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Given a set of objects $O$ in the plane, the corresponding intersection graph is defined as follows. Each object defines a vertex and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit…
An $\epsilon$-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most $\epsilon$ from each other. Given a set of points and a set of objects,…
The aim of this paper is to present a new method of approximation of planar data set using only arcs or segments. The first problem we are trying to solve is the following: the CNC machines can work only with simple curves (arcs or…
We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has…
We propose algorithms and software for computing projections onto the intersection of multiple convex and non-convex constraint sets. The software package, called SetIntersectionProjection, is intended for the regularization of inverse…
Drawings of non-planar graphs always result in edge crossings. When there are many edges crossing at small angles, it is often difficult to follow these edges, because of the multiple visual paths resulted from the crossings that slow down…
Under conditions that prevent tangential intersection, we prove quadratic convergence of a projection algorithm for the feasibility problem of finding a point in the intersection of a smooth curve and line in $\mathbb{R}^2$. This nonconvex…
Intersection detection between three-dimensional bodies has various applications in computer graphics, video game development, robotics as well as military industries. In some respects, entities do not want to disclose sensitive information…
Presented paper describes the method for finding the intersection of class space rational Bezier curves. The problem curve/curve intersection belongs among basic geometric problems and the aim of this article is to describe the new…
We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…
This contribution presents a brief survey of clipping and intersection algorithms in E2 and E3 with a nearly complete list of relevant references. Some algorithms use the projective extension of the Euclidean space and vector-vector…
A comparison of a new algorithm for line clipping in E2 and E3 by convex polygon and/or polyhedron with O(1) processing complexity and Cyrus- Beck algorithm is presented. The new algorithm in E2 is based on dual space representation and…
The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to…
Computing on graphics processing units (GPUs) has become standard in scientific computing, allowing for incredible performance gains over classical CPUs for many computational methods. As GPUs were originally designed for 3D rendering, they…
The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP)…
This paper presents a state-of-the-art filter that reduces the complexity in object detection, tracking and mapping applications. Existing edge detection and tracking methods are proposed to create suitable autonomy for mobile robots,…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives, e.g., splines, triangles, and hexahedra,…
We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact…
A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and…