Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

Computing the Minkowski sum of ruled surfaces

The boundary of the Minkowski sum of two geometric objects is part of the so-called convolution surface of the boundary surfaces of the two input objects. In most cases, convolution surfaces can be computed only by numerical algorithms. The present paper studies convolution surfaces of ruled surfaces. There, explicit parameterizations for the convolution surface can be derived. Moreover, we study the rational convolution surface of two rational ruled surfaces and the connection to rational parameterizations of offsets of rational ruled surfaces.     

Proper real reparametrization of rational ruled surfaces

Let K⊆R be a computable field. We present an algorithm to decide whether a proper rational parametrization of a ruled surface, with coefficients in K(i), can be properly reparametrized over a real (i.e. embedded in R) finite field extension of K. Moreover, in the affirmative case, the algorithm provides a proper parametrization with coefficients in a real extension of K of degree at most 2.     

Improved positioning of cylindrical cutter for flank milling ruled surfaces

An optimized positioning procedure for flank milling ruled surfaces with cylindrical cutter is described in the paper. The tool axis trajectory surface is a ruled surface, which is generated by moving the tool axis. The proposition that the envelope surface of cylindrical cutter is the offset surface of tool axis trajectory surface is proved using kinematics approach. It is a complement of Bedi's [Bedi S, Mann S, Menzel C. Flank milling with flat end cutter. Comput Aided Des 2003; 35:293-300] analysis about the envelope surface of cylindrical cutter. Subsequently, we get another proposition that the deviation at extremum point between the designed surface and the envelope surface of cylindrical cutter is equal to that between the offset surface of designed surface and the tool axis trajectory surface. Based on this proposition, we propose three points offset (TPO) strategy to approximate the offset surface. In order to reduce errors further, a simple least square approximation scheme is established to make the tool axis trajectory surface fit the offset surface of designed surface as much as possible. By solving the linear system of equations, the tool axis trajectory surface is deformed. Simultaneously, the corresponding envelope surface is deformed to approximate the designed surface better. Two examples are given to verify the effectiveness of the developed 5-axis flank milling technique.     

Triple tangent flank milling of ruled surfaces

This paper presents a positioning strategy for flank milling ruled surfaces. It is a modification of a positioning method developed by Bedi et al. [Comput Aided Des 35 (2003) 293]. A cylindrical cutting tool is initially positioned tangential to the two boundary curves on a ruled surface. Optimization is used to move these tangential points to different curves on the ruled surface to reduce the error. A second optimization step is used to additionally make the tool tangent to a rule line, further reducing the error and resulting in a tool position, where the tool is positioned tangential to two guiding rails and one rule line. The resulting surface has 88% less under cutting than the method of Bedi et al.     

A new radiosity approach for regular objects: application to ruled surfaces

This paper introduces a new approach in the radiosity method. The main principle applied here, is the improvement of form factor computation by the knowledge of the model properties. More precisely, if surfaces describing a scene are “regular”, the values of the form factor and its “derivatives” between any patch B and a patch A intervening in the meshing of a given surface, enable us to evaluate (with a required precision) the form factors values between B and the neighboring patches of A. We will also show that the mathematical relation we have obtained, 1) is especially efficient with ruled surfaces, 2) significantly decreases the computation time, 3) is well-adapted to refinement or subdivision techniques and 4) gives us an interesting variety of surfaces. Calculation times are equivalent to those obtained with a projective method (hemi-cube for example) but with an efficient control of the generated errors.     

Analytical estimation of error in flank milling of ruled surfaces

This article introduces a method to estimate geometric error during flank milling of a ruled surface. The various positioning schemes developed by researchers are intended to reduce this geometric error in order to mill with larger sized milling cutters while respecting the tolerance interval. There are two trends in positioning: either positioning is simple and right from the start it is easy to determine design of the maximum allowed milling cutter radius, or positioning is complex and determination of the maximum milling cutter dimensions can only be conducted after digital calculations of the error. It will then be necessary to choose another milling cutter radius and recommence the positioning procedure and error calculation in order to validate the tool. In the present study, a method to estimate error in the scope of complex positioning is presented. The aim is to be capable of choosing a maximum cutting tool radius that respects the tolerance interval.     

Complexity of the consistency problem for certain Post classes

The complexity of the consistency problem for several important classes of Boolean functions is analyzed. The classes of functions under investigation are those which are closed under function composition or superposition. Several of these so-called Post classes are considered within the context of machine learning with an application to breast cancer diagnosis. The considered Post classes furnish a user-selectable measure of reliability. It is shown that for realistic situations which may arise in practice, the consistency problem for these classes of functions is polynomial-time solvable.     

Collision and intersection detection of two ruled surfaces using bracket method

Collision and intersection detection of surfaces is an important problem in computer graphics and robotic engineering. A key idea of our paper is to use the bracket method to derive the necessary and sufficient conditions for the collision of two ruled surfaces. Then the numerical intersection curve can be characterized. The cases for two bounded ruled surfaces are also discussed.     

Tool path planning for 5-axis flank milling of ruled surfaces considering CNC linear interpolation

This paper investigates tool path planning for 5-axis flank milling of ruled surfaces in consideration of CNC linear interpolation. Simulation analyses for machining error show insights into the tool motion that generates a precision machined surface. Contradicting to previous thoughts, the resultant tool path does not necessarily produce minimal machining error when the cutter contacts the rulings of a developable surface. This effect becomes more significant as the distance between two cutter locations is increased. An optimizing approach that adjusts the tool position locally may not produce minimal error as far as the entire surface is concerned. The optimal tool path computed by a global search scheme based on dynamic programming supports this argument. A flank milling experiment and CMM measurement further validate the findings of this work.     

Analytic inversion for a general case of certain classes of matrices

The explicit inverse and determinant of a class of matrices are presented. The class under consideration is defined by 4n ? 2 parameters, analytic expressions of which form the elements of the upper Hessenberg type inverse. These analytic expressions enable recursion formulae to be obtained, which reduce the arithmetic operations to O(n ²). The Hadamard product of two specific structures' matrices provides the class presented, special cases of which are already known classes of test matrices.     

Computing the intersection of two ruled surfaces by using a new algebraic approach

In this paper a new algorithm for computing the intersection of two rational ruled surfaces, given in parametric/parametric or implicit/parametric form, is presented. This problem can be considered as a quantifier elimination problem over the reals with an additional geometric flavor which is one of the central themes in V. Weispfenning research. After the implicitization of one of the surfaces, the intersection problem is reduced to finding the zero set of a bivariate equation which represents the parameter values of the intersection curve, as a subset of the other surface. The algorithm, which involves both symbolic and numerical computations, determines the topology of the intersection curve as an intermediate step and eliminates extraneous solutions that might arise in the implicitization process.