Michael Bartoň

Computing roots of polynomials by quadratic clipping

We present an algorithm which is able to compute all roots of a given univariate polynomial within a given interval. In each step, we use degree reduction to generate a strip bounded by two quadratic polynomials which encloses the graph of the polynomial within the interval of interest. The new interval(s) containing the root(s) is (are) obtained by intersecting this strip with the abscissa axis. In the case of single roots, the sequence of the lengths of the intervals converging towards the root has the convergence rate 3. For double roots, the convergence rate is still superlinear (32). We show that the new technique compares favorably with the classical technique of Bezier clipping.

Precise Hausdorff distance computation between polygonal meshes

We present an exact algorithm for computing the precise Hausdorff distance between two general polyhedra represented as triangular meshes. The locus of candidate points, events where the Hausdorff distance may occur, is fully classified. These events include simple cases where foot points of vertices are examined as well as more complicated cases where extreme distance evaluation is needed on the intersection curve of one mesh with the medial axis of the other mesh. No explicit reconstruction of the medial axis is conducted and only bisectors of pairs of primitives (i.e. vertex, edge, or face) are analytically constructed and intersected with the other mesh, yielding a set of conic segments. For each conic segment, the distance functions to all primitives are constructed and the maximum value of their low envelope function may correspond to a candidate point for the Hausdorff distance. The algorithm is fully implemented and several experimental results are also presented.

Gaussian quadrature for splines via homotopy continuation

We introduce a new concept for generating optimal quadrature rules for splines. To generate an optimal quadrature rule in a given (target) spline space, we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one, while preserving the number of quadrature points and therefore optimality. The quadrature nodes and weights are, considered as a higher-dimensional point, a zero of a particular system of polynomial equations. As the space is continuously deformed by changing the source knot vector, the quadrature rule gets updated using polynomial homotopy continuation. For example, starting with C 1 cubic splines with uniform knot sequences, we demonstrate the methodology by deriving the optimal rules for uniform C 2 cubic spline spaces where the rule was only conjectured to date. We validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen.

Towards efficient 5-axis flank CNC machining of free-form surfaces via fitting envelopes of surfaces of revolution

Abstract We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet tangentially along a smooth curve. We seek both an optimal shape of the milling tool as well as its optimal path in 3D space and propose an optimization based framework where these entities are the unknowns. We propose two initialization strategies where the first one requires a user’s intervention only by setting the initial position of the milling tool while the second one enables to prescribe a preferable tool-path. We present several examples showing that the proposed method recovers exact envelopes, including semi-envelopes and incomplete data, and for general free-form objects it detects envelope sub-patches.

Detection and reconstruction of freeform sweeps

We study the difficult problem of deciding if parts of a freeform surface can be generated, or approximately generated, by the motion of a planar profile through space. While this task is basic for understanding the geometry of shapes as well as highly relevant for manufacturing and building construction, previous approaches were confined to special cases like kinematic surfaces or “moulding” surfaces. The general case remained unsolved so far. We approach this problem by a combination of local and global methods: curve analysis with regard to “movability”, curve comparison by common substring search in curvature plots, an exhaustive search through all planar cuts enhanced by quick rejection procedures, the ordering of candidate profiles and finally, global optimization. The main applications of our method are digital reconstruction of CAD models exhibiting sweep patches, and aiding in manufacturing freeform surfaces by pointing out those parts which can be approximated by sweeps.

Gauss–Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently derived [5] act on spaces of the smallest odd degrees and, therefore, are still slightly sub-optimal. In this work, we derive optimal rules directly for even-degree spaces and therefore further improve our recent result. We use optimal quadrature rules for spaces over two elements as elementary building blocks and use recursively the homotopy continuation concept described in [6] to derive optimal rules for arbitrary admissible number of elements. We demonstrate the proposed methodology on relevant examples, where we derive optimal rules for various even-degree spline spaces. We also discuss convergence of our rules to their asymptotic counterparts, these are the analogues of the midpoint rule of Hughes et al. [16], that are exact and optimal for infinite domains.

Precise gouging-free tool orientations for 5-axis CNC machining

We present a precise approach to the generation of optimized collision-free and gouging-free tool paths for 5-axis CNC machining of freeform NURBS surfaces using flat-end and rounded-end (bull nose) tools having cylindrical shank. To achieve high approximation quality, we employ analysis of hyper-osculating circles (HOCs) (Wang et?al., 1993a,b), that have third order contact with the target surface, and lead to a locally collision-free configuration between the tool and the target surface. At locations where an HOC is not possible, we aim at a double tangential contact among the tool and the target surface, and use it as a bridge between the feasible HOC tool paths. We formulate all such possible two-contact configurations as systems of algebraic constraints and solve them. For all feasible HOCs and two-contact configurations, we perform a global optimization to find the tool path that maximizes the approximation quality of the machining, while being gouge-free and possibly satisfying constraints on the tool tilt and the tool acceleration. We demonstrate the effectiveness of our approach via several experimental results. We present an algorithm for generating optimized gouging-free tool path for 5-Axis CNC machining.We employ analysis of hyper-osculating circles that provides third order approximation of the surface.Double tangential contact between the tool and the target surface is employed to connect feasible hyper-osculating tool paths.A robust collision and gouging detection algorithm is provided.We introduce a global optimization algorithm that maximizes the geometric matching between the tool and the target surface.

Construction of rational curves with rational rotation-minimizing frames via möbius transformations

We show that Mobius transformations preserve the rotation-minimizing frames which are associated with space curves. In addition, these transformations are known to preserve the class of rational Pythagorean-hodograph curves and rational frames. Based on these observations we derive an algorithm for G1 Hermite interpolation by rational Pythagorean-hodograph curves with rational rotation-minimizing frames.

Explicit Gaussian quadrature rules for C 1 cubic splines with symmetrically stretched knot sequences

We provide explicit expressions for quadrature rules on the space of C 1 cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires minimal number of nodes. Numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented. The paper provides expressions for quadrature rules on the space of C 1 cubic splines with non-uniform, symmetrically stretched knot sequences.Any function from the space is exactly integrated by the rule.The quadrature nodes and weights are derived via explicit recursion formulae, with no intervention of any numerical solver.The rule is optimal, i.e. it requires minimal number of nodes.

Constrained multi-degree reduction with respect to Jacobi norms

We show that a weighted least squares approximation of Bezier coefficients with factored Hahn weights provides the best constrained polynomial degree reduction with respect to the Jacobi L 2 -norm. This result affords generalizations to many previous findings in the field of polynomial degree reduction. A solution method to the constrained multi-degree reduction with respect to the Jacobi L 2 -norm is presented. We study the problem of constrained degree reduction with respect to Jacobi norms.Our results generalize several previous findings on polynomial degree reduction.We explore the space of Jacobi parameters on the reduced polynomial approximation.