The Geometry of the Active Part of a Cutting Tool 253 active part of a cutting tool in many cases is measured in reference planes con- guration of which depends upon tangent planes to the surfaces R s and C s . The effect of the cutting edge torsion onto the material removal process in metal cutting has not yet been profoundly investigated. A cutting edge can be considered as a line of intersection of three surfaces: the generating surface T of the cutting tool, the rake surface R s , and the clearance surface C s . The equation of the cutting edge can be derived as a result of mutual consideration of the equation of one of three pairs of surfaces: (a) r rs rs rs U V( , ) and r cs cs cs U V( , ) , (b) r T T T U V( , ) and r rs rs rs U V( , ) , or (c) r T T T U V( , ) and r cs cs cs U V( , ) . The solution to any of three pairs of equations can be reduced to the equation of the cutting edge, which yields representation in matrix form: r r ce ce ce T ce T ce T t X t Y t Z t ce = =         ( ) ( ) ( ) ( ) 1       (6.63) where t ce denotes the parameter of the cutting edge. In a particular case, length S ce of the cutting edge can be chosen as the parameter of the cutting edge (that is, t S ce ce ≡ ). Under such a scenario, tor- sion τ ce of the cutting edge can be computed from τ ρ ce ce ce ce ce ce ce ce d dS d dS d dS = ⋅ ⋅      2 2 2 3 3 r r r          -1 (6.64) where the sign of the torsion τ ce is not in compliance with the direction of the angle of inclination λ s . 6.2.6 Diagrams of Variation of the Geometry of the Active Part of a Cutting Tool Analytical methods for the computation of actual values of the geometry of the active part of a cutting tool are accurate. They are capable of computing the dis- tribution of a geometrical parameter of a cutting tool both at a given point of the cutting edge in different reference cross-sections or within the active part of the cutting edge in similar cross-sections. Results of such computations are accurate and are of critical importance to a tool designer. For the preliminary analysis of the geometry of the active part of a cutting tool, the implementation of diagrams of variations of the geometrical parameters have proven useful. Distribution of the function tan γ of the rake angle in different refer- ence planes through the point M within the cutting edge of a form-cutting tool is shown in Figure 6.14. Once the rake angle in two different reference planes is determined, then the distribution of the function tan γ follows the circle. The circle constructed on any two known vectors through the point M enables easy determination of the function tan γ i in any direction through © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 255 For analysis of the distribution of the function cot g, Figure 6.16 is helpful [4]. Two known values of the function cot g in known corresponding direc- tions yield construction of the straight line AB. Ultimately, the actual value of the function cot g i in the reference plane through a current direction is specied by the corresponding point within the straight line AB. All the diagrams are in perfect correlation with the results of the analyti- cal computations. 6.3 Geometry of the Active Part of Cutting Tools in the Tool-in-Use System When machining a part surface, the actual direction of the primary motion, as well as of the feed-rate motion, can differ from the assumed directions of these motions, say in the tool-in-hand system. Moreover, the actual kinemat- ics of a machining operation can be made up not only of the primary and the feed-rate motions, but also of motions of another nature (for example, vibra- tions, orientation motions of the cutting tool [see Chapter 2], etc.). In order to cot γ o cot γ C cot γ o cot γ E cot γ F cot γ D λ s M 2 M 1 M 3 A 1 A 2 B 1 B 3 h = 1 γ o h = 1 M F E A D C B cot λ s cot γ B cot λ s FIGURE 6.16 Example of the diagram of distribution of the function cot g of a cutting tool rake angle g . © 2008 by Taylor & Francis Group, LLC 256 Kinematic Geometry of Surface Machining precisely specify geometric parameters of the active part of a cutting tool, all the elementary motions that compose the resultant motion of the cutting tool relative to the work must be taken into consideration. There are two possible ways to represent the machined surface P. First, the machined surface P can be considered as an enveloping surface to consecu- tive positions of the generating surface T of the cutting tool when the cutting tool is moving relative to the blank. Second, the machined surface P can be considered as a set of discrete surfaces of cut P se . At an instance of time when the surface P is generated, both the surface T and the surface P se are tangent to P either at point K or along the characteris- tic curve. Because of this, the tool-in-hand reference system can be associated either with the assumed surface of the cut or with the cutting tool. The two options are identical in the sense of the tool-in-hand reference system. When machining a part surface, it is necessary to consider the kinematic geometric parameters of the active part of the cutting tool in a reference sys- tem associated with the surface of the cut. For this purpose, the tool-in-use reference system is used. Commonly, rake surface R s as well as clearance surface C s of a cutting tool are shaped in the form of three-dimensional surfaces having complex geometry. Due to this, consideration of the surfaces R s and C s at a distinct point of the cutting edge is required. Contact of the cutting wedge with the work is considered at a distinct point of the cutting edge. Because size of the area of contact of the cutting edge and the work is small, the rake surface as well as the clearance surface are locally approximated by corresponding planes, by the planes that are tangent to the surfaces R s and C s at the point of interest of the cutting edge. Generally speaking, the geometry of the active part of a cutting tool must be determined for an elementary cutting edge of length dl (that is, in dif- ferential vicinity of the point M within the cutting edge). It is also necessary to consider the geometry of the active part at a given instant of time, say for the vector V Σ of known magnitude and direction. Such an approach would enable one to determine the distribution curves of geometric parameters within the cutting edge and the distribution curves of geometric parameters in time. In order to perform such an analysis, a generalized method of com- putation of geometry of the active part of a cutting tool is necessary. In particular cases, actual values of geometric parameters of the active part of a cutting tool can impose certain constraints onto parameters of kinematics of the machining operation. For example, variation in the actual value of geometric parameters either within the cutting edge or in time may impose restrictions on the parameters of feed-rate motion, of orienta- tion motion of the cutting tool, and so forth. If parameters of kinematics of the machining operation exceed the limits, then the machining operation is not feasible. The capability to determine critical feasible values of parameters of geom- etry of the active part of a cutting tool is critically important for the tool designer. © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 257 6.3.1 The Resultant Speed of Relative Motion in the Cutting of Materials As follows from the above analysis, the direction of the resultant speed V Σ of relative motion in the cutting of materials is a critical issue for establishing the tool-in-use reference system. Usually, relative motion of the cutting tool is of a complex nature. In the general case of surface machining, this motion is composed of the actual primary motion V p , the surface generation motion V gen , one or more feed-rate motions V f i. , the orientation motions of the rst V or I and of the second V or II kinds, and of other motions. This yields the fol- lowing equation for vector V Σ : V V V V V V V Σ Ι ΙΙ = + + + + + = = = ∑ ∑ K gen f i or or i n j j m . 1 1 (6.65) where V p is the vector of the primary motion, V gen is the vector of the motion of surface generation, V f i. is the vector of the feed-rate motion, n is the total num- ber of feed-rate motions, V or I is the vector of the orientation motion of the rst kind of the cutting tool, V or II is the vector of the orientation motion of the second kind of cutting tool, V j is the j elementary relative motion of the cutting tool, and m is the total number of elementary relative motions of the cutting tool. When determining the vector V Σ , vectors of all particular relative motions, those that signicantly affect the V Σ must be taken into account. Relative motions, those that cause sliding of the surface P or the generating surface T of the cutting tool over itself must be incorporated as well. Motions V or I and V or II of orientation of the cutting tool, as well as the feed- rate motions V f i. , are usually signicantly smaller compared to the primary motion V p . However, all must be incorporated for determination of the vector V Σ . In particular cases, some of these motions are comparable with the motion V Σ . Moreover, in special cases, they can even exceed the primary motion V p . When cutting a material, vibration of the cutting tool is often observed. The vibration may result in positive and negative clearance angle (Figure 6.17a). For certain frequencies and magnitudes of the vibration, neglecting the vector of vibration V vib is not allowed [1,3,13]. Due to vibrations, the rake and the clear- ance angles vary within a certain interval ± σ o . The current value of the angle σ σ o o vib p is = arctan | | | | . V V When the vector V vib is pointed toward the part surface P, then the corre- sponding rake angle γ o raises to the range of ′ = + γ γ δ o o o . At this instant, the clearance angle α o reduces to ′ = - α α δ o o o . If the vector V vib is directed oppo- sitely, then the corresponding rake angle γ o and the clearance angle α o can be computed from the equations ′′ = - γ γ δ o o o and ′′ = + α α δ o o o (see Figure 6.17b). When a partly worn cutting tool is used, then the clearance angle within a narrow land on the clearance surface next to the cutting edge reduces to 0°. © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 259 The surface of cut P se can be represented as a locus of consecutive posi- tions of the cutting edge that travels with the resultant speed V Σ relative to the work. The plane of cut is tangent to the surface of cut P se at the point of interest within the cutting edge. In a particular case, the surface of the cut and the plane of the cut are congruent. The last case is the degenerated one. The main reference plane P re is perpendicular to the vector V Σ . The work- ing plane P fe is the plane through the directions of the primary motion, and of the feed-rate motion. Due to this, the working plane P fe is perpendicular to the main reference plane P re . The tool back plane P pe is perpendicular to the reference planes P re and P fe . Other reference planes are of importance for the tool-in-use system. They are the plane of cut P se , the rake surface plane R s , and the clearance plane C s . For the purpose of determining the geometry of the active part of the cutting tool, it is convenient to employ three reference planes P se , R s , and C s in con- junction with the vector of the resultant cutting tool motion V Σ . The current orientation of the reference planes P se , R s , and C s is specied by unit normal vectors n rs , n cs , and c e . Prior to running the analysis, it is necessary to represent equation r P = r r P P P U= ( of the part surface P as well as equation r r T T T T U V= ( , ) of the gen- erating surface T of the cutting tool in a common coordinate system X Y Z T T T . For this purpose, implementation of the operator Rs( )T P→ of the resultant coordinate system transformation is helpful. Equations of tangent planes r P tp. and r T tp. to the surfaces P and T at the point of interest M can be represented in vectorial form: r r n P tp P M P. ( ) - ( ) × = 0 (6.66) and r r n T tp T M T. ( ) - ( ) × = 0 (6.67) The kinematic method can be employed for the derivation of the equation of the surface of cut P se . For this purpose, it is necessary to know the equation of the cutting edge and the parameters of the resultant relative motion of the cutting tool with respect to the work. The equation of the surface of cut P se can be obtained in the following way. Consider a form-cutting tool. The cutting edge of the form-cutting tool is determined as the line of intersection of the face rake surface R s by clear- ance surface C s . Therefore, in the coordinate system X Y Z T T T , the cutting edge of the form-cutting tool can be described analytically by a set of two vectorial equations: r r r r rs rs rs rs cs cs cs cs U V U V = =      ( , ) ( , ) (6.68) © 2008 by Taylor & Francis Group, LLC 260 Kinematic Geometry of Surface Machining An auxiliary Cartesian coordinate system X ce Y ce Z ce is associated with the cutting edge. Initially, axes of the coordinate system X ce Y ce Z ce align with cor- responding axes of the coordinate system X T Y T Z T . Then consider the motion that the cutting edge together with the coordinate system X ce Y ce Z ce is per- forming in the coordinate system X T Y T Z T . Parameters of this relative motion of the cutting edge are identical to the corresponding parameters of motion of the cutting tool relative to the work. The equation of the cutting edge in a current location of the coordinate system X ce Y ce Z ce with respect to the coordi- nate system X T Y T Z T can be represented in the form r r r r rs rs rs rs cs cs cs cs U V U V = =      ( , , ) ( , , ) Ξ Ξ Σ Σ (6.69) where Ξ Σ designates the parameter of the resultant relative motion of the cutting tool. On the premises of Equation (6.69), one of the two curvilinear parameters, either the U rs or V cs parameter can be expressed in terms of another parameter. For example, the U rs parameter is expressed in terms of the V cs parameter. This relationship yields analytical representation in the form U U V rs rs cs = ( ) . Ulti- mately, this results in the vectorial equation of the surface of cut P se in the form r r r se se cs cs cs se cs U V V V= =[ ( ), , ] [ , ]Ξ Ξ Σ Σ (6.70) Similarly, the equation of the surface of cut P se can be expressed in terms of V cs and Ξ Σ parameters. For many purposes, the generating surface T of the form- cutting tool can be considered as a good approximation to the surface of cut P se . In order to compose the tool-in-use system for machining a surface on a conventional machine tool, two vectors are of principal importance: vector V Σ of resultant relative motion of the cutting tool with respect to the work and unit normal vector to the surface of cutting n se . The vector V Σ is computed from Equation (6.65). The unit normal n se can be computed as the cross-product n u v se se se = × . For the derivation of the unit tangent vectors u se and v se , Equation (6.70) of the surface of cut P se can be used. That same unit normal vector n se can also be computed as the cross-product (Figure 6.18): n v c se e = × Σ (6.71) where the unit vector v Σ is equal to v Σ = V Σ /|V Σ |. Equation (6.71) for the computa- tion of the unit normal vector n se is convenient for performing computations. Other equations for the computation of the unit normal vector n se can be used as well: n V n n V n n se rs cs rs cs = × × × × Σ Σ [ ] | [ ]| (6.72) © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 261 It is useful to keep in mind that the approximation n n se T ≅ is valid in most practical cases of the computations. Unit vectors v Σ (or V Σ ) and n se are helpful for the analytical representation of the tool-in-use system. 6.3.3 Reference Planes Investigation of the impact of kinematics of a machining operation on actual (kinematical) values of geometry of the active part of a cutting tool can be traced back to research done by Pankin [6] or even to earlier works. A proper tool-in-use system is necessary but not sufcient for determining geometric parameters of the active part of a cutting tool. The specication of the conguration of reference planes is also of critical importance. For free orthogonal cutting, the reference plane for the rake angle g, the clearance angle a, the cutting wedge angle b, and the angle of cutting d is the plane through the vector V Σ . This reference plane is orthogonal to the plane of cut P se . For free oblique cutting, there are several reference planes for specication of the angles g, a, b, and d. The conguration of reference planes for nonfree cutting cannot be speci- ed in general terms. The mechanics of non-free cutting has not yet been thoroughly investigated. 6.3.3.1 The Plane of Cut Is Tangential to the Surface of Cut at the Point of Interest M For specication of the conguration of the plane of cut r se tp. , the vector of the resultant motion V Σ of the cutting tool relative to the work, and the unit vec- tor c e that is tangent to the cutting edge at M can be employed (Figure 6.19). dl P se M n cs C s R s n se c e n rs V Σ FIGURE 6.18 Elementary cutting wedge of the innitesimal length dl. © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 263 cutting edge. The angle of inclination λ se is measured within the plane of cut r se tp. . This is the angle between the vector V Σ and the unit normal vector n ce . The vector n ce is orthogonal to the cutting edge (Figure 6.19a), and is within the plane of cut r se tp. . If observing from the end of the unit normal vector n se to the surface of cut P se , then the positive angle λ se is measured in a counter- clockwise direction, and the negative angle λ se is measured in a clockwise direction (see Figure 6.19b). When the equality λ se = °0 is valid, then the cut- ting is the orthogonal cutting. Otherwise, when λ se ≠ °0 , then the more general case of cutting — the oblique cutting — is observed. Major frictions of the cut- ting tool (that is, chip deformation, direction of chip ow over the rake surface, etc.) depend upon the actual value of the angle of inclination λ se . The algebraic value of the angle of inclination λ se can be computed from the following equation (Figure 6.19b): λ se e e e = ∠ - ° = - ⋅ × ( , ) arctan | | c v c v c v Σ Σ Σ 90 (6.74) For the cutting tools of various designs the optimal value of the angle of inclination λ se varies within the interval λ se = ± °80 . 6.3.3.2 The Normal Reference Plane Conguration of the normal reference plane P ne of a cutting tool in the tool- in-use system is identical to its conguration in the tool-in-hand system. The normal plane is orthogonal simultaneously to the rake surface R s , to the clearance surface C s of the cutting wedge, to the plane of cut P se , and ultimately, to the cutting edge (Figure 6.20). The unit normal vector n ce to the cutting edge is within the normal reference plane P ne . Therefore, congu- ration of the normal reference plane P ne can be specied in terms of any two unit vectors n rs , n cs , n se , and n ce at the point M (Figure 6.20), or by the point M and the unit vector c e along the cutting edge. Evidently, there are many more options for the specication of conguration of the normal reference plane in the tool-in-use system rather than in the tool-in-hand system. 6.3.3.2.1 Normal Rake Angle Orientation of the rake surface of a cutting tool relative to the plane of cut depends upon the actual value of normal rake angle γ ne . The normal rake angle is measured in the normal reference plane. This is the angle that forms the unit normal vector n se to the plane of cut P se and the rake surface R s . The value of the angle γ ne is measured from the vector n se toward the rake surface R s . The normal rake angle γ ne is positive when the unit normal vector n se does not pass through the cutting wedge of the tool, and it is negative when the vec- tor n se is passing through the cutting wedge of the tool (Figure 6.20b). It is convenient to determine the normal rake angle γ ne as the angle that complements to 90° the angle between the unit normal vectors n se and n rs © 2008 by Taylor & Francis Group, LLC The Geometry of the Active Part of a Cutting Tool 265 6.3.3.2.2 Normal Clearance Angle Orientation of the clearance surface C s with respect to the plane of cut P se depends upon the normal clearance angle α ne . This angle is measured in the normal reference plane. The normal clearance angle α ne is the angle that the unit normal vector n se forms with the opposite direction of the unit normal vector — the clearance surface C s . The value of the clearance angle α ne is measured from the plane of cut P se toward the clearance surface C s . The normal rake angle α ne is always positive ( ) α ne > °0 . Only within a narrow land along the cutting edge can the normal clearance angle α ne be equal to zero or even be negative ( ) α ne ≤ °0 . It is convenient to determine the normal clearance angle α ne as the angle that complements to 180° the angle between the unit vectors n se and n cs (see Figure 6.20b): α ne cs se cs se cs se = °- ∠ = - × ⋅ 180 ( , ) arctan | | n n n n n n (6.76) For cutting tools of various designs, the optimal value of the normal clear- ance angle α ne is usually within the interval α ne = ° ÷ °10 30 . The uncut chip thickness a is the predominant factor that affects the opti- mal value of the clearance angle. On the premises of the analysis of impact of chip thickness a, Larin [23] proposed an empirical formulae α ne a = arcsin . . 0 13 0 3 (6.77) for the computation of reasonable value of the clearance angle. After a short period of cutting, a zero clearance angle α ne = °0 is observed within a narrow worn land along the cutting wedge. 6.3.3.2.3 The Mandatory Relationship For a workable cutting tool, satisfaction of the relationship N N se ce ⋅ < 0 (or the equivalent relationship n n se ce ⋅ = -1 ) is necessary (see Figure 6.20). Vio- lation of the relationship is allowed only within a narrow land along the cutting wedge. The normal cutting wedge angle is measured in the normal reference plane. The normal cutting wedge angle is the angle that forms the rake plane R s and the clearance plane C s . The value of the angle β ne can be computed from a simple equation (see Figure 6.20b): β α γ ne ne ne = ° - +90 ( ) (6.78) The normal cutting angle is measured in the normal reference plane. The normal cutting angle is the angle that forms the plane of cut P se and the clearance plane C s . The value of this angle δ ne is equal (see Figure 6.20b): δ γ ne ne = ° -90 (6.79) © 2008 by Taylor & Francis Group, LLC 266 Kinematic Geometry of Surface Machining Denitely, both the angles β ne and δ ne can be expressed in terms of unit normal vectors to the corresponding planes of the cutting wedge, and to the reference surfaces. 6.3.3.3 The Major Section Plane Conguration of the major section plane P ve is determined by two directions through the point M. One of the directions is specied by the unit normal vector n se to the plane of cut P se , and another direction is specied by the vector of the resultant motion of the cutting tool V Σ with respect to the work (Figure 6.21a). The major section plane P ve is perpendicular to the plane of cut P se . The equation of the major section plane P ve in terms of the vectors V Σ and n se yields representation in vectorial form: r r n v ve tp se M se. ( ) [ ]- ( ) × × = Σ 0 (6.80) where r ve tp. designates the position vector of a point of the major section plane. The rake angle γ ve is measured in the major section plane P ve (Figure 6.21b). The rake angle γ ve is equal to the angle between the unit normal vector n se to the plane of cut, and the unit vector b is tangent to R s and is located within the reference plane P ve : γ ve se se se = ∠ = × ⋅ ( , ) arctan | | n b n b n b (6.81) The rake angle γ ve is positive when the vector n se does not penetrate the cutting wedge, and it is negative when it does (Figure 6.21b). The clearance angle α ve is the angle that the unit normal vector n se makes with the unit vector c. Here, the unit vector c is tangent to the line of intersection of the clearance surface C s by the major section plane P ve (see Figure 6.21b): α ve se se se = ∠ = × ⋅ ( , ) arctan | | n c n c n c (6.82) The cutting wedge angle β ve is the angle between the unit vectors b and c (see Figure 6.21b): β ve = ∠ = × ⋅ ( , ) arctan | | b c b c b c (6.83) In the major section plane P ve , the equality β α γ ve ve ve = ° - +90 ( ) is always observed. The angle of cutting δ ve is the angle that the unit vector b makes with the vector V Σ of the resultant motion of the cutting tool relative to the surface of © 2008 by Taylor & Francis Group, LLC [...]... Capabilities of the Analysis of Geometry of the Active Part of Cutting Tools Implementation of the vectorial method for the computation of geometry of the active part of a cutting tool is very helpful In order to demonstrate the creative capabilities of the method, consider a few practical examples 6.4.1 Elements of Geometry of Active Part of a Skiving Hob The geometry of active parts of skiving hobs... Generating Surface T of the Cutting Tool Gauss introduced the notion of mapping of surface normals onto the surface of a unit sphere by means of parallel normals, in which a point on a map is the result of the intersection of the surface normal vector, translated so as to emanate from the center of a unit sphere, with the surface of the unit sphere [4] © 2008 by Taylor & Francis Group, LLC 291 Conditions of. .. to the surface P is minimized Use of both approaches reduces the mean difference in condition of machining of each small portion of the given sculptured surfaces P and thereby increases tool life For evaluation of the degree of deviation of actual sculptured surface orientation from its desired orientation, a measure of the deviation is required 7.1.2 Gaussian Maps of a Sculptured Surface P and of the... orientation of the workpiece so as to minimize the number of setups in the multi-axis NC machining of a given sculptured surface P, or to allow the maximal number of surfaces to be machined in a single setup Here, a method for computing such an optimal workpiece orientation is developed based on the geometry of the sculptured part surface P to be machined, on the geometry of the generating surface T of the... the rake surface of hob teeth could enhance capabilities of the cutting tool It would be possible to use the hob having a modified rake surface of the teeth for machining of gears having a modified tooth profile Modification of the gear tooth profile is recognized as a powerful tool for improving the performance of gear drives For this purpose, the rake surface of the hob teeth is composed of two portions... Patent 1.017.444) © 2008 by Taylor & Francis Group, LLC 280 Kinematic Geometry of Surface Machining Consider a Cartesian coordinate system XYZ associated with the hob tooth Origin O of the coordinate system is within the line of intersection of the two portions of the hob rake surfaces The axis Y aligns to the line of intersection of the two portions of the rake surface The axis Z is parallel to the vector VΣ... peculiarities of shape of the part surface to be machined, of the generating surface of the cutting tool, and of the kinematics of the machining operation, the shape of the machined part surface could deviate from its desired shape When a portion of stock on the part surface remains uncut, then an overcut is observed When the cutting tool removes material beneath the part surface, then an undercut is observed... this kind of surface mapping for the purpose of investigating the surface topology Since Gauss’ publication [4], the mapping of a surface on a unit sphere is usually referred to as Gauss’ mapping of the surface Later Gauss’ idea of surface mapping received wide implementation both in science [1] and engineering As early as in 198 7, Radzevich [9] applied Gauss’ idea to the sculptured part surface orientation... of actual values of the angles ϕ e and ϕ 1e, projections of these angles onto the rake plane can be used © 2008 by Taylor & Francis Group, LLC 272 Kinematic Geometry of Surface Machining The tool tip (nose) angle ε e is determined for the tip of a cutting tool The angle ε e can be computed from the equation ε e = 180° - (ϕ e + ϕ 1e ) (6 .94 ) The tip of a form-cutting tool coincides with the point of. .. No.1, 193 6 [7] Pat. No. 99 0.445, USSR, A Precision Involute Hob./S.P. Radzevich, Int. Cl B23F 21/16, Filed October 8, 198 1 [8] Pat. No. 1.017.444, USSR, An Involute Hob./S.P. Radzevich, Int. Cl B 23f21/16, Filed April 26, 198 2 [9] Pat. No. 1.114.543, USSR, A Gear Hob./S.P. Radzevich, Int Cl B 23f21/16, Filed September 7, 198 2 © 2008 by Taylor & Francis Group, LLC 286 Kinematic Geometry of Surface Machining . of the angle of inclination λ s . 6.2.6 Diagrams of Variation of the Geometry of the Active Part of a Cutting Tool Analytical methods for the computation of actual values of the geometry of. representation of the tool-in-use system. 6.3.3 Reference Planes Investigation of the impact of kinematics of a machining operation on actual (kinematical) values of geometry of the active part of a cutting. plane of projections V (Figure 6.25). 6.4 On Capabilities of the Analysis of Geometry of the Active Part of Cutting Tools Implementation of the vectorial method for the computation of geometry of