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Computational Line Geometry

http://www.amss.ac.cn/xwdt/kydt/201305/P020130509334302251871.pdf

Certified rational parametric approximation of real algebraic space curves with local generic position method

The past decade has witnessed sustained interest in elucidating the basic theory of Pythagorean–hodograph (PH) curves, developing construction algorithms, formulating generalizations, and investigating applications. The rapid pace of this activity, encompassing diverse lines of inquiry, makes it desirable at this point to take a broad perspective of these recent developments, and assess their relationships. In the present article, we aim to address this need by categorizing recent results into a number of broad themes—extensions and specializations of the basic polynomial PH curves; rational orthonormal frames along spatial PH curves; construction and analysis algorithms for PH curves; surface design based on PH curves; and the use of PH curves in practical applications.

New Developments in Theory, Algorithms, and Applications for Pythagorean–Hodograph Curves

In this paper, an algorithm to compute a certified $G^1$ rational parametric approximation for algebraic space curves is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the same topology and their Hausdauff distance is smaller than a given precision. Thus, the method also gives a new algorithm to compute the topology for space algebraic curves. The main advantage of the algorithm, inhering from the local generic method, is that topology computation and approximation for a space curve is directly reduced to the same tasks for two plane curves. In particular, the error bound of the approximation space curve is obtained from the error bounds of the approximation plane curves explicitly. Nontrivial examples are used to show the effectivity of the method.

Certified Rational Parametric Approximation of Real Algebraic Space Curves with Local Generic Position Method

Morphogenesis of surfaces with planar lines of curvature and application to architectural design

http://www.karlin.mff.cuni.cz/~sir/papers/biarc.pdf

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

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.

/pdf/geometry-and-tool-motion-planning-for-curvature-adapted-cnc-pcrds6jejr.pdf

Geometry and tool motion planning for curvature adapted CNC machining

A symbolic-numerical approach to approximate parameterizations of space curves using graphs of critical points

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.

Michal Bizzarri

Papers

C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

Geometry and tool motion planning for curvature adapted CNC machining

A symbolic-numerical approach to approximate parameterizations of space curves using graphs of critical points

Manufacturing of Screw Rotors Via 5-axis Double-Flank CNC Machining

Computing projective equivalences of special algebraic varieties