We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted...

Valiant’s famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a proc...

We present Wowtao, an easy-to-use pottery manufacture system to take digital creativity into the real world. Our approach is able create customized virtually on mobile phones or tablet computers interactively and quickly. work takes integration of customer in product personalization consideration support personal design field. Different from other virtual systems, our simulates entire physical ...

We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points w...

In algebraic approaches to geometric computing, the general procedure is as follows [11]: first, the geometric configuration, including both the hypotheses and the conclusion, is translated into an algebraic formulation in a prerequisite algebraic language; second, algebraic computations are carried out to the conclusion by utilizing the computational rules of the algebra and the given hypothes...

In this paper we consider the problem of efficiently constructing k-vertex fault-tolerant geometric tspanners in R (for k ≥ 0 and t > 1). Vertex fault-tolerant spanners were introduced by Levcopoulus et. al in 1998. For k = 0, we present an O(n log n) method using the algebraic computation tree model to find a t-spanner with degree bound O(1) and weight O(ω(MST )). This resolves an open problem...

As a direct consequence of the Galton-Pearson-McCartin Theorem [10, Theorem 2], the concentration ellipse provides a unifying thread to the Euclidean construction of various lines of regression. These include lines of coordinate regression [7], orthogonal regression [13], λ-regression [8] and (λ, μ)-regression [9] whose geometric constructions are afforded a unified treatment in the present pap...