Cutting Path Planning for Ruled Surface Impellers

At present, most commercial computer-aided manufacturing （CAM） systems are deficient in efficiency and performances on generating tool path during machining impellers. To solve the problem, this article develops a special software to plan cutting path for ruled surface impellers. An approximation algorithm to generate cutting path for machining integral ruled surface impellers is proposed. By fitting sampling data points of an impeller blade into a curve, a model of ruled surface blade of an impeller is built up. Furthermore, by calculating the points where the cutter axis vector intersects the free-form hub surface of an impeller, problems about, for instance, the ambiguity in calculation and machining the wide blade surface with a short flute cutter are solved. Finally, an integral impeller cutting path is planned by way of an integrated cutter location control algorithm. Simulation and machining tests with an impeller are performed on a 5-axis computer numerically controlled （CNC） mill machine, which shows the feasibility of the proposed algorithm.

On the Ruled surfaces in Minkowski 3-space R1^3

Izumiya and Takeuchi （2003） obtained some characterizations for Ruled surfaces. Turgut and Haclsalihoglu （1998） defined timelike Ruled surfaces and obtained some characterizations in timelike Ruled surfaces. Choi （1995） and Jung and Pak （1996） studied Ruled surfaces. This study uses the method in （lzumiya and Takeuchi, 2003） to investigate cylindrical helices and Bertrand curves as curves on timelike Ruled surfaces in Minkowski 3-space R1^3. We have studied singularities of the rectifying developable （surface） of a timelike curve. We observed that the rectifying developable along a timelike curve a is non-singular if and only if a is a cylindrical helice. In this case the rectifying developable is a cylindrical surface.

RULED SURFACE MACHINING BY 4-AXES SIMULTANEOUS CONTROL WIRE-EDM

On the tasis of study in the mathematical model of 3-dimensional ruled surface (RS),this paper introduces a new concept of distance paramcter (DP) and also puts forward that themethod of modeling a RS depends on not only two boundary curves but also DP. According toabove theory, the formulas to calculate corresponding point coordinates to any kind of top and bot-tom profile of a workpiece and formulas to calcuate the maximum inclination angle of ruling linehave been obtained. Then a different top and bottom RS mathining method including profile withline-are combination as well as parametric curves has been achieved by 4-axes simultancous con-trol programming proposed.

Tool path generation for 5-axis flank milling of ruled surfaces with optimal cutter locations considering multiple geometric constraints

Ruled surfaces found in engineering parts are often blended with a constraint surface,like the blade surface and hub surface of a centrifugal impeller.It is significant to accurately machine these ruled surfaces in flank milling with interference-free and fairing tool path,while current models in fulfilling these goals are complex and rare.In this paper,a tool path planning method with optimal cutter locations(CLs)is proposed for 5-axis flank milling of ruled surfaces under multiple geometric constraints.To be specific,a concise three-point contact tool positioning model is firstly developed for a cylindrical cutter.Different tool orientations arise when varying the three contact positions and a tool orientation pool with acceptable cutter-surface deviation is constructed using a meta-heuristic algorithm.Fairing angular curves are derived from candidates in this pool,and then curve registration for cutter tip point on each determined tool axis is performed in respect of interference avoidance and geometric smoothness.On this basis,an adaptive interval determination model is developed for deviation control of interpolated cutter locations.This model is designed to be independent of the CL optimization process so that multiple CLs can be planned simultaneously with parallel computing technique.Finally,tests are performed on representative surfaces and the results show the method has advantages over previous meta-heuristic tool path planning approaches in both machining accuracy and computation time,and receives the best comprehensive performance compared to other multi-constrained methods when machining an impeller.

Positioning method of a cylindrical cutter for ruled surface machining based on minimizing one-sided Hausdorff distance

Motivated by the definition of the machining errors induced by tool path planning methods, a mapping curve of the tool axis of a cylindrical cutter is constructed on the tool surface. The mapping curve is a typical one that can be used to express the closeness between the tool surface and the surface to be machined. A novel tool path planning method is proposed for flank or plunge milling ruled surfaces based on the minimization of the one-sided Hausdorff distance （HD） from the mapping curve to the surface to be machined. It is a nonlinear optimization problem in best uniform approximation （BUA） or Chebyshev sense. A mathematical programming model for computing the minimum one-sided HD is proposed. The linearization method of the programming model is provided and the final optimal solutions are obtained by simplex method. The effectiveness of the proposed BUA method is verified by two numerical examples and compared with the least squares （LS） and double point offset （DPO） methods. The variation in tool orientation induced by the optimization of the tool positions is also evaluated.

Proper Reparametrization of Rational Ruled Surface

In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.

Kinematic generation of ruled surface based on rational motion of point-line

This paper studies representation of rigid combination of a directed line and a reference point on it （here referred to as a ＂point-line＂） using dual quatemions. The geometric problem of rational ruled surface design is viewed as the kinematic prob- lem of rational point-line motion design. By using the screw theory in kinematics, mappings from the spaces of lines and point-lines in Euclidean three-dimensional space into the hyperplanes in dual quaternion space are constructed, respectively. The problem of rational point-line motion design is then converted to that of projective Bezier or B-spline image curve design in hyperplane of dual quatemions. This kinematic method can unify the geometric design of ruled surfaces and tool path generation for five-axis numerical control （NC） machining.

A Symbolic-Numeric Approach for Parametrizing Ruled Surfaces

This paper presents symbolic algorithms to determine whether a given surface(implicitly or parametrically defined)is a rational ruled surface and find a proper parametrization of the ruled surface.However,in practical applications,one has to deal with numerical objects that are given approximately,probably because they proceed from an exact data that has been perturbed under some previous measuring process or manipulation.For these numerical objects,the authors adapt the symbolic algorithms presented by means of the use of numerical techniques.The authors develop numeric algorithms that allow to determine ruled surfaces"close"to an input(not necessarily ruled)surface,and the distance between the input and the output surface is computed.

A Survey of the Representations of Rational Ruled Surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the authors revisit the representations of ruled surfaces including the parametric form,algebraic form,homogenous form and Plucker form.Moreover,the transformations between these representations are proposed such as parametrization for an algebraic form,implicitization for a parametric form,proper reparametrization of an improper one and standardized reparametrization for a general parametrization.Based on these transformation algorithms,one can give a complete interchange graph for the different representations of a rational ruled surface.For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces,the characterization methods are recalled to identify the ruled surfaces from them.

Bonnet Ruled Surfaces

We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.

The local classifications of the ruled surfaces of normals and binormals of a regular space curve

The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.

Physical origins of ruled surfaces on the reduced density matrices geometry

The reduced density matrices （RDMs） of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetry- breaking phase. We show that, apart from being signatures of symmetry-brealdng, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti＇s theorem--in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetry- breaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.

CAD/CAM ALGORITHM OF RULED SURFACE BASED ON LINE GEOMETRY

A new concept of design and manufacturing of ruled surface based on line geometry is proposed. Some practical algorithm for CAD system is derived. Some problems in design and manufacturing of ruled surface can be solved by using the algorithm.

Constant angle surfaces constructed on curves

The Frenet-Serret formula is used to characterize the constant angle ruled surfaces in R3. When the surfaces are the tangent developmental and normal surfaces, that is, r（s, v） = tr（s） ＋v（cosα（s） . t（s） ＋sina（s） . n（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a certain special surface. When the surfaces are normal and binormal surfaces, that is, r （ s, v ） = σ （ s ） ＋ v （ cosa （ s ） . n（s） ＋ since（s） . b（s））, it is shown that each of these surfaces is locally isometric to a piece of a plane or a cylindrical surface.

Existence Criteria of Blending Surfaces of Three Quadric Surfaces——Wu Wen-tsun's Formulae

Smoothly blending several implicit quadric surfaces with surfaces of the lowest degree is very useful in engineering technology.

Surface borehole synthesis tension deformation fracture time-space rule

In order to release the tension and shear effect of the superjacent rock strata movement during excavation in coal mine,protect the surface borehole case from fracturing fast and make a good use of the surface borehole during goaf methane drawing,a common synthesis tension deformation fracture model was set up based on the synthesis tension effect of the rock strata,and the deformation rule of the surface borehole case with time and space was researched.The results suggest that,to reduce the deformation the surface borehole should be built between the boundary of the stope and the knee of subsidence curve.At the same time,a 3DEC simulation model and an engineering example were carried out to examine the rules of theoretical model.The result suggests that the model and the rules accord to the test and have good building and protection engineering application values to the surface borehole.

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space G_(3)

In this paper,we define the curve rλ=r+λd at a constant distance from the edge of regression on a curve r(s)with arc length parameter s in Galilean 3-space.Here,d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s)in 3-dimensional Galilean space.We build the Frenet frame{Tλ,Nλ,Bλ}of the constructed curve rλwith respect to two types of the vector d and we indicate the properties related to the curvatures of the curve rλ.Also,for the curve rλ,we give the conditions to be a circular helix.Furthermore,we discuss ruled surfaces of type A generated via the curve rλand the vector D which is defined as tangent of the curve rλin 3-dimensional Galilean space.The constructed ruled surfaces also appear in two ways.The first is constructed with the curve rλ(s)=r(s)+λT(s)and the non-isotropic vector D.The second is formed by the curve rλ=r(s)+λ2N+λ3B and the non-isotropic vector D.We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable.Finally,we provide examples and visuals to back up our research.

Blaschke Hypersurfaces with Symmetric Shape Operator

A Blaschke hypersurface admits S symmetry if and only if S（X, Y） = S（Y, X）. We prove that the shape operator has only one eigenvalue. And such Blaschke surfaces are classified as affine spheres or ruled surfaces.

Joint product numerical range and geometry of reduced density matrices

The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection is convex in R3. The boundary of may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces： symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti＇s theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range II of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that a ruled surface emerges naturally when taking a convex hull of ∏. We show that, a ruled surface on sitting in ∏ has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of , with two boundary pieces of symmetry breaking origin separated by two gapless lines.