In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of ...Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.
We investigate a recently introduced methodology for 5-axis flank computer numerically controlled (CNC) machining, called double-flank milling (Bo et al., 2020). We show that screw rotors are well ...suited for this manufacturing approach where the milling tool possesses tangential contact with the material block on two sides, yielding a more efficient variant of traditional flank milling. While the tool’s motion is determined as a helical motion, the shape of the tool and its orientation with respect to the helical axis are unknowns in our optimization-based approach. We demonstrate our approach on several rotor benchmark examples where the pairs of envelopes of a custom-shaped tool meet high machining accuracy.
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•The method for 5-axis double-flank CNC machining of screw rotor sis presented.•The shape of the milling tool is an unknown in our optimization-based framework.•For symmetric profiles, there exists a custom-shaped tool that is provably exact.•For non-symmetric profiles, our approximation results meet fine machining tolerances.•We validate our approach on several screw rotor benchmarks.
In this paper, the problem of Hermite interpolation by clamped Minkowski Pythagorean hodograph (MPH) B-spline curves is considered. Using the properties of B-splines, our intention is to use the MPH ...curves of degrees lower than in algorithms designed before. Special attention is devoted to C1∕C2 Hermite interpolation by MPH B-spline cubics/quintics. The resulting interpolants are obtained by exploiting properties of B-spline basis functions and via solving special quadratic and linear equations in Clifford algebra Cℓ2,1. All the presented algorithms are purely symbolic. The results are confirmed by several applications, in particular we use them to generate an approximate conversion of a given analytic curve to MPH B-spline curve with a high order of approximation, then to an efficient approximation of the medial axis transform of a planar domain leading to NURBS representation of the (trimmed) offsets of the domain boundaries, and to skinning of systems of circles in plane.
A new method for 5-axis flank milling of free-form surfaces is proposed. Existing flank milling path-planning methods typically use on-market milling tools whose shape is cylindrical or conical, and ...is therefore not well-suited for meeting fine tolerances for manufacturing of benchmark free-form surfaces like turbine blades, gears, or blisks. In contrast, our optimization-based framework incorporates the shape of the tool into the optimization cycle and looks not only for the milling paths, but also for the shape of the tool itself. Given a free-form reference surface and a guiding path that roughly indicates the motion of the milling tool, tangential movability of quadruplets of spheres centered along a straight line is analyzed to indicate possible shapes and their motions. This results in G1 Hermite data in the space of rigid body motions that are interpolated and further optimized, both in terms of the motion and the shape of the milling tool itself. We demonstrate our algorithm on synthetic free-form surfaces and industrial benchmark datasets, showing that the use of custom-shaped tools is capable of meeting fine industrial tolerances and outperforms the use of classical, on-market tools.
This paper is devoted to the investigation of selected situations when computing projective (and other) equivalences of algebraic varieties can be efficiently solved via finding projective ...equivalences of finite sets of points on the projective line. In particular, we design a method that finds for two algebraic varieties X,Y from special classes an associated set of automorphisms of the projective line (the so called good candidate set) consisting of suitable candidates for the subsequent construction of possible mappings X→Y. The functionality of the designed approach is presented for computing projective equivalences of rational curves, determining projective equivalences of rational ruled surfaces, detecting affine transformations between planar algebraic curves, and computing similarities between two implicitly given algebraic surfaces. When possible, symmetries of given shapes are also discussed as special cases.
Analyzing the symmetries present in point clouds, which represent sets of 3D coordinates, is important for understanding their underlying structure and facilitating various applications. In this ...paper, we propose a novel decomposition-based method for detecting the entire symmetry group of 3D point clouds. Our approach decomposes the point cloud into simpler shapes whose symmetry groups are easier to find. The exact symmetry group of the original point cloud is then derived from the symmetries of these individual components. The method presented in this paper is a direct extension of the approach recently formulated in Bizzarri et al. (2022a) for discrete curves in plane. The method can be easily modified also for perturbed data. This work contributes to the advancement of symmetry analysis in point clouds, providing a foundation for further research and enhancing applications in computer vision, robotics, and augmented reality.
•We present a theoretical framework to assess exact point clouds symmetries via decomposition-based method.•The exact symmetry group of the 3D point cloud is derived from the symmetries of simpler components.•The method can be easily modified also for perturbed data.
We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in the plane. The method is based on a suitable trigonometric interpolation of vertices of the given ...polyline and consequent computation of the symmetry group of the obtained trigonometric curve. The algorithm exploits the fact that the introduced unique assignment of the trigonometric curve to each closed discrete curve commutes with isometries. For understandable reasons, an essential part of the paper is devoted to determining rotational and axial symmetries of trigonometric curves. We also show that the formulated approach can be easily applied on unorganized clouds of points. A functionality of the designed detection method is presented on several examples.
CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric ...computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between "dual" classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software.
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