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An adapted orthonormal frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent f1(ξ)=r′(ξ)/|r′(ξ)| and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal-plane vectors. Employing the Euler-Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean-hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal-plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal-plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples. [ABSTRACT FROM AUTHOR]

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results. [ABSTRACT FROM AUTHOR]

Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of G1 and G2 continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design. [ABSTRACT FROM AUTHOR]

Treating trauma to the cranio-maxillofacial region is a great challenge and requires expert clinical skills and sophisticated radiological imaging. The aim of reconstruction of the facial fractures is to rehabilitate the patient both functionally and aesthetically. In this article we employed B-spline curves to construct the occipital bone fracture using Digital Imaging and Communications in Medicine (DICOM) format data. The construction of occipital bone fracture starts with the boundary extraction followed by corner detection, construction of fractured part inner outer curve for each DICOM data using B-spline curves and finally the construction of fractured part in DICOM format. Method used in this article is based on DICOM data only and does not require any technique such as mirror imaging, technical help, reference skull, or to take average thickness of skull bone. Using the proposed method, the constructed fractured implant is custom made for every individual patient. At the end of this article we present a real case, in which we have constructed the occipital bone fracture using B-spline. The proposed method has been validated using post-operation DICOM data. For practical application, Graphical User Interface (GUI) has been developed. [ABSTRACT FROM AUTHOR]

Non-Uniform Rational B-Spline (NURBS) is the multidisciplinary topic in Mathematics, Computer Science, and Engineering. The NURBS curves together with their geometric properties are widely used in Computer Aided Design, Computer Aided Geometric Design, and Computational Mathematics. When a single weight approaches infinity, the limit of a NURBS curve tends to the corresponding control point. In this paper, a kind of control structure of a NURBS curve, called regular control curve, is defined. We prove that the limit of the NURBS curve is exactly its regular control curve when all of weights approach infinity, where each weight is multiplied by a certain one-parameter function tending to infinity, different for each control point. Moreover, some representative examples are presented to show this property and indicate its application for shape deformation. [ABSTRACT FROM AUTHOR]

A particular class of planar two-patch geometries, called bilinear-like G 2 two-patch geometries, is introduced. This class includes the subclass of all bilinear two-patch parameterizations and possesses similar connectivity functions along the patch interface. It is demonstrated that the class of bilinear-like G 2 two-patch parameterizations is much wider than the class of bilinear parameterizations and can approximate with good quality given generic two-patch parameterizations. We investigate the space of C 2 -smooth isogeometric functions over this specific class of two-patch geometries. The study is based on the equivalence of the C 2 -smoothness of an isogeometric function and the G 2 -smoothness of its graph surface (cf. Groisser and Peters (2015) and Kapl et al. (2015). The dimension of the space is computed and an explicit basis construction is presented. The resulting basis functions possess simple closed form representations, have small local supports, and are well-conditioned. In addition, we introduce a subspace whose basis functions can be generated uniformly for all possible configurations of bilinear-like G 2 two-patch parameterizations. Numerical results obtained by performing L 2 -approximation and solving Poisson’s equation indicate that already the subspace possesses optimal approximation properties. [ABSTRACT FROM AUTHOR]

The paper presents an unambiguous modelling of imprecisely defined shapes of closed curves using classical and directed interval arithmetic. The authors focus on the development of an effective strategy of modelling interval smooth closed curves (which enforce C2 continuity in points at which adjacent interval segments join) using interval cubic Bézier segments. For this purpose, algebraic relationships between Bézier and de Boor control points, formerly known for precisely defined curves, are generalized. We obtain interval control points that define interval closed curves. Additionally, the reliability of such way of modelling of closed curves is examined. We directly apply classical and directed interval arithmetic to mentioned relationships and try to solve obtained interval systems of algebraic equations. However, we obtain ambiguous solutions. Therefore, we propose our new strategy of modification of directed interval arithmetic to obtain reliable and unambiguous shapes of interval closed curves. [ABSTRACT FROM AUTHOR]

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 92d-6c+1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage. [ABSTRACT FROM AUTHOR]

To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bézier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bézier rotation surfaces are constructed using a generalized Bézier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bézier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bézier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bézier rotation surfaces provide a valuable way for the design of rotation surfaces. [ABSTRACT FROM AUTHOR]

In this paper, two methods for approximation of offset curves of H-Bézier curves and H-B spline curves are presented, which are based on the approximation of shifting control points and norm of parametric speed. Firstly, after calculating the perturbed vectors, the offset curve can be obtained by shifting the control points of the base curve. Then, both the Chebyshev approximation and optimal trigonometric polynomials approximation of parametric speed of base curve are presented, and two approximation functions of offset curves are also obtained. Finally, some examples are used to demonstrate their practicality. [ABSTRACT FROM AUTHOR]

In order to tackle the problem of shape design and shape adjustment of complex surfaces in engineering, continuity conditions between generalized Bézier-like surfaces with multiple shape parameters are studied in this paper. Firstly, the geometric model of the generalized Bézier-like surfaces is built by blending a number of Bézier-like curves with independent shape parameters. Secondly, based on the terminal properties and linear independence of Bernstein-like basis functions, the conditions for G continuity between two adjacent generalized Bézier-like surfaces are derived, and then simplified by choosing appropriate shape parameters. Finally, some properties and applications of the smooth continuity between generalized Bézier-like surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which can greatly improve the ability to construct complex surfaces by using the generalized Bézier-like surfaces. [ABSTRACT FROM AUTHOR]

NURBS surface is very useful in geometric modeling, animation, image morphing and deformation. Constructing non-self-intersecting (injective) NURBS surfaces is an important process in surface and solid modeling. In this paper, the injective conditions of tensor product NURBS surface are studied, based on the geometric positions of control points, which are equivalent to the surface to be non-self-intersecting for all positive weights. Finally, some representative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized Refine-Smooth algorithm to design a family of a -point b -ary approximating subdivision schemes with bell-shaped mask, where a ≥ 3 and b ≥ 2. We use the combination of corner cutting b -ary subdivision scheme and weighted average of ( b + 1 ) -points to construct the proposed family. We demonstrate that the proposed family has smaller complexity and support width and higher continuity than the existing Refine-Smooth subdivision schemes. We also study the shape preserving properties of the proposed family. In addition, it is observed that the proposed family is suitable for fitting the locally noisy, oscillatory, and irregular data. [ABSTRACT FROM AUTHOR]

Composite basis functions for pyramidal elements on the spaces H 1 ( Ω ) , H ( curl , Ω ) , H ( div , Ω ) and L 2 ( Ω ) are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart–Thomas–Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, etc.) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions. [ABSTRACT FROM AUTHOR]

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc´ıa-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface. Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster. [ABSTRACT FROM AUTHOR]

As a new method of representing curves, Q-Bézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for Q-Bézier curves of degree n. Following the analysis of basis functions and terminal properties of Q-Bézier curves of degree n, the continuity conditions of $\mathrm{G}^{1}$ and $\mathrm{G}^{2}$ between two adjacent Q-Bézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for Q-Bézier curves and analyze the influence rules of shape parameters for Q-Bézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design. [ABSTRACT FROM AUTHOR]

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (e.g. Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user. Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it. This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan. Numerical results show that this hybrid method is efficient and robust. [ABSTRACT FROM AUTHOR]

A rotation-minimizing frame ( f , f , f ) on a space curve r( ξ) defines an orthonormal basis for $\mathbb {R}^{3}$ in which $\mathbf {f}_{1}=\mathbf {r}^{\prime }/|\mathbf {r}^{\prime }|$ is the curve tangent, and the normal-plane vectors f , f exhibit no instantaneous rotation about f . Polynomial curves that admit rational rotation-minimizing frames (or RRMF curves) form a subset of the Pythagorean-hodograph (PH) curves, specified by integrating the form $\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )$ for some quaternion polynomial $\mathcal {A}(\xi )$ . By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial $\mathcal {A}(\xi )$ , a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms. [ABSTRACT FROM AUTHOR]

The paper presents an investigation study of the mechanical properties of metal foam. In this investigation, a numerical simulation by finite element method is used. For this purpose, the foam structure is modelled by a regular network of Kelvin cells. In these cells, the strands are modelled as a three-dimensional (3D) finite element beam. In particular, we consider four types of strands cross sections: (1) circular (2) square (3) triangular and (4) Plateau shape. Our numerical results are in agreement with the experimental results obtained on a real Nickel metallic foam. In addition, we perform a parametric analysis to study the effect of some geometrical characteristics on the elasticity of a metal foam. Among these geometrical characteristics, we consider the shape and dimensions of strands cross section as well as the inertia variation. [ABSTRACT FROM AUTHOR]

The injectivity of NURBS curve implies the curve has no self-intersection. In this paper, we propose a geometric condition on the control polygon which guarantees the NURBS curve to be injective for all possible choices of positive weights. The proof is based on the degree elevation algorithm and toric degeneration theory of NURBS curve. [ABSTRACT FROM AUTHOR]

The current study presents rational cubic trigonometric Bézier curve with two shape parameters, which is a novel technique for drawing free form curves. The proposed curve retains most of the geometric properties of conventional rational cubic Bézier but is flexible due to the presence of shape parameters. The shape of the curve can be adjusted locally by altering the values of shape parameters as well as the weights. Using the given end point curvatures, conditions on shape parameters and weights have been driven so that the curve always lies in the convex hull of its control point. Later, this curve is used to generate piecewise rational trigonometric curves stitched together using parametric and geometric Hermite continuity conditions which can facilitate a designer to generate a smooth composition of curves in a situation where single curve does not work. The piecewise curves thus constructed are used to draw the layout of certain English and Arabic alphabets as an application in font designing. [ABSTRACT FROM AUTHOR]

We present a framework for analyzing nonlinear $$\mathbb {R}^d$$ -valued subdivision schemes which are geometric in the sense that they commute with similarities in $$\mathbb {R}^d$$ . It allows us to establish $$C^{1,\alpha }$$ -regularity for arbitrary schemes of this type, and $$C^{2,\alpha }$$ -regularity for an important subset thereof, which includes all real-valued schemes. Safe bounds on the domain of convergence of the scheme and on the Hölder exponent of the limit curves can be found by determining the range of certain real-valued functions. This task can be executed automatically and rigorously by a computer when using interval arithmetics. [ABSTRACT FROM AUTHOR]

We construct G 1 Hermite interpolating curves in biangular coordinates, and provide sufficient conditions for their convexity. In a biangular coordinate system, the problem reduces to that of choosing suitable functions interpolating the biangular coordinates of the curve at its end points. The simplest linear equations, γ = ( ( 1 − t ) α , t β ) , in biangular coordinates correspond to a sectrix of Maclaurin, which we extend by introducing two shape parameters that pull the curve towards the sides of its triangular envelope. In addition, we consider a class of curves whose biangular coordinates have a constant sum, and we analyze their shape and curvature. [ABSTRACT FROM AUTHOR]

Let $${\Sigma}$$ be a strictly convex (hyper-)surface, S an optimal triangulation (piecewise linear in ambient space) of $${\Sigma}$$ whose m vertices lie on $${\Sigma}$$ and $${\tilde{S}_m}$$ an optimal triangulation of $${\Sigma}$$ with m vertices. Here we use optimal in the sense of minimizing $${d_H(S_m, \Sigma)}$$ , where $${d_H}$$ denotes the Hausdorff distance. In 'Lagerungen in der Ebene, auf der Kugel und im Raum' Fejes Tóth conjectured that the leading term in the asymptotic development of $${d_H(S_m, \Sigma)}$$ in m is twice that of $${d_H(\tilde{S}_m, \Sigma)}$$ . This statement is proven. [ABSTRACT FROM AUTHOR]

QSAR modeling is a method for predicting properties, e.g. the solubility or toxicity, of chemical compounds using machine learning techniques. QSAR is in widespread use within the pharmaceutical industry to prioritize compounds for experimental testing or to alert for potential toxicity during the drug discovery process. However, the confidence or reliability of predictions from a QSAR model are difficult to accurately assess. We frame the application of QSAR to preclinical drug development in an off-line inductive conformal prediction framework and apply it prospectively to historical data collected from four different assays within AstraZeneca over a time course of about five years. The results indicate weakened validity of the conformal predictor due to violations of the randomness assumption. The validity can be strengthen by adopting semi-off-line conformal prediction. The non-randomness of the data prevents exactly valid predictions but comparisons to the results of a traditional QSAR procedure applied to the same data indicate that conformal predictions are highly useful in the drug discovery process. [ABSTRACT FROM AUTHOR]

Many works have investigated the problem of reparameterizing rational Bézier curves or surfaces via Möbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after Möbius transformation. What's more the users of computer aided design softwares may require some guidelines for designing rational Bézier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational Bézier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal parametric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational Bézier surfaces with compact derivative bounds. [ABSTRACT FROM AUTHOR]

In this paper, we research the thermo-mechanical reliability of lead-free solder joints in WLCSP assembly subjected to an accelerated thermal cyclic loading based on finite element simulation and Taguchi method. Effects of different control factors, including solder materials, height of solder ball, chip thickness and substrates thickness on the reliability of the lead-free solder joints were examined and calculated. The corner solder joint is the most critical solder joints in the assembly, the top surface of the solder joint is the crack propagated location. The optimal design in the WLCSP assembly has the combination of the SnAgCuAl, height of solder ball at 0.25 mm, chip thickness at 0.6 mm, substrate thickness at 1.0 mm. Moreover, the solder materials is the most important factor among the control factors in the WLCSP assembly. [ABSTRACT FROM AUTHOR]

The problem of covering a convex 3D polytope by the minimal number of congruent spheres is reduced to a sequence of problems of minimising sphere radius when fixing the number of the spheres. We form a mathematical model of the problem using the Voronoi polytopes. Characteristics of the model are investigated. Extrema are attained at the vertices of the Voronoi polytopes constructed for sphere centres. To search for local minima, a modification of the Zoutendijk feasible directions method in a combination with random search is developed. Some numerical results for a cube and a non-regular octahedron are obtained. [ABSTRACT FROM AUTHOR]

With the addition CNTs, the properties of SnAgCu solders and solder joints can be improved remarkably. In this paper, based on the creep testing, the Dorn creep model was used to represent the creep behavior of SnAgCuCNT solder, and the parameters were determined with the fitting analysis. Moreover, the Dorn creep model of SnAgCu and SnAgCuCNT was incorporated into finite element code for calculating the creep strain response of solder joints in FCBGA (Flip Chip Ball Grid Array) device during thermal cycle loading. The results indicated that the CNTs can enhance the creep resistance of SnAgCu solder, which demonstrates the SnAgCuCNT solders can replace the SnAgCu solder in electronic industry for highreliability requirement. [ABSTRACT FROM AUTHOR]

This paper describes the implementation of a reconfigurable hardware-based genetic algorithm (HGA) accelerator using the hardware-software (HW/SW) co-design methodology. This HGA is coupled with a unique TRNG that extracts random jitters from a phase lock loop (PLL) to ensure proper GA operation. It is then applied and benchmarked with several case studies, which include the optimization of a simple fitness function, a constrained Michalewicz function, and the tuning of parameters in finger-vein biometrics. A HGA solution is necessary in systems that demand high performance during the optimization process. However, implementations that are completely designed in hardware will result in a very rigid architecture, making it difficult to reconfigure the system for use in different applications. This paper aims to solve this issue by proposing a HGA design that provides reconfigurability and flexibility by moving problem-dependent processes into software. The prototyping platform used is an Altera Stratix II EP2S60 FPGA prototyping board with a clock frequency of 50 MHz. The HW/SW co-design technique is applied, and system partitioning is done based on aspects such as system constraints, operational intensity, process sequencing, hardware logic utilization, and reconfigurability. Experimental results show that the proposed HGA outperforms equivalent software implementations compiled with an open-sourced C++ GA component library (GAlib) running on the same prototyping platform by 102 times at most. In the final case study, the application of the proposed HGA in tunable parameter optimization in finger-vein biometrics improved the matching rate, reducing the equal error rate (EER) value from 1.004% down to 0.101%. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how to use the Gauss map to produce efficient triangulations of parametric surfaces. The central idea is to invert canonical triangulations of the sphere using the Gauss map. The main problem is that the inverse of the Gauss map is not in general well defined. We show how to overcome this problem and formulate an algorithm to generate the desired triangulation. [ABSTRACT FROM PUBLISHER]

A new web service can be built by combining existing web services and coordinating their actions by a 'mediator'. This work presents a formal design methodology based on model checking that generates all the mediator designs that meet requirements specified by a user. The methodology allows the user to explore the designs in search of the mediator that offers the best theoretical performance. Markov models of various measures of performance are considered. Each measure may result in a different ranking of the mediator designs (from best to worst for example). From these rankings the user can select the design that at least theoretically best fits the purpose. The novelty of this work is the use of model checking and Markov analysis in a single formal framework to generate, explore, and select from a set of provably-correct mediators-while still in the design stage of development. [ABSTRACT FROM AUTHOR]

Dynamic Geometry is the field of interactively performing geometric construction on a computer. In addition to simulating ruler-and-compass constructions we allow a drag mode. This drag mode allows to move geometric objects that have at least one degree of freedom. The remaining part of the construction should adjust automatically. Thus, during the motion, we have to trace the resulting paths of all geometric objects. This path tracking problem is known as the Tracing Problem from Dynamic Geometry. It combines the step-by-step procedure of doing geometric constructions with the continuous concept of motions. This study is based on the model for Dynamic Geometry used in the interactive geometry software Cinderella. We give a numerical solution to the Tracing Problem based on continuation methods and a reliable algorithm based on real and complex interval arithmetic. Degenerate situations like the intersection of two identical lines lead to critical points in the configuration space and are treated separately. [ABSTRACT FROM AUTHOR]