Conditions of Proper Part Surface Generation 329 can be generated. The form transition surface is machined instead. This is due to violation of the fth necessary condition of proper PSG. 7.2.6 The Sixth Condition of Proper Part Surface Generation When machining a sculptured surface, point contact of the surfaces P and T is usually observed. Due to point contact of the surfaces, the discrete genera- tion of the sculptured surface often occurs. Representation of the generating surface T by distinct cutting edges of the form-cutting tool is the other rea- son the discrete generation of the sculptured surface takes place. In an instant of time, it is physically impossible to generate the sculptured surface P by a single moving point. When the discrete surface generation occurs, the nominal smooth, regular sculptured surface P n( ) and the actual machined surface P a( ) are not identical. The actual part surface P a( ) can be interpreted as the nominal sculptured surface P n( ) that is covered by cusps (Figure 7.26) or may have other deviations from P n( ) . The sixth necessary condition of proper PSG is formulated as follows: The Sixth Condition of Proper PSG: The actual part surface P with cusps, if any, must remain within the tolerance on surface accuracy. Cusps on the machined sculptured surface P must be within the tolerance on the surface accuracy. Then maximal height h Σ of the cusps must not exceed the tolerance [ ]h on the sculptured surface accuracy. Consider a Cartesian coordinate system X Y Z P P P associated with the sculp- tured surface P. The sixth necessary condition of proper PSG is satised if and only if the following condition is satised at every point of the nominal sculptured surface P: n r r r n P n P a P n P t P n h h ( ) ( ) ( ) ( ) ( ) [ ]⋅ = − ≤ = ⋅ Σ (7.40) Z P Y P X P (n) P r (t) P r (a) P r h Σ (n) n P P (t) [h] h Σ K P (a) P (n) FIGURE 7.26 Cusps on the machined sculptured part surface P. (From Radzevich, S.P., Computer-Aided Design, 34 (10), 727–740, 2002. With permission.) © 2008 by Taylor & Francis Group, LLC 330 Kinematic Geometry of Surface Machining where the position vector of a point of a nominal sculptured surface P n ( ) is designated as r P n( ) ; the position vector of the corresponding point * of the actual part surface P a ( ) is designated as r P a( ) , the position vector of a point of the surface of tolerance P t( ) is designated as r P t( ) , and the unit normal vector to surface P n( ) is designated as n P n( ) . If the sixth necessary condition of proper PSG is satised, then the actual part surface P a( ) is entirely located within the nominal sculptured part surface P n( ) and the surface of tolerance P t( ) . In the example (Figure 7.26), the surface of tolerance P t( ) is depicted over the surface P n( ) at the distance n P n h ( ) [ ]⋅ . Fulllment of the set of six conditions of proper part surface generation is necessary and sufcient to insure machining of the part surface in compli- ance with the requirements indicated in the part blueprint. 7.3 Global Verification of Satisfaction of the Conditions of Proper Part Surface Generation When machining a sculptured surface on a multi-axis NC machine, it is important to get to know whether the entire part surface can or cannot be machined on the given machine. It is also important to detect the sculptured surface regions, those that are not accessible by the cutting tool of a given design. In other words, it is necessary to detect regions on the sculptured surface P which the cutting tool cannot reach without being obstructed by another portion of the part. Certainly, such regions (if any) are due not just to the geometry of the sculptured surface P, but also to the geometry of the generating surface T of the cutting tool. The particular problem under con- sideration is now referred to as the cutting-tool-dependent partitioning of a sculptured surface onto the cutting-tool-accessible and onto the cutting-tool- not-accessible regions. 7.3.1 Implementation of the Focal Surfaces For solving the problem of cutting-tool-dependent partitioning (CT-dependent partitioning) of a sculptured surface, the third necessary condition of proper PSG is the most critical issue. The geometry of contact of the surfaces P and T in the innitesimal vicinity of a cutter-contact-point (CC-point) K is a vital link for verication of whether the third necessary condition of proper PSG is globally satised or not. Within the cutting-tool-accessible portions of the sculptured surface, the proper correspondence is observed between the normal curvature k P of the * Two points on the surfaces r P n( ) and r P a( ) are corresponding to each other if they share a com- mon straight line, which aligned with the perpendicular n P n( ) to the surface r P n( ) . © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 331 surface P, and the normal curvature k T of the generating surface T of the cutting tool (Table 7.1). The normal curvatures k P and k T are measured in the same direction specied by the unit tangent vector t P . Otherwise, when the correspondence between the normal curvatures k P and k T is improper (Table 7.1), interference of the surfaces P and T occurs. Such regions of the surface P cannot be machined properly. Implementation of the indicatrix of conformity Cnf P T R ( / ) (see Equa- tion 4.59) enables detection of local, not global, interference of the surfaces P and T. If negative diameters d cnf of the indicatrix of conformity Cnf P T R ( / ) are observed, this immediately indicates that a certain portion of the sur- face P is not machinable with the cutting tool of a given design. It is easy to conclude that within the bordering curve between the cutting-tool-accessible and the cutting-tool-not-accessible portions of the surface P, the identity d cnf  0 is observed. * Ultimately, the problem of partitioning of a sculptured sur- face reduces to the problem of nding those lines on the part surface P within which the identity d cnf  0 is valid. For solving the problem, various approaches can be used. The implementation of focal surfaces is promising in this concern. 7.3.1.1 Focal Surfaces The geometry of contact of the surfaces P and T in the innitesimal vicinity of a CC-point K, turns our attention to the normal curvatures of the surfaces P and T, and to the location of centers of normal curvature of these surfaces. The direction of feasible tool approach to a surface point is dened as the direction along which a cutting tool can reach a part surface without being obstructed by another portion of the part. For a part design to be machinable, every feature of the part design should have at least one such feasible direc- tion. For a sculptured surface, if a point on the surface does not have at least one such feasible direction, it is not machinable. Global analysis and detection of the surface P regions, those that are cutting- tool-accessible, as well as those that are cutting-tool-not-accessible, and a visual interpretation of the global accessibility of the surface can be per- formed by means of focal surfaces for the surfaces P and T. For generating the focal surfaces, it is necessary to recall that there are two principal plane sections C P1. and C P2. through a point M of smooth, regular sculptured surface P. Principle surfaces C P1. and C P2. are passing through the surface P unit normal vector n P , and through the directions specied by the principal unit tangent vectors t 1.P and t 2.P . Principal radii of curvature R P1. and R P2. of the surface P are measured in the principal plane sections C P1. and C P2. . Centers of curvature O P1. and O P2. of the sculptured sur- face at point M (Figure 7.27) are located within the straight line through the unit normal vector n P erected at the point M. Points of this kind are usually referred to as the focal points of a surface P at M. * The same is true with respect to the Ar R P T( / ) -indicatrix (see Chapter 4). © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 333 terms of unit normal vector n P to the surface P; and in terms of corresponding radii of principal curvature, either R P1. or R P2. (Figure 7.27): f r n 1 1. . ( , ) ( , ) P P P P P P P P U V U V R= − ⋅ (7.41) f r n 2 2. . ( , ) ( , ) P P P P P P P P U V U V R= − ⋅ (7.42) Elementary substitution R k P P1 1 1 . . = − and R k P P2 2 1 . . = − yields expression of the focal surfaces f 1.P , f 2.P (Equation 7.41 and Equation 7.42) in terms of principal curvatures: f r n 2 2 1 . . ( , ) ( , ) P P P P P P P P U V U V k= − ⋅ − (7.43) f r n 1 1 1 . . ( , ) ( , ) P P P P P P P P U V U V k= − ⋅ − (7.44) Radii of principle curvatures R .P1 and R .P2 in Equation (7.41) and Equa- tion (7.42) are computed using one of the equations represented in Chapter 1 (Equation 1.14 and Equation 1.19). Radii of principle curvatures R .P1 and R .P2 can be expressed in terms of the mean curvature % M P and of the Gaussian curvature % G P of the surface P: R M M G P P P P1 2 1 . = + − ( ) − % % % (7.45) R M M G P P P P2 2 1 . = − − ( ) − % % % (7.46) n P a r P r 1 M 1 S 1 S P M FIGURE 7.28 On representation of a focal surface as an enveloping surface to perpendiculars to the surface P. (From Radzevich, S.P., Computer-Aided Design, 37 (7), 767–778, 2005. With permission.) © 2008 by Taylor & Francis Group, LLC 334 Kinematic Geometry of Surface Machining Ultimately, equations for the focal surfaces f 1. ( , ) P P P U V and f 2. ( , ) P P P U V can be represented in the form f r n 1 2 . ( , ) P P P P P P P P U V M M G = − + − % % % (7.47) f r n 2 2 . ( , ) P P P P P P P P U V M M G = − − − % % % (7.48) The focal surfaces f 1.P and f 2.P for a saddle-like patch of a sculptured surface P are plotted in Figure 7.29a. Such a patch of the surface P can be machined, for example, with the convex generating surface T of a cutting tool. Focal surfaces f 1. ( , ) T T T U V , and f 2. ( , ) T T T U V for this surface T are depicted in Figure 7.29b. In Figure 7.29, the respective lines of curvature are designated as ( C P 1. , ( C P 2. and ( C T1. , ( C T2. , correspondingly. Points O T1. and O T2. are the points of the cor- responding focal surfaces f 1.T and f 2.T for the surface T at point M: f r n 1 1 1 . . ( , ) ( , ) T T T T T T T T U V U V k= − ⋅ − (7.49) f r n 2 2 1 . . ( , ) ( , ) T T T T T T T T U V U V k= − ⋅ − (7.50) n T T ( )b Y P Z P X P М f 2 P С 1 P    С 2. P С 1. T  С 2. T t 1. T t 2. T O 1. T f 2. T f 1. T O 2. T f 1 P P ( )a М Y P Z P X P O 2 P t 2. P t 1. P O 1 P n P FIGURE 7.29 Examples of focal surfaces constructed for a local patch of the sculptured surface P (a), and for a local patch of the generating surface T of a cutting tool (b). (From Radzevich, S.P., Computer- Aided Design, 37 (7), 767–778, 2005. With permission.) © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 335 Focal surfaces f 1.P and f 2.P intersect the sculptured surface P along parabolic curved lines on it — that is, along lines at which Gaussian curvature % G P of the sculptured surface is equal to zero ( % G P  0 ). In order to use focal surfaces for the verication of whether or not the third necessary condition of proper PSG is satised globally, it is necessary to plot both of the focal surfaces f 1.P and f 2.P for the sculptured surface P and the similar focal surfaces f 1.T and f 2.T for the generating surface T of the cutting tool in a common coordinate system. An example of the relative conguration of the focal surfaces f 1.P , f 2.P and f 1.T , f 2.T at the point K of contact of the given surfaces P and T is illustrated in Figure 7.30. The saddle-like ( ) G P < 0 local patch of a sculptured surface P is machined with a convex patch ( G T > 0 , M T > 0 ) of the generating surface T of the cutting tool. In the case under consideration, angle µ of the local P Y P Z P X P n T T f 2.T f 1.T f 2.P O 2.T K n P O 1.T Č 2.T Č 1.P f 1.P Č 2.P Č 1.T O 2.P O 1.P FIGURE 7.30 Conguration of the focal surfaces f 1.P , f 2.P for the sculptured surface P relative to the focal surfaces f 1.T and f 2.T for the generating surface T of the cutting tool. (From Radzevich, S.P., Com- puter-Aided Design, 37 (7), 767–778, 2005. With permission.) © 2008 by Taylor & Francis Group, LLC 336 Kinematic Geometry of Surface Machining relative orientation of surfaces P and T is equal to zero ( µ = 0 ). Inspect- ing Figure 7.30, it is easy to realize that the rst principle planes C C P T1 1. .  (the identity is due to µ = 0 ) intersect the surfaces P and T. The lines of the intersection ( C P1. and ( C T1. are convex lines ( k P1 0 . > ; k T1 0 . > ). Therefore, no problem is observed to satisfy the third necessary condition of proper PSG in this plane section. The second principle planes C C P T2 2. .  (the identity is due to µ = 0 , these plane sections are also congruent to each other) intersect the surfaces P and T. The line ( C P2. of the intersection is a concave curve. The line ( C T2. of the intersection is the convex curve. Because the distance KO T2. exceeds the distance KO P2. (i.e., KO KO T P2 2. . > ), the principle curvatures k P2. and k T2. correspond to each other as | | . . k k P T2 2 > . Because the inequal- ity | | . . k k P T2 2 > is valid, the third necessary condition of proper PSG in the second principal section of the surfaces P and T is not satised. Summariz- ing, one can conclude that the third necessary condition of proper PSG is not satised in the innitesimal vicinity of the CC-point K (Figure 7.30). Analysis of Table 7.1 allows for analytical expression of the criterion for verication of whether the third necessary condition of proper PSG is satis- ed or not: sgn sgn sgn( )k k k k P T P T ⋅ ⋅ + = −      0 1 (7.51) In order to globally satisfy the third necessary condition of proper PSG, it is necessary to ensure satisfaction of Equation (7.51) at every point K, and in every cross-section of the surfaces P and T by a plane through the unit normal vector n P . The third condition of proper PSG could be satised globally when each of the focal surfaces f 1.T and f 2.T is entirely located between the convex surface P and the corresponding focal surface f 1.P or f 2.P . Focal surfaces f 1.T and f 2.T can touch one or both focal surfaces f 1.P or f 2.P . In a similar way, location of the focal surfaces f 1.T and f 2.T , for concave and for saddle-like local patches of surface P can be specied. Focal surfaces f 1.T and f 2.T must not intersect the sculptured surface P and the corresponding focal surfaces f 1.P and f 2.P for the generating surface of the form-cutting tool. Otherwise, the third nec- essary condition of proper PSG would be violated. Focal surfaces f 1.P and f 2.P are the bounding surfaces of space, within which the centers of principal curvatures of the generating surface T of the cutting tool have been located. The portions of space bounded by the focal surfaces f 1.P and f 2.P are referred to as the cutting-tool-allowed (CT-allowed) zones. The rest of the space is referred to as the cutting-tool-prohibited (CT-prohibited) zones. 7.3.1.2 Cutting Tool (CT)-Dependent Characteristic Surfaces When the third necessary condition of proper PSG is globally satised, then certain constraints are imposed on the actual conguration of the focal sur- faces. For the purpose of verication of accessibility of the surface P by the © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 337 cutting tool, the CT-dependent characteristic surfaces can be used. It is convenient to illustrate the concept of the CT-dependent characteristic surfaces with an example of generation of a concave patch of the sculptured surface P. First, the current point of the rst focal surface f 1.T of the generating sur- face T of the cutting tool is located within the straight line along the unit normal vector n P that is erected at the corresponding point of the surface P. Second, within the straight line there exists a straight-line segment. Location of the current point of the focal surface f 1.T is allowed within the straight- line segment, as well as at its endpoints. Therefore, without loss of generality, instead of two focal surfaces f 1.P and f 1.T , just one CT-dependent character- istic surface ) f 1 ( , )U V T T can be employed. This surface features the summa ( ) . . R R P T1 1 + of the rst principal radii of curvature. The locus of points, determined in the above way, forms the rst CT-dependent characteristic surface ) f 1 ( , )U V T T of the sculptured surface P and of the gen- erating surface T of the cutting tool. The position vector of a point of the rst CT-dependent characteristic surface ) f 1 can be expressed in terms of the parameters r P , n P , R P1. , and R T1. : ) f r n 1 1 1 ( , ) ( , ) ( ) . . U V U V R R T T P T T P T P = − + ⋅ (7.52) A similar analysis can be performed for the second focal surface f 2.T of the generating surface T of the cutting tool. Ultimately, the position vector of a point of the second CT-dependent char- acteristic surfaces ) f 2 can be expressed in terms of the parameters r P , n P , R P 2. , and R T 2. : ) f r n 2 2 2 ( , ) ( , ) ( ) . . U V U V R R T T P T T P T P = − + ⋅ (7.53) Summarizing, one can conclude that the CT-dependent characteristic sur- face is a surface, each point of which is remote from the sculptured surface P perpendicular to it at a distance that is equal to the algebraic sum of the corresponding radii of principal curvature of the surfaces P and T. When the CT-dependent characteristic surfaces ) f 1 and ) f 2 do not intersect the sculptured surface P, then the third necessary condition of proper PSG is satised globally. Under such a scenario, the sculptured surfaces P can be machined properly in compliance with the surface blueprint. Otherwise, if the CT-dependent characteristic surfaces ) f 1 and ) f 2 intersect the surface P, or they are entirely located within the interior part of the body, the third neces- sary condition of proper PSG cannot be satised. In this case, the surface P cannot be machined properly. Application of the CT-dependent characteristic surfaces for the purposes of resolving the problem of partitioning the sculptured surface onto the cutting-tool-accessible and cutting-tool-not-accessible regions reduces the number of surfaces to be considered from four focal surfaces ( f 1.P , f 2.P and f 1.T , f 2.T ) to two CT-dependent characteristic surfaces ( ) f 1 and ) f 2 ). © 2008 by Taylor & Francis Group, LLC 338 Kinematic Geometry of Surface Machining The cutting-tool-accessible regions are separated from the cutting-tool- not-accessible regions of the sculptured surface P by a corresponding boundary curve. 7.3.1.3 Boundary Curves of the CT-Dependent Characteristic Surfaces The boundary curve for cutting-tool-accessible region of the sculptured sur- face P is the line of intersection of the part surface by the corresponding CT-dependent characteristic surfaces ) f 1 and ) f 2 . Therefore, every point of the boundary curve r bc satises the corresponding set of two equations: ) f f r n r r 1 1 1 1 = = − + ⋅ = ˆ ( , ) ( ) ( , . . U V R R U V P P P P T P P P P P ))      (7.54) ) f f r n r r 2 2 2 2 = = − + ⋅ = ˆ ( , ) ( ) ( , . . U V R R U V P P P P T P P P P P )).      (7.55) Equations for the two-surface intersection curve can be derived from the condition r n r P P T P P P P R R U V− + ⋅ =( ) ( , ) . .1 1 (7.56) r n r P P T P P P P R R U V− + ⋅ =( ) ( , ) . .2 2 (7.57) The approach for determining the boundary curves which is based on the solutions to Equation (7.56) and Equation (7.57) can be signicantly simplied taking into consideration Equation (7.51). After inserting the previously derived Equation (7.51) and rearranging Equation (7.56) and Equation (7.57) cast into r n r P P T P T P P P k k k k U− ⋅ ⋅ ⋅ + ⋅ =sgn sgn sgn( ) ( . . . .1 1 1 1 ,, )V P (7.58) r n r P P T P T P P P k k k k U− ⋅ ⋅ ⋅ + ⋅ =sgn sgn sgn( ) ( . . . .2 2 2 2 ,, )V P (7.59) Equation (7.58) and Equation (7.59) represent an analytical description of the boundary curves that separate the cutting-tool-accessible regions of the sculptured surface P from the cutting-tool-not-accessible regions on it. Derivation of the boundary curves of the CT-dependent characteristic sur- faces is illustrated below with two examples. Consider generation of the torus surface P. A computer model of a torus surface is widely used as a convenient test case. It is proven [16,17,20] that the torus surface provides signicantly higher accuracy of approximation and thus is preferred for local approximation of the surfaces P and T over quadrics. This is because the principal radii of curvature R P1. and R P2. of the © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 339 surface P (and the similar principal radii of curvature R T1. and R T2. of the surface T) uniquely specify the torus surface. The rst principal radius of curvature R P1. is equal to the radius of the generating circle of the torus surface, and the second principal radius of cur- vature R P2. is equal to the radius of the outside circle of the torus surface (and therefore, the radius R of the directing circle is equal to the difference R R R P P = − 2 1. . ). A similar condition is valid with respect to the generating surface T of the cutting tool. For both examples below, Equation (7.16) of the torus surface P from Exam- ple 7.1 is implemented. Example 7.2 Consider machining of a torus surface P with the at-end milling cutter (Figure 7.31). The radius r of the generating circle of the surface P is equal to r mm= 50 , and the radius R of the directing circle of the surface P is equal to R mm= 90 . Gaussian (curvilinear) coordinates θ P and ϕ P of a point on the surface P vary in the range of 0 180 ≤ ≤ θ P and 0 360 ≤ ≤ ϕ P . Using Equa- tion (7.58) and Equation (7.59) in the commercial software MathCAD allows the equation r bc P P P ( ) sin cos θ θ θ = ⋅ ⋅ ±               90 90 50 1 (7.60) T 1 e Boundary Curves ω T R r X P P Y P ω T ω T ω T K 2 K 1 Z P T 2 T 3 T 4 K 3 K 4 FIGURE 7.31 Partitioning of the torus surface P that is machining with a at-end milling cutter. © 2008 by Taylor & Francis Group, LLC [...]... Group, LLC 364 Kinematic Geometry of Surface Machining principal curvatures of the surface P The global KGR-map of the surface P is located inside the allowed rectangle and is sharing at least one point with each of its sides The obtained KGR-map of the surface P helps to visualize major restrictions that are imposed on the parameters of the generating surface of the cutting tool by the surface P topology... the satisfaction of the third necessary condition of proper PSG corresponds to the case of the maximal rate of conformity of the generating surface of the tool to the sculptured surface — when normal radii of curvature of these surfaces in a current section by normal plane are of the same value and of the opposed sign 7.3.3.4 Selection of an Optimal Cutting Tool for Sculptured Surface Machining Further,... Surface Machining 15 10 5 ZP 0 00 0 2− 1 2 2 Surface 3 3 Figure 7.40 An example of the characteristic ℜ2 -surface of the second kind matrix representation. A visualized image of the R 2 -surface is depicted in Figure 7.40 The line of intersection of the R 2 -surface exactly corresponds to the boundary ℜ 0 - curve on the surface P (Figure 7.38) So, application of only one R 2 -surface (and not two surfaces... characteristic surfaces of the discovered kind 7.3.3 Selection of the Form-Cutting Tool of Optimal Design For the verification of satisfaction of the third necessary condition of proper PSG, global use of K-mapping is useful [17,21] Two kinds of K-mapping are of importance in this concern First, the local KLR-mapping of the surfaces P and T that gives insight into the global KGR-mapping of the surfaces P... 346 Kinematic Geometry of Surface Machining If the minimal diameter d min of the indicatrix of conformity CnfR ( P/T ) is cnf equal to zero d min = 0 (Figure 7.34c), then in the direction of t max along d min, cnf cnf cnf radii of normal curvature of the surfaces P and T are of the same magnitude and of opposite sign ( R P = − R T ) A point at which the equality d min = 0 is cnf observed is a point of. .. characteristic surfaces of the first kind, it is convenient to express all the elements of local topology of the R 1 -surface in terms of the corresponding elements of local topology of the surface P and min of the minimal radius rcnf of the indicatrix of conformity CnfR ( P/T ) at the current CC-point Consider a sculptured surface P that is given by vector equation rP = rP (U P , VP ) Components gR ij of the... unit normal vector n R to the R 1 -surface can be analytically represented as nR =  A n | A| P (7.77) This derivation immediately follows from the definition of principal curvatures of a surface [17] © 2008 by Taylor & Francis Group, LLC 350 Kinematic Geometry of Surface Machining The second fundamental tensor of the R 1 -surface can be expressed in terms of the components of the fundamental tensors g ij... marking these curves on the surface P to identify the surface regions that require additional care Two approaches for the derivation of equation of the boundary ℜ 0-curve can be employed First, the equation of the boundary curve  ℜ 0  can be derived from consideration of the line of intersection of the characteristic R 1 -surface and of the surface P itself Coordinates of points of the boundary © 2008 by... characteristic surfaces of the first kind, the characteristic surfaces of the second kind are considered 7.3.2.3.1 Determination of the Characteristic Surface of the Second Kind In order to globally satisfy the third necessary condition of proper PSG, the characteristic R 1 -surface of the first kind has to be located outside the part body, and it does not intersect the sculptured surface P Only tangency of the surfaces...   dC2 P The surfaces above could be used in the way that the focal surfaces f1)P , f2.P , ) f1.T , and f2.T (and the CT-dependent characteristic surfaces f1 and f2 ) are used for the cases of regular tangency of the surfaces P and T In cases of local-extremal tangency of the surfaces P and T, implementation of the DF-surfaces, and inplementation of the DCT-dependent characteristic surfaces is helpful . used for the cases of regular tangency of the surfaces P and T. In cases of local-extremal tangency of the surfaces P and T, implementation of the DF-surfaces, and inplementation of the DCT-dependent. & Francis Group, LLC 350 Kinematic Geometry of Surface Machining The second fundamental tensor of the R 1 -surface can be expressed in terms of the components of the fundamental tensors g ij . another portion of the part. Certainly, such regions (if any) are due not just to the geometry of the sculptured surface P, but also to the geometry of the generating surface T of the cutting