Vector fields for five-axis machining. A survey

Abstract

Toolpath generation (TPG) for multi-axis milling machines using vector fields (VF) and vector field analysis (VFA) is becoming increasingly popular in the manufacturing industry. Therefore, the paper presents a survey of algorithms and methods of TPG based on the vector fields of preferred directions (VFPD) for five-axis CNC machining. Two hundred relevant citations in top manufacturing and optimization journals during 1995–2021 have been presented and discussed. Additional 79 references in Appendices are related to a classification of five-axis machines, the theory and recent advances in the area of the vector and tensor fields.

Appendices

Appendix A. Design of the five-axis machines

There are 120 possible combinations of the joints of a five-axis mechanism. Each combination of the joints has 6 alternative locations of the axes on the tool and the workpiece chain. Furthermore, there are three possible choices of the two rotary axes. Therefore, the total number of the possible configurations is $3 \times 120\times 6 = 2160$ (Bohez [199] and Zhou et al. [200]).

My and Bohez [201] argue that in practice the position of the rotary axes is usually near to the tilting table or to the spindle. Using this principle and ignoring the 3 possible choices of the 2 rotary axes reduces the number of feasible configurations to 108. The design of the five-axis machine implies that three translation axes are mutually orthogonal and are aligned with the MCS. The tool axis is always parallel to Z axis of the MCS. Given these assumptions, let us denote the rotation axis by R and the translation axis by T. The survey by Bohez [199] introduces the five-axis mechanisms as follows. “Four main groups can be distinguished: (i) three T axes and two R axes; (ii) two T axes and three R axes; (iii) one T axis and four R axes and (iv) five R axes. Nearly all existing five-axis machine are in group (i). Also a number of welding robots, filament winding machines and laser machining centers fall in this group. Only limited instances of five-axis machines in group (ii) exist for the machining the ship propellers. Groups (iii) and (iv) are used in the design of robots. Usually more degrees of freedom are added.”

Let us consider the five-axis machines in group (i). The kinematic chain is composed of three joints corresponding to the translation axes denoted by X, Y and Z and two rotary joints A, B or A, C or B, C, where A, B and C denote the rotations around vectors parallel to X, Y and Z respectively. Note that the linear axes are mutually orthogonal. However, the rotation axes are not necessarily orthogonal. Therefore, if the rotations are designed around vectors parallel to the linear axes (such as A, B or C), the configuration is called the orthogonal rotary axis, otherwise it is called the non-orthogonal rotary axis. The model of a five-axis machine is often thought of as two kinematic chains (the tool and the workpiece chain) linked in the world coordinate system). In the R-T notations the kinematic chain is given by $T^\prime T^\prime R^\prime R^\prime T$, where the prime denotes an axis carrying the workpiece. In the XYZABC notations it is given by $X^\prime Y^\prime A^\prime C^\prime Z$.

The classification introduced in the recent literature is based on the following criteria

• 1. Relationship between the two kinematic chains. The five-axis machines are classified into six groups by Bohez [199]: (i) 0/5 - all axes are on the workpiece chain while the tool is attached to a fixed frame; (ii) 5/0 - all axis on the tool chain and the workpiece is attached to a fixed clamping device; (iii) 1/4 - one axis on the tool and four axes are on the workpiece; (iv) 4/1 - four axes are on the tool and one axis is attached to the workpiece; (v) similarly, 2/3 and 3/2 are the machines with two axes attached to the tool and three axes attached to the workpiece and vice versa.

• 2. Relative location of the rotary axes in the two kinematic chains (i) 0/2 the two rotary axes are both attached to the table (table-tilting type); (ii) 2/0 both rotary axes are attached to the tool (spindle-tilting type), and (iii) 1/1 the rotary axes are distributed on both the tool and the table chain (table/spindle tilting type). This simple classification is included in Makhanov and Anotaipaiboon [7], Bohez [199], My and Bohez [201], Lee and She [202], Mahbubur et al. [203], She and Lee [204], Jung et al. [205], She and Chang [206], She and Huang [207], Yang and Altintas [208], Farouki et al. [209], and Tang et al. [210].

• 3. The tool axis. According to Moriwaki [211], five-axis machines are classified into vertical, horizontal, and the double column configuration machine types. In vertical machines, the tool moves vertically. The tool of horizontal machines is attached to a horizontal wall-frame. Finally, machining large industrial parts is often performed by a tool attached to a beam constructed from two columns. This machine is defined as the double column machine. In this case, the tool can move along the vertical axis as well as along the horizontal axis. Nakaminami et al. [212] note that the actual structure of the machine is determined by the required accuracy, rigidity, thermal deformation property, and the ease of manufacture The orthogonal machines have gained wide-spread acceptance in modern manufacturing. Their kinematic models have been thoroughly studied and experimented with by My and Bohez [201]. However, the non-orthogonal machines have certain advantages. In particular, a nutating head or a nutating table can change the position and orientation of the tool relative to the workpiece towards any admissible angle. In addition, the tool is able to traverse continuously between the horizontal and vertical positions (see Cui et al. [213]). The mathematical models of the non-orthogonal machines are presented and analyzed in She and Huang [207], Yang and Altintas [208], Chen et al. [214], Son et al. [215], and Wu et al. [216]. Finally, My and Bohez [201], Lee and She [202] and She and Huang [207] introduce classification based on

• 4. Relative location of the rotary axes and orthogonality, allowing one non-orthogonal axis based on the following:

  • Type I: (0/2) the rotary axes are on the workpiece carrying chain. Both axes are orthogonal (see Bohez [199], She and Chang [206], Yang and Altintas [208], and Xu et al. [217]).

  • Type II: (0/2N) the rotary axes are on the workpiece carrying chain. One rotary axis is orthogonal and the other one is non-orthogonal (see She and Huang [207], Sørby [218], and Liu et al. [219]).

  • Type III: (2/0) the rotary axes are orthogonal and are on the tool (see She and Chang [206], Yang and Altintas [208], Farouki et al. [209], and Tutunea-Fatan and Feng [220]).

  • Type IV: (2N/0) One rotary axis is non-orthogonal. Both rotary axes are on the tool (see She and Chang [206], She and Huang [207], and Tutunea-Fatan and Feng [220]).

  • Type V: (1/1) One rotary axis on the tool, the second rotary axis is on the workpiece. Both the axes are orthogonal (see Bohez [199], She and Chang [206], Yang and Altintas [208], and Farouki et al. [209]).

  • Type VI: (1N/1). The first rotary axis is on the tool; the second rotary axis is on the workpiece. The rotary axis on the tool is non-orthogonal (see She and Huang [207] and Wang et al. [221]).

  • Type VII: (1/1N): Observe that the My and Bohez [201] ignore the 7-th possible type (1/1N). In order to complete the classification, Type VII: (1/1N) is suggested: the first rotary axis is on the tool, whereas the second rotary axis is on the workpiece. The axes on the workpiece are non-orthogonal.

A comprehensive report on the recent and future CNC machine market is presented by Marketsandresearch.biz [222]. According to the report the most popular five-axis CNC machine manufacturers are: HAAS Automation, Hurco, Makino, Okuma, Shenyang Machine Tools, CMS North America, Fanuc, Jyoti CNC Automation, Yamazaki Mazak, Mitsubishi Electric and Siemens AG. Figures 18, 19, and 20 are examples of different types of five-axis machines developed by these manufacturers. Figure 21 shows a machine with a non-orthogonal axis on the tool.

Fig. 18

Mazak (INTEGREX-i-400S-2), classification 1) 2/3, 2) 1/1, 3) Vertical 4) Type V

Fig. 19

Fanuc (VMC850), classification 1) 1/4, 2) 0/2, 3) Vertical, 4) Type I

Fig. 20

Hendrick (HSLR-V), classification 1) 4/1, 2) 2/0, 3) Double Column 4) Type III

Fig. 21

Maho (DMU60P), classification 1) 3/2, 2) 1/1, 3) Horizontal 4) Type VI

Currently, commercial five-axis software such as Vericut, Fusion360, HSMWorks, PowerMill, SolidEdge, SolidWorks CAM, Solidcam, MasterCam, NX, Catia, Rhino CAM offer an excellent simulation of machining in the WCS. However, a simulation in the WCS does not produce a realistic outcome. In order to obtain an accurate result, the solid model of the actual machine must be built and incorporated into the CAM software. Moreover, the postprocessor to translate the toolpath into the MCS must comply with the virtual machine. Fortunately, the virtual controllers are available for the majority of commercially available machines. Moreover, if the controller is not available, it is often possible to modify the available controller. Building a system for machining experiments is not an easy task. Most start-up companies do not have this knowledge and rely on software from the vendor.

Appendix B. Vector fields in science and engineering

The goal of this section is to review the vector flow analysis for the professionals in five-axis manufacturing. The ideas and citations available in other fields such as image processing and pattern recognition are discussed.

1.1 B1 Classical vector fields

In this section the theory of VFA and selected applications of VF which are or could be potentially relevant to five-axis research are reviewed. VF are indispensable for many applications ranging from weather and ocean studies, physics, chemistry and automobile design, to the analysis of medical images and virtual reality. Unfortunately, many VFA techniques are not scalable to the increasing size and complexity of the real data. An obvious example is a five-axis G-code which can include tens or even hundreds of thousands of CC points. VFA is challenging from the viewpoint of human perception since no natural representation of the VF exists. Unlike, a surface, color or texture the VF are difficult to depict. Besides, specific mathematical techniques are required to develop algorithms to analyze and classify their topology. The readers interested in the general VFA are referred to surveys by Jiang et al. [223], Laramee et al. [224], Pobitzer et al. [225], and Edmunds et al. [226].

VF is a mathematical abstraction used in classical field theory to describe electromagnetism, gravitation, flows in fluid dynamics, aerodynamics, forces in mechanics and many other physical phenomena. Vector Calculus considers a VF as a continues field of n-dimensional vectors assigned to each point of a domain $R^n$. A VF in $R^2$ and $R^3$ are visualized by a collection of arrows with a certain size and direction attached to every point of a subset in space such as a surface in $R^3$. The generalizations of VF are tensor fields and tensor bundles analyzed by Biagi and Bonfiglioli [227].

It should be noted that the available terminology suffers from inconsistencies—some terms are synonymous, some are homonyms, and others are simply ambiguous and context-dependent. For instance, Vaxman et al. [228] suggest that if the magnitude is irrelevant and the vectors are normalized, then the VF should be called the direction field (DF) rather than the VF. Further, the VF can be multivalued, assigning a set of vectors rather than one vector at every point. Of a particular interest are rotational-symmetric DFs. Common variants are opposite ($\pi$-different) directions (2-DF), four $\pi /2$-different directions (4-DF), or two independent ($\pi$-different) pairs of directions ($2^2$-DF). This type of symmetry appears in many applications, e.g. when evaluating the principal curvature, the principal directions of stress, or strain tensors, conjugate directions or Langer’s lines (see Liu et al. [229], Diamanti et al. [230], and Bommes et al. [231]).

Note that the VFs applied to five-axis machining are usually 2-DF, i.e. the opposite or $\pi$-different directions. They are often called orientation field (OF) and depicted as a set of strokes rather than a set of paired vectors. However, in five-axis research such OFs (2-DFs) are usually called the VFs, implying that the opposite direction is also included (Dang and Makhanov [181]).

Another reason to consider one-vector VFs is that they are one of the most well-studied types. In particular, many important features such as parallel flows, shock waves (unsteady flows), convergent or divergent flows, vortex flows, irrotational flows and sources or sinks characterizing the behavior of the VF are considered as a representation of the physical flow. Vector Calculus operators such as grad, div, curl, def, $\Delta$ (Laplacian), the famous Green Theorem and many other relevant properties are used to analyze the VF and solve the corresponding partial differential equations (PDE). The well-established fundamental disciplines applying the VFA are Fluid Dynamics, Aerodynamics, Theory of Electromagnetism, Theory of Gravitation, Classical Mechanics and many others. In particular, VFA is used in oceanography Zhu and Moorhead [232], hydrodynamics Jones and Ma [233], aerodynamics Kenwright and Haimes [234], turbo machinery and engineering mechanics Roth and Peikert [235], temperature of heated surfaces Laramee et al. [236], Wiebel et al. [237], and Palke et al. [238], weather forecast Roy et al. [239], Pepper et al. [240], Pepper and Waters [241], and Cervantes et al. [242], CAD Liu et al. [229], mesh generation Knupp [243], surface deformation Huang et al. [244], etc. The gradient of a function creates a VF which shows the direction of the steepest descent. Therefore, the gradient-based methods are fundamental in numerical minimization and optimization. The divergence is a linear operator mapping VFs to functions. The sign of an integral of the divergence over a domain shows a possible source or sink. The point at which the flux through an enclosing surface is zero has zero divergence. The operator curl maps a 3D VF onto another 3D VF which measures the amount that the VF circulates. Analogous to sources and sinks, the curl of the VF is generated by vortexes. The operator divergence measures the flow in and out of the domain, while the curl measures the flow along the boundary. If the integral of the curl is zero, the VF is irrotational (conservative). A div-free VF is also known as Hamiltonian, solenoidal, or incompressible.

Koenderink and Van Doorn [245] write that def is a VF which describes a contraction in a certain direction and an expansion in an orthogonal direction. The vector Calculus makes it possible to analyze geometric structures of VFs. Although, a general VF is neither curl-free nor div-free, it is often possible to find subregions having the above properties (see Zhang et al. [246]).

Recently, VFA has been applied to various contemporary areas of computer science and engineering such as pattern recognition (face recognition, fingerprint detection, character recognition, etc.), image segmentation, computerized data visualization (the analysis of flows in industrial and natural systems), computer animations, texture mapping and synthesis, non-photorealistic rendering, anisotropic shading, shape analysis, traffic and crowd simulation and many others. It is often the case that the VF is defined on a rectangular grid or even at an irregular set of points. Besides, the VF is often distorted by noise (see Cervantes et al. [242]). Therefore, many applications are required to interpolate or approximate the discrete VF and evaluate its features using the continuous VFA Cervantes et al. [242], Azencot et al. [247], and Huang and Ju [248]. Some common methods to approximate the discrete VF are the least-square method by Kee et al. [249], neural networks by Kuroe et al. [250], radial basis functions by Cabrera et al. [251], Majdisova and Skala [252], and Smolik et al. [253], and the second-order Taylor series by Smolik and Skala [254].

Note that VF can be represented not only by the Cartesian coordinates, but also by the angle-based approach, complex numbers, tensors, composite 1-Forms, complex polynomials, linear operators, spherical harmonics, etc. The details of these representations and relevant citations are compiled by Vaxman et al. [94].

1.2 B2 Singular points and separatrices

If we consider a two dimensional VF given by $V=V(p) \equiv (V_x(x,y), V_y(x,y))$, the VFA is often performed in the neighbourhood of the critical (singular) points, where $V(p)=0$. These points can be extracted using the winding number $I_\Gamma$ defined as the number of revolutions of the VF along a closed curve $\Gamma$. The winding number in a region of 2D VF is evaluated by the Cauchy Theorem as follows:

$$\begin{aligned} I_\Gamma =\int \limits _{\Gamma }\frac{\mathrm {d}z}{z}. \end{aligned}$$

(72)

If $I_\Gamma \ne 0$ there is at least one critical point inside the domain bounded by the curve. The flows around the critical points are reviewed by Theisel et al. [255] as follows 1) Parabolic sector: either all streamlines end, or all streamlines originate at the critical point. 2) Hyperbolic sector: all stream lines pass by the critical point, except for two stream lines being the boundaries of the sector. One of these two stream lines ends at the critical point while the other one originates at it. 3) Elliptic sector: all stream lines originate and end at the critical point. Figure 22 illustrates the classification. Consider, the corresponding Jacobian given by

$$\begin{aligned} J=\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \equiv \begin{pmatrix} \frac{\partial V_x}{\partial x} &{} \frac{\partial V_x}{\partial y} \\ \frac{\partial V_y}{\partial x} &{} \frac{\partial V_y}{\partial y} \end{pmatrix}. \end{aligned}$$

(73)

Using the Taylor series around $p_0=(x_0,y_0)$ yields $V(p)=V(p_0)+J(p_0)(p-p_0)+O((p-p_0)^{2})$. Linearizing and assuming that $p_0=(x_0,y_0)$ is a critical point, i.e. $V(p_0)=0$, we have $V(p)=J(p_0)(p-p_0)$. The eigenvalues $\lambda _{i}$ of the Jacobian provide a classification of the first-order critical points given in Table 2 and Fig. 23, where $J(p_0) \ne 0$, $R_i=\text {Re}(\lambda _i),\, I_i=\text {Im}(\lambda _i),\,i=1,2.$

Fig. 22

From left to right: parabolic, hyperbolic, elliptic sector, an example of a critical point: p-parabolic sector, h-hyperbolic sector, e-elliptic sector, a stream line T separating two h-sectors

Fig. 23

Critical points of a 2D VF, the first row from left to right: attracting focus, center, repelling focus, repelling node; the second row: attracting node, saddle, attracting star, repelling star

Table 2 Types of 2D critical points

The analysis of the critical points is applied to analyze the phase portrait of systems of differential equations. Therefore, this classification is often called the phase portrait analysis (PPA). The PPA assumes that $V(p_0)=0$. Therefore, a linear VF is given by $V(p)=Jp$. If V(p) is discrete, J can be approximated by the linear least square method by minimizing $\sum _{i}(V(p_i)-J(a,b,c,d))^2$ with regard to a, b, c, d in the neighborhood of p. A nonlinear approximation of the VF could also be used. A constraint non-linear approximation Yau et al. [256] could be used to retain the first-order coefficients suitable for PPA. Examples of early applications of PPA are a classification of the flows around the airfoil by Helman and Hesselink [257] and detecting independent flow patterns by Kass and Witkin [258].

Shu and Jain [259] show the relationship between the Jacobian-based PPA and the classical operators grad, div, curl and def proving the equivalence of the two schemes. For example, if the VF topology is the star, then $(\textbf {def}\,V)^2=(\textbf {curl}\,V)^2=0$ and $\textbf {div}\,V>0$. If $\textbf {div}\,V>0$ the star is repelling, otherwise it is attractive.

The generalization of the Rankin model of vortex proposed by Acheson [260] is a method of singular points (see Corpetti et al. [261]).

The approximation of the VF using the Hessian matrix, i.e. by the second-order Taylor series, is computationally expensive, but it provides additional options for PPA proposed by Smolik and Skala [254]. In this case

$$\begin{aligned} V_L=J_L(p_0)(p-p_0)+\frac{1}{2}(p-p_0)H_L(p-p_0), \end{aligned}$$

(74)

where $L=x,y$,

$$\begin{aligned} H_L= \begin{pmatrix} \frac{\partial ^2 V_L}{\partial x^2} &{} \frac{\partial ^2 V_L}{\partial x\partial y} \\ \frac{\partial V_L}{\partial y \partial x} &{} \frac{\partial ^2 V_L}{\partial y^2} \end{pmatrix}, \end{aligned}$$

(75)

is the Hessian matrix and $J_x,J_y$ are the first and the second columns of the Jacobian. Writing the above system as equations of a conic section, yields

$$\begin{aligned} V_L=\frac{1}{2} \begin{pmatrix} x-x_0,&y-y_0,&1 \end{pmatrix} \begin{pmatrix} \frac{\partial ^2 V_L}{\partial x^2} &{} \frac{\partial ^2 V_L}{\partial x\partial y} &{}\frac{\partial V_L}{\partial x} \\ \frac{\partial V_L}{\partial y \partial x} &{} \frac{\partial ^2 V_L}{\partial y^2} &{} \frac{\partial V_L}{\partial y} \\ \frac{\partial V_L}{\partial x} &{} \frac{\partial V_L}{\partial y} &{} 0 \end{pmatrix} \begin{pmatrix} x-x_0\\ y-y_0\\ 1\\ . \end{pmatrix}, \end{aligned}$$

(76)

This makes it possible to classify the region as parabolic, elliptic, parallel or intersecting lines. Apparently, the combination of the first and the second-order classification has potential benefits. Usually, the VF is analyzed using a moving window. Therefore, the second-order model is able to detect whether the window contains single or multiple critical points. Scheuermann et al. [262] observe that “the piecewise linear or bilinear interpolation destroys the topology of the VF if there are higher order critical points present.” Therefore, they suggest detecting the presence of high order singularities and using a local polynomial approximation based on Clifford Algebra. The PPA of the OF has been effectively used for fingerprint analysis by Yau et al. [256] and Wang et al. [263], architectural distortion in mammograms by Rangayyan and Ayres [264], feature extraction and simplification of the flow fields by Post et al. [265], etc. During the last decades, readily distinguishable from other viewpoints was the idea to use the gradient VF in image processing. The analysis of $|\textbf {grad}I|$, where I is the gray level of the image, gave a rise to many prominent edge detection methods designed to generate the edge maps and segment the objects of interest. We refer the interested readers to numerous textbooks on image processing (see, for instance, El-Sayed [266]). The gradient VF is used in combination with active contours by Kass et al. [267] and Xu and Prince [268] and level set methods by Osher and Sethian [269] designed for image segmentation. Interested readers are referred to the vast and well developed segmentation algorithms based on deformable shapes. The above methods require smooth, simplified VF which drive the deformable shapes to their destinations. Therefore, a variety of pre-processing procedures have been developed. In particular, the Gradient Vector Flow by Xu and Prince [268] based on the solution of system of elliptic PDE became standard to simplify the original VF.

The pre-processed VF is a solution of the system given by

$$\begin{aligned} \lambda \nabla ^2V-(V^\prime -V)|V|^2=0, \end{aligned}$$

(77)

where $\lambda$ controls the diffusion and $V^\prime$ is the original VF. The system is solved by iterations with regard to the pseudo-time t, i.e.

$$\begin{aligned} V^{n+1}=V^n+\tau (\lambda \nabla ^2V-(V^\prime -V)|V|^2)^n, \end{aligned}$$

(78)

where n is the iteration number and $\tau$ is the pseudo-time step. The initial condition is $V=V^\prime$. The boundary conditions vary depending on the application. The importance of this model has been recognized in the image processing community. However, it is not well known in the material science and manufacturing. Note that one of the most popular modifications of this idea proposed by Xu and Prince [182] is Generalized Gradient Vector Flow.

The variety of features on the different scales of the VF makes their classification mathematically and computationally hard problem. Along with the critical points, the VF can be characterized by separatrices defined as the curves separating regions (basins) with the different behavior of the streamlines (see Scheuermann et al. [262]). Theisel et al. [255] classify separatices as 1) curves originating or/and ending at the critical point, 2) curves from inbound boundary switch points, and 3) isolated closed curves (Fig. 24).

Fig. 24

left: inbound boundary switch point, second left: outboundary switch point, third left: separatrix from inbound boundary divides the domain into 3 regions, second right: isolated closed stream line divides the domain into regions, right: singular points and closed separatices in a complex stream

The separatrices from the boundary switch points have been analyzed by de Leeuw and van Liere [270]. Wischgoll and Scheuermann [271] and Theisel et al. [272] offer algorithms designed to detect closed separatrices. Skraba et al. [273, 274] propose a simplification of the VF based on the topological robustness, representing the stability of the critical points and their significance. Recent theoretical results on the existence and structure of separatrices have been compiled in Branner and Dias [275], Mol and Sánchez [276], and Novello et al. [277].

There exists a variety of the directional fields (DF). The classic VF specifies a magnitude and a direction, while others consider a single or multiple directions per point, often with symmetry. Such DF appear in the literature under several names, such as VF, DF, line fields, cross fields, poly-vector fields or tensor fields. Notable examples are the principal directions of a surface, stress/strain tensors or multiple cutting directions associated with the toolpath of a five-axis machine. Of a particular interest are the rotational-symmetric DF. Common variants are two directions with $\pi$-symmetry, four directions with $\pi /2$- symmetry, or two independent pairs of directions with $\pi$-symmetry within each pair (Vaxman et al. [94]). We recall that the index of singularity of the VF is the winding number of a small curve around the singularity. The Poincaré–Hopf theorem states that the sum of all the indices of a VF on a surface without boundary is $2-2g$ where g is the genus of the surface. The concept of indices of singularities can be generalized to other types of DF. However, the index is not an integer anymore. For instance, the N-VF generates an index which is a multiple of 1/N (Ray et al. [278]).

Tensors of rank 2 are being used in various contexts relevant to differential geometry, engineering mechanics, dynamics, etc. The examples are curvature, metric, strain and stress. A tensor on a 2-manifold is a real-valued 2x2 matrix given in the local coordinates by

$$\begin{aligned} T= \begin{pmatrix} T_{11}&{} T_{12}\\ T_{21}&{} T_{22} \end{pmatrix}. \end{aligned}$$

(79)

Symmetric tensors ($T_{12} = T_{21}$) are useful due to their straightforward relation to directional information. A symmetric matrix has either two distinctive real eigenvalues or a single eigenvalue corresponding to a double root of the characteristic polynomial. Since the eigenvectors are defined up to the sign, a rank-2 tensor field defines two orthogonal 2-DF. This representation is unique only when the eigenvalues $\lambda _1 \ne \lambda _2$. Further, directional information is extracted from the so called traceless deviatoric part given by $T-\textbf{Tr}(D)/2$. A special case of symmetric tensors which is particularly interesting for five-axis research is the structure tensors applied to represent 2-DF in an arbitrary dimension. The key idea is to represent a line L in $R^n$ as the eigenspace of the largest eigenvector $v_L$ of an $n\times n$ matrix, where $v_l$ parallel to L. The topology-based visualization of symmetric, second-order, planar tensor fields has been analyzed by Delmarcelle and Hesselink [157].

The method selects one DF corresponding to the minor or the major eigenvalue. This allows evaluation of the tensor lines considered as an extension of the classical stream lines. The foundations of this technique have been laid down by Helman and Hesselink [279]. The theoretical background is provided by the theory of dynamical systems and differential geometry The tensor approach results in a graph representation, where the edges are separatrices and the nodes are singularities of the tensor field. The tensor field is composed of curves everywhere tangent to the eigenvectors. Hence, these curves have no inherent orientation as opposed to stream lines. At a degenerate point, the deviator is the zero matrix. If the tensor field is linear these singularities are represented by trisectors and wedge points . Note that such structures would be impossible to detect in the oriented VF case. Delmarcelle and Hesselink [157] show how the standard VF configurations follow from the tensor field (Fig. 25).

Fig. 25

left column: trisector and wedge(parabolic or hyperbolic), middle and right column: merging singular points into saddle, attracting/repelling node and focus

The technique proved to be suitable for tensor fields with a simple structure and has been used for TPG for five-axis machines (Fig. 5). Nevertheless, the VF/DF of the optimal (or possibly optimal) directions on a complex shaped STL surface create cluttered depictions that are of little help for interpretation. The topology of such fields includes a large number of features on different scales. For instance, a single directional flow on a small scale can be a part of a complex configuration such as a flow around a saddle or a trisector flow. The configurations depend on the method of classification and may result in a totally different decomposition. These shortcomings of VF simplification must be carefully considered when they are used for TPG for multi-axis machining. The algorithm should be able to prune insignificant features, driven by qualitative and quantitative criteria specific for the required surface, tool, speed and prescribed accuracy. The features can be understood in different ways and the corresponding VFs can be analyzed, simplified and decomposed using a variety of numerical algorithms.