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## On Algebraic Surfaces Meeting with Geometric Continuity

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**Author**

Warren, Joe D.

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**Abstract**

An increasingly prominent area of computer science is Computer Aided Geometric Design or CAGD. The main task of CAGD is to automate, to the greatest extent possible, the process of designing physical objects. A designer typically models an object as a collection of surfaces. For many objects, design specifications indicate only a few critical surfaces, with the remaining surfaces to be chosen so as to make the surface of the resulting object smooth. Smoothness is important because, in many mechanical objects, sharp edges are undesirable for functional or aesthetic reasons. For example, sharp edges on the interior surface of a gate valve retard fluid flow. Automatically calculating these remaining surfaces, called blending surfaces is an important task in any CAGD system. Therefore, an understanding of the mathematics of surfaces that meet smoothly is fundamental to CAGD. Specifically, this thesis investigates the following problem: given a surface V and a point or curve W on that surface, construct surfaces that meet with V with a specified degree of smoothness along W. Working from a measure of smoothness known as geometric continuity, the first half of this thesis establishes that the space of all surfaces meeting V with the k-th order geometric continuity along W is directly related to certain algebraic structures called ideals. For example, let Z(M) (the set of point for which a polynomial M is zero) be an irreducible surface that intersects another surface Z(N) transversally (nontangentially) in an irreducible curve Z(M) $\frown$ Z(N). It is shown that a surface Z(F) meets Z(M) with order k geometric continuity along Z(M) $\frown$ Z(N) if and only if F is a polynomial of the form $AM = BN^{k+1}$ where A and B are free polynomials and A is nonzero on Z(M) $\frown$ Z(N). The second half of this work applies these results to the problem of generating blending surfaces. Using the geometric properties of blending surfaces, it is shown that any surface Z(F) that smooths the intersection of two surfaces Z(M) and Z(N) must have certain algebraic properties. In particular, the degree of F must be greater than or equal to the maximum of the degrees of $M^{2}$ and $N^{2}$. Finally, existing methods for constructing blending surfaces are shown to be consistent within the above algebraic framework. In fact, we demonstrate that these distinct methods are instances of a single general method for generating blending surfaces. Moreover, this general method is shown to generate blending surfaces of the lowest possible degree.

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**Date Issued**

1986-08#####
**Publisher**

Cornell University

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**Subject**

computer science; technical report

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**Previously Published As**

http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR86-770

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**Type**

technical report