x, y, t) = 0 is called a moving curve. A moving curve F (x, y, t) is said to follow a rational curveA new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y, whereas standard resultant methods generate the implicit equation as the determinant of a 2n × 2n matrix where each entry is a linear polynomial in x and y. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.

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In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G , compute a representation of the function H = F o G . We examine this problem in detail for the case when F and G are given in either Be´zier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric-modeling problems can be solved with a small amount of additional software.

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