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We present a fast and memory efficient algorithm that generates a manifold triangular mesh S with or without boundary passing through a set of unorganized points P ⊂ R³ with no other additional information. Nothing is assumed about the geometry...

Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-standard analysis, as implemented in ACL2(r). Since Taylor series of a given order have less accuracy than Chebyshev series in general, we used hundreds of Taylor series generated by ACL2(r) to evaluate the error of a Chebyshev series. 1

In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.

Computing the visible portions of curved surfaces from a given viewpoint is of great interest in many applications. It is closely related to the hidden surface removal problem in computer graphics and machining applications in manufacturing. Most of the earlier work has focused on discrete methods based onpolygonization or ray-tracing and hidden curve removal. In this paper we present an algorithm for decomposing a given surface into regions such that each region is either completely visible or hidden from a given viewpoint. Initially, it decomposes the domain of each surface based on silhouettes and boundary curves. To compute the exact visibility, we introduce a notion of visibility curves obtained byprojection of silhouette and boundary curves and decomposing the surface into nonoverlapping regions. 1

This thesis describes a probabilistic approach to the definition of the quality of reaction plans, as a basis for anytime reactive planning for domains that are not fully predictable. The reaction plans are generated by a planner based on a possible models representation of actions, which allows coping with missing, ambiguous and context-dependent information. These plans have the structure of deterministic automata whose states represent actions to be executed, and whose transitions are labeled by possible models representing properties of the world. Due to the use of probabilistic logic, the probability to make a transition from one state to another can be computed from the data of a-priori probabilities of events in the domain, and from the probability of actions achieving different possible effects. Markov chain theory enables us to study properties of these reaction plans considered as stochastic processes, and provides a framework for a dynamic evaluation of their quality with r...

Abstract. This paper provides an example of the implicit function theorem to the accuracy of global positioning system (GPS) navigation. The implicit function theorem allows one to approximate the timing accuracy required by the GPS navigation system to locate a user within a certain degree of precision.

ACL2(r) supports the definition of real-valued functions. In this paper, we introduce a theory of inverse functions into description of inverse functions, to a still abstract but more tractable treatment of the inverse of continuous functions. A macro is introduced to simplify the introduction of concrete inverse functions. We illustrate the approach by defining some inverse functions in ACL2, including the square root, natural logarithm, inverse sine, and inverse cosine functions.