292 Kinematic Geometry of Surface Machining form, say in terms of the rst and the second fundamental forms of the GMap P( ) of the given surface P. The rst fundamental form Φ 1 0.P of the GMap P( ) is given by expression [2] Φ 1 0 0 2 2 2 2 .P P P P P P P P P ds e du f du dv g dv⇒ = + + (7.2) where ds P0 is the differential of an arc of a curve on the unit sphere; e P , f P , g P , are the rst-order Gaussian coefcients of the GMap(P); and u P , v P are the parametric coordinates of an arbitrary point of the GMap(P). Omitting bulky derivations, one can write the following equations for the rst fundamental form Φ 1 0.P of the GMap P( ) of the part surface P: Φ Φ Φ 1 0 0 2 2 1.P P .P P .P P ds⇒ = −M G (7.3) where M P designates mean curvature of the sculptured surface P (see Equa- tion 1.15), and G P designates Gaussian curvature of that same surface P (see Equation 1.16). The second fundamental form Φ 2 0.P of the GMap P( ) of a given patch of the surface P is calculated as [2] Φ 2 0 0 0 2 2 2 .P P P P P P P P P P d d l du m du dv n dv⇒ − ⋅ = + +r n (7.4) and is derived in a similar manner. In Equation (7.4), the values l P , m P , n P are the second-order Gaussian coefcients of the GMap P( ) of the surface unit sphere. Skipping the proofs, some useful properties of the GMap P( ) and GMap T( ) can be noted: The GMap P( ) of an orthogonal net on a sculptured surface P for which mean curvature M P is not equal to zero ( M P ≠ 0 ) is also an orthogonal net if and only if the initial net is made up of lines of curvature. If the mean curvature M P of the surface P is equal to zero ( M P = 0 ), then the net of coordinate lines on the GMap P( ) will be orthogonal as well. Points on the boundaries of the surface P and on its GMap P( ) are not necessarily in one-to-one correspondence. GMap P( ) is a many-to-one map: Each point on a smooth part sur- face P has a corresponding point on the GMap P( ) , but each point on GMap P( ) may correspond to more than one point on the part surface P. This means that in particular cases, GMap P( ) can be interpreted as having more than one layer. GMap P( ) of this kind are often referred to as the multilayer GMap P( ) . For example, GMap P( ) of a torus surface is of two layers. © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 293 The general solution of the problem of design of a form-cutting tool for machin- ing sculptured surfaces on a multi-axis NC machine (see Chapter 5) reveals that the generating surface T of the cutting tool can be as complex as a sculptured surface P can be. The generating surface T of the cutting tool of any design has the corresponding GMap T( ) . Figure 7.3 illustrates examples of GMap T( ) of cut- ting tools that are commonly used in industry. Computation of the parameters of the GMap T( ) of the generating surface T of the cutting tool is similar to the calculation of the parameters of the GMap P( ) of a sculptured surface P. 7.1.3 The Area-Weighted Mean Normal to a Sculptured Surface P The efciency of machining a sculptured surface on a multi-axis NC machine can be extremely high when the workpiece orientation is optimal. It is con- venient to calculate the parameters of the (two-criterion) optimal workpiece orientation taking into consideration the orientation of the area-weighted mean normal to the surface P. A point on the part surface P at which the surface normal is parallel to the area-weighted mean normal of the surface P is referred to as the central point of the surface P. To calculate the parameters of the area-weighted mean normal, the sur- face P can be subdivided into a large number of reasonably small patches S U V Pi Pi Pi = ×∆ ∆ . Here “i” indexes the small patches on the surface P. At the central point M i inside of each small patch of the surface P, the parameters of the perpendicular N Pi to the surface P can be computed: N r r Pi P P i P P i U V = ∂ ∂ × ∂ ∂ (7.5) The perpendicular N Pi may be considered to be an area vector element with magnitude equal to the innitesimal area of part surface P at a point i. FIGURE 7.3 Examples of the form-cutting tools of various designs. © 2008 by Taylor & Francis Group, LLC 294 Kinematic Geometry of Surface Machining For the computation of parameters of orientation of the area-weighted mean normal vector % N P , the following formula is employed: % N N N P Pi Pi i n P Pi Pi i n Pi P S S U V S = = = = ∑ ∑ ∆ ∆ ∆ 1 1 (7.6) where n designates the number of small patches on surface P, and S P desig- nates the area of surface P to be machined. ( S P is equal to the sum of all the areas of the separate workpiece surfaces to be machined in one setup.) Allowing the number of small patches on the surface P to approach inn- ity yields % N N N P P P P P S P P P P P P S P U V dS S U V dU dV S P P = = ∫ ∫ ( ; ) ( ; ) (7.7) Differential of the surface P area is dS E G F dU dV P P P P P P = − 2 . Accordingly, Equation (7.7) casts into % N N P P P P P P P P P P P E G F U V dU dV S = − ∫∫ 2 ( ; ) (7.8) In cases when several part surfaces P i are to be machined on a multi-axis NC machine in one setup, Equation (7.8) yields the more general formula % N N P P P P P P P P P P i k Pi i k E G F U V dU dV S i = − ∫∫ ∑ = = 2 1 1 ( ; ) ∑∑ (7.9) where k is the total number of the part surfaces P i to be machined in one setup. In the latter case, the area-weighted mean normal to the part surface P is not considered, but the area-weighted mean normal to the several surfaces P i is considered. The last is referred to as the area-weighted mean normal to all part surfaces P i . In this case, instead of a central point of the surface, a central point of the entire part to be machined is considered. Denitely, this is a considerably more general approach. © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 295 The area-weighted mean normal to a at portion of surface P is N Pi Pi S . This result can be used in Equation (7.9). The parameters of the area-weighted mean normal to a sculptured sur- face P as calculated above allow alteration of the initial orientation of the sculptured surface P to the desired orientation in the coordinate system of the multi-axis NC machine. By rotations of the workpiece, for example, through the angle of nutation ψ , through the angle of precession θ , and through the angle of pure rotation ϕ , the workpiece can be reoriented to its optimal orientation. In its optimal orientation, the workpiece allows machin- ing of all surfaces with a single setup. 7.1.4 Optimal Workpiece Orientation In the initial orientation of the workpiece, the angles that the area-weighted mean normal to the surface P makes with the coordinate axes of the NC machine are denoted α , β , γ (Figure 7.4). It is convenient to show these angles on the GMap P( ) of the part surface P (remembering that the area- weighted mean normal to the part surface P has the same direction as the position vector of the point on the GMap P( ) corresponding to the point on surface P at which the perpendicular is erected). In the case under consider- ation, the problem of optimal workpiece orientation reduces to a problem of coordinate system transformation. Consider two Cartesian coordinate systems X Y Z P P P and X Y Z NC NC NC . The rst coordinate system is associated with the workpiece. Another is con- nected to the multi-axis NC machine. In the initial orientation of the workpiece, orientation of the coordinate system X Y Z P P P relative to the coordinate system X Y Z NC NC NC is dened M * P0 γ β α Z P0 Y P0 X P0 M P0 r P0 r * P0 FIGURE 7.4 Spherical map of a point of the surface P in the initial orientation of the work and after its optimal orientation. © 2008 by Taylor & Francis Group, LLC 296 Kinematic Geometry of Surface Machining by the angles α , β , and γ . Actual values of these angles can be computed using one of the above-derived Equation (7.6) through Equation (7.9). The computed value of the area-weighted mean normal vector % N P immediately yields computation of the angles α , β , and γ . For this purpose, the follow- ing formulae cos α = ⋅i N % P , cos β = ⋅j N % P , and cos γ = ⋅k N % P can be used. All the computations must be performed in a common reference system. Use of the coordinate system X Y Z NC NC NC is preferred. In the optimal workpiece orientation, corresponding axes of these coor- dinate systems are parallel to each other and are of the same direction. To put the workpiece into the optimal orientation means to make three suc- cessive rotations, for example, by the Euler angles — that is, to rotate the workpiece in the coordinate X Y Z NC NC NC through the angle of nutation ψ , through the angle of precession θ , and nally, through the angle of pure rotation ϕ . The resultant coordinate system transformation using Euler’s angles can be analytically represented with the operator Eu ( , , ) ψ θ ϕ of Eulerian trans- formation (see Equation 3.11). In the optimal workpiece orientation, it is possible to rotate the part sur- face P about the area-weighted mean normal % N P . Under such a rotation, the optimal orientation of the workpiece is preserved, but the orientation of part surface P relative to the NC machine coordinate axes changes. This feasible rotation of the surface P can be used for satisfying additional requirements to the part surface orientation on the worktable of the multi- axis NC machine. For example, the workspace of the multi-axis NC machine is the bounded plane or volume within which the cutting tool and the workpiece can be positioned and through which controlled motion can be invoked. When NC instructions are generated by a part programmer, the geometry of the work- piece must be transformed into a coordinate system that is consistent with the workspace origin and coordinate reference frame. That is why after the workpiece is turned to a position at which its area-weighted mean normal has an optimal orientation, it is necessary to rotate it about the weighted normal to a position in which the projection of the part surface P (or of the part surfaces P i ) to be machined is within the largest closed contour traced by the cutting tool on the plane of the NC machine worktable. In addition, the vertical position of the workpiece must conform to the capabilities of the NC machine to move in the vertical direction. Proper location of the workpiece on the worktable of a multi-axis NC machine can be specied in terms of (a) the joint space, which is dened by a vector whose components are the relative space displacements at every joint of a multi-axis NC machine; (b) the working envelope, which is understood as a surface or surfaces that bound the working space; (c) the working range, which means the range of any variable for normal operation of a multi-axis NC machine; and (d) the working space that includes totality of points that can be reached by the reference point of the cutting tool. © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 297 7.1.5 Expanded Gaussian Map of the Generating Surface of the Cutting Tool An ordinary Gauss’ map can be constructed for any generating surface of a cutting tool. Examples of the GMap T( ) of the form-cutting tools of various designs are shown in Figure 7.3. However, when machining a sculptured surface, the cutting tool is traveling with respect to the surface P. Corre- spondingly, the GMap T( ) moves over the surface of the unit sphere covering in such a motion that the area exceeds the area of the original GMap T( ) . Gauss’ map that is constructed for the moving generating surface T of the cutting tool in all its feasible positions is referred to as the expanded Gauss’ map GMap T e ( ) of the generating surface of the cutting tool. Actually, when machining a sculptured surface on a multi-axis NC machine, the workpiece and the cutting tool perform certain relative motions. For further analysis, it is convenient to implement the principle of inversion of relative motions. On the premises of the principle of inversion of relative motions, consider the resultant motion of the cutting tool relative to the sta- tionary workpiece. At every point K of contact of the surfaces P and T, the unit normal vectors n P and n T to these surfaces are of opposite directions. (Remember that a nor- mal to the part surface P is pointed outward from the part body, and a normal to the generating surface T of the cutting tool is pointed outward from the generating body of the cutting tool. Therefore, the equality n n P T = − must be satised.) Then, employ the concept of antipodal points [5]. Those points on the Gauss’ map are usually referred to as the antipodal points, which are the pairs of diametrically opposed points on the unit sphere. Implementation of the antipodal points yields introduction of the centro-symmetrical image of the GMap T( ) of cutting tool surface T. The last is referred to as the antipodal GMap T a ( ) of the generating surface T of the cutting tool. Analysis of possible relative positions of the GMap P( ) of the sculptured surface P and of the antipodal GMap T a ( ) of the generating surface T of the cutting tool yields the following intermediate conclusions: Conclusion 7.1: If GMap P( ) of the part surface P is entirely located within the antipodal GMap T a ( ) of the generating surface T of the cutting tool (that is, the GMap P( ) contains no points outside GMap T a ( ) ), then machining of the surface P is possible. This is the necessary but not sufcient condition for the machinability of the part surface with the given cutting tool. Conclusion 7.2: If any portion of the GMap P( ) is located outside the antipodal GMap T a ( ) , then machining of the surface P is impossible. This is the sufcient condition that the part surface P cannot be machined with the given cutting tool. © 2008 by Taylor & Francis Group, LLC 298 Kinematic Geometry of Surface Machining When machining a sculptured surface on a multi-axis NC machine, the cutting tool is capable of moving along three axes of the coordinate system X Y Z NC NC NC , and rotating about one or more of the axes. These additional degrees of freedom (rotations) allow the antipodal indicatrix GMap T a ( ) of the generating surface T of the cutting tool to expand around the surface of the unit sphere, while the GMap P( ) remains xed. Similar to the spherical indicatrix GInd T( ) of the surface T that serves as the boundary curve for the corresponding GMap T( ) , the antipodal indica- trix GInd T a ( ) serves as the boundary curve to the antipodal GMap T a ( ) . For example, consider machining of a sculptured surface P on a three-axis NC machine. The antipodal GMap T a ( ) of the generating surface T of the cutting tool occupies the xed area ABCD (Figure 7.5). Then, assume that one more NC-axis is added somehow to the articulation capabilities of the NC machine. The additional fourth NC-axis (say, rotation of the cutting tool about an axis not coinciding with the axis of its cutter-speed rotation) causes the antipodal GMap T a ( ) to extend in direction 1 from the initial location ABCD to encompass A B CD 1 1 . If the fth and the sixth NC-axes are added, then these additional NC-axes cause the antipodal GMap T a ( ) to extend in direction 2 and to rotate about an axis through the center of the unit sphere and through a point within the antipodal GMap T a ( ) of the generating sur- face T of the cutting tool. A surface patch on the unit sphere is covered by the antipodal GMap T a ( ) such that its motion over the unit sphere is referred to as the expanded antipo- dal GMap T ae ( ) of generating surface T of the cutting tool. GMap a (T)GMap ae (T) C 2 B 2 B A 2 D 1 C 1 B 1 A 1 D C A M T0 Z P0 Y P0 X P0 2 D 2 1 FIGURE 7.5 An example of the expanded antipodal of the generating surface T of the cutting tool. © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 299 The expanded antipodal indicatrix GMap T ae ( ) is a useful tool for the investi- gation of workpiece orientation on the worktable of a multi-axis NC machine. The boundary curve of the expanded antipodal GMap T ae ( ) of the generat- ing surface T of the cutting tool serves as the expanded antipodal indicatrix GInd T ae ( ) of the tool surface T. Because the surface P is considered motion- less, the expanded antipodal indicatrix GInd T ae ( ) of the generating surface T of the cutting tool as well as its expanded antipodal GMap T ae ( ) cannot rotate about an axis through the origin of the coordinate system X Y Z P P P0 0 0 . When the parameters of the relative motion of the given sculptured sur- face P and of the given generating surface T are known, then the parameters of the expanded initial GInd T e ( ) and of the expanded antipodal GInd T ae ( ) indicatrices of the tool surface T can be calculated using the developed meth- ods of spherical trigonometry [5]. For machining a sculptured surface on a multi-axis (four or more axes) NC machine, the following two statements hold: Conclusion 7.3: If GMap P( ) of the part surface P is contained entirely inside the expanded antipodal GMap T ae ( ) of generating surface T of the cutting tool, then the surface P can be machined. This is the necessary but not sufcient condition for the machinability of a sculptured surface on a give multi-axis NC machine with the given cutting tool. Conclusion 7.4: If GMap P( ) of the part surface P contains at least one point outside the expanded antipodal GMap T ae ( ) of generating surface T of the cutting tool, then machining of the surface P is not feasible. This condition is sufcient for the sculptured surface that cannot be machined on a given multi-axis NC machine with the given cutting tool. 7.1.6 Important Peculiarities of Gaussian Maps of the Surfaces P and T In particular cases, a sculptured surface P, as well as the generating sur- face T of the form-cutting tool can have two or more points, at which unit normal vectors are parallel to each other and are pointed in that same direction. Points of this sort can be easily found out, for example, on the torus surface. When parallel and similarly directed unit normal vectors are observed, then the GMap P( ) of the sculptured surface P becomes “multilayered.” The num- ber of layers of the the GMap P( ) is equal to the number of points with parallel and similarly oriented unit normal vectors. For example, parallel and similarly oriented unit normal vectors occur on the part surface P (Figure 7.6). The surface P is bounded by the bordering line ABCDEFG . Gauss’ map GMap P( ) for this portion of the surface P is represented by the portion A B G D E F 0 0 0 0 0 0 of the unit sphere. Figure 7.6 reveals that the area B 0 C 0 D 0 G 0 on the unit sphere corresponds to the GMap P( ) of the portion BCDG of the © 2008 by Taylor & Francis Group, LLC Conditions of Proper Part Surface Generation 301 the surface P 1 occupies a portion of the unit sphere within the closed contour ABDCFE . The area-weighted mean normal n r P P0 0 * * passes through the point M P0 * . In such an orientation, the surface P 1 can be machined on the three- axis NC machine in one setup. In the second case, a sculptured surface P 2 has a multilayer (two-layer) GMap P m ( ) . The subscript m here indicates that the Gauss’ map is multilayer. Likewise, in the rst case, GMap P m ( ) of the surface P 2 occupies the portion within the closed contour ABDCFE on the unit sphere. In addition to the portion ABDCFE , the GMap P m ( ) is represented with the portion CDEF on the unit sphere (Figure 7.7). When the portion CDEF is added, then the Gauss’ map is a two-layer map, so it is twice as heavily weighted. Due to the increase in weight of the GMap P m ( ) , the area-weighted mean normal n r P P0 0 of the surface P 2 turns about the center of the unit sphere through a certain angle ε . In this position, the unit normal vector passes through the point M P 0 . In such a position of the workpiece, it is infeasible to machine the sculp- tured surface P 2 on the three-axis NC machine in one setup. It is neces- sary to consider the trade-offs between the orientation of the part surface P 2 in compliance with the position of its area-weighted mean normal and the orientation of the surface P 2 to avoid shadowed areas. One could con- sider, for example, whether it is preferred to machine the sculptured part surface P 2 on a cheaper, three-axis NC machine with nonoptimal work- piece orientation vis-à-vis cutting conditions, or to machine the part sur- face P 2 in the optimal workpiece orientation but on a more costly four (or more) axis NC machine. Generally, machining of a part surface in a single setup with some loss of optimality of cutting condition is preferable to machining in two or more setups. Thus, the generally favored situation is to orient the workpiece such that the difference in angle between the area- weighted normal to the part surface to be machined and the tool axis of rotation changes as little as possible, without requiring more setups than necessary. After the analysis of Figure 7.7 is performed, it is important to focus again on the properties of Gauss’ mapping of the surfaces. Figure 7.6 provides a good example to illustrate the property (b) of the GMap P( ) (see Section 7.1.2). Gauss’ map of the bordering contour ABCDE of the surface P is represented by the circular arc A B C D E 0 0 0 0 0 (Figure 7.6). In this case, all points of the bordering contour ABCDE of the surface P and all points of the boundary A B C D E 0 0 0 0 0 of the Gauss’ map are in one-to-one correspondence. On the other hand, Gauss’ map of the bordering contour AFE of the surface P is represented by the circular arc A F E 0 0 0 . It is evident that the Gauss’ map A F E 0 0 0 of the bordering contour AFE is not a border for the GMap P( ) of the sculptured surface P. Moreover, boundary arc B G D 0 0 0 of the GMap P( ) of the surface P is just an image of the curve BGD on the surface P. However, the BGD is not a boundary of the surface P. This example illustrates that a boundary of the GMap P( ) of a sculptured surface P may or may not be a boundary of its GMap P( ) , and vice versa. © 2008 by Taylor & Francis Group, LLC 302 Kinematic Geometry of Surface Machining 7.1.7 Spherical Indicatrix of Machinability of a Sculptured Surface The above-considered Gauss maps of the sculptured part surface and of the generating surface of the cutting tool provide engineers with a power- ful analytical tool. Among others, implementation of this tool is helpful for determining whether or not a given sculptured surface P can be machined with the given cutting tool on the NC machine with the given articulation. For this purpose, spherical indicatrices GInd P( ) and GInd T( ) can be used. It is inconvenient to treat simultaneously two separate characteristic curves GInd P( ) and GInd T( ) to determine whether the part surface P can or cannot be machined in one setup with the given cutting tool on the NC machine with the given articulation. For this purpose, a characteristic curve of another nature is proposed. This characteristic curve is referred to as the spherical indicatrix of machinability Mch P T( / ) of a given sculptured surface P with the given cutting tool T on the NC machine with the given articulation. For CAD/CAM applica- tion, it is necessary to represent this characteristic curve analytically. Without loss of generality, one can consider for simplicity the machining of a sculptured surface P with a ball-end milling cutter. For this case, the GMap T( ) of the generating surface of the cutting tool occupies a hemisphere of the unit sphere (Figure 7.8). GMap P( ) of the sculptured surface P is rep- resented with a certain patch on the unit sphere. The great circle of the unit sphere serves as the GInd T( ) of the generating surface T of the cutting tool. Ultimately, GInd P( ) is represented by the boundary of the GMap P( ) . An arbitrary point M P 0 is chosen within the GMap P( ) of the surface P. A cross-section of the unit sphere by the plane Σ i through the origin of the coordinate system X Y Z P P P 0 0 0 and the chosen point M P 0 is considered. The plane Σ i intersects the spherical indicatrices GInd P( ) and GInd T( ) at the points A i0 and B i0 , respectively. The angle between the position vector r Ai of the point A i0 and the position vector r M of the chosen point M P 0 is des- ignated as ς Ai . A similar angle between the position vector r Bi of the point B i0 and the position vector r M is designated as ς Bi . The difference of the (a) (b) (c) B 0i r Mi GMap a (T) GMap(P) Z P0 Y P0 X P0 r Bi M P0 i Σ r Ai Σ i r Mi r Ai r Bi Mi ς ς Ai A 0i Bi ς B 0i Z P0 Y P0 M P0 * M P0 r Mi Σ 2 Σ 1 Z P0 X P0 Σ 3 r Mi M P0 A 0i ς Mi = ς Mi * FIGURE 7.8 Derivation of equation of the spherical indicatrix of machinability of a surface P with the given cutting tool. © 2008 by Taylor & Francis Group, LLC [...]... effectiveness of the first necessary condition of proper PSG with an example of machining of an involute working surface of a cam (Figure 7.14) Working surface P of the cam is shaped in the OT ωT P ± VT TG P* K w* w w** P P ** d ** w ± ωP dw = db d* w OP Figure 7.14 Examples of satisfaction and of violation of the first necessary condition of proper part surface generation when machining the involute working surface. .. analysis of the actual correspondence between the radii of normal curvature of the surfaces, the normal cross-sections of all possible kinds of contact of the surfaces P and T have been analyzed (Table 7.1) © 2008 by Taylor & Francis Group, LLC 316 Kinematic Geometry of Surface Machining RP T RT RP RT T RT RP RT RP P P K K (a ) (b) Figure 7.16 Examples of satisfaction (a) and of violation (b) of the... Condition of Proper Part Surface Generation When machining, the sculptured part surface is generated by the generating surface of the cutting tool A cutting tool of a certain design is necessary for the machining of a given sculptured surface on a multi-axis NC machine A cutting tool of any design can be designed on the premises of the generating surface T of the cutting tool This means that existence of. .. P of a cam (From Radzevich, S.P., Computer-Aided Design, 34 (10) , 727–740, 2002 With permission.) © 2008 by Taylor & Francis Group, LLC 312 Kinematic Geometry of Surface Machining form of an involute surface that has the base circle of diameter db When machining the surface P, the cam is swinging about the axis OP with an angular velocity ±ω P The grinding wheel has a flat working surface When machining... cutting tool This means that existence of the generating surface T of the cutting tool is a prerequisite for the feasibility of machining a given sculptured part surface The principal methods for generation of the generating surface T of the cutting tool are disclosed in Chapter 5 © 2008 by Taylor & Francis Group, LLC 310 Kinematic Geometry of Surface Machining Z Z ωπ π2 π4 π4 π5 π1 π3 π2 M Y nπ VM X VM... equation of this characteristic curve can also be interpreted as an analytical representation of the third necessary condition of proper PSG Implementation of this characteristic curve is of particular importance for the development of software for the machining of a sculptured part surface on a multi-axis NC machine The scenario when the radii of normal curvature RP and RT of the surfaces P and T are of. .. Examples of violation (a) and (b) and of satisfaction (c) and (d) of the first necessary condition of proper part surface generation (From Radzevich, S.P., Computer-Aided Design, 34 (10) , 727–740, 2002 With permission.) Regardless of whether the generation of the surface T as an enveloping surface is convenient or not, the generating surface T of the cutting tool can be considered as the enveloping surface. .. Condition of Proper PSG: The condition of proper contact of a sculptured surface P to be machined and of the generating surface T of the form-cutting tool without their mutual penetration (that is, without their mutual interference in differential vicinity of the point of contact) is the third necessary condition of proper PSG As an illustration of the necessity of proper correspondence between the radii of. .. center of symmetry of the curve © 2008 by Taylor & Francis Group, LLC 319 ρT.c ρT.b OP γt T αt P ωP β ωT RT.c a OT b c ρT ρP a rP.c ρP.a ρT.a b ρP.b c ρP.c Conditions of Proper Part Surface Generation Figure 7.18 Example of violation of the third necessary condition of proper part surface generation when regrinding a broach © 2008 by Taylor & Francis Group, LLC 320 Kinematic Geometry of Surface Machining... that absence of the cutting tool leads to the machining operation of a sculptured surface P not being able to be performed To design an appropriate form-cutting tool, the machining surface T of the cutting tool must exist This allows formulation of the following: The First Condition of Proper PSG: Existence of the generating surface T of a form-cutting tool which is conjugate to a given part surface P . LLC 302 Kinematic Geometry of Surface Machining 7.1.7 Spherical Indicatrix of Machinability of a Sculptured Surface The above-considered Gauss maps of the sculptured part surface and of the. 292 Kinematic Geometry of Surface Machining form, say in terms of the rst and the second fundamental forms of the GMap P( ) of the given surface P. The rst fundamental form Φ 1 0.P of the. worktable of a three- axis numerical control milling machine. © 2008 by Taylor & Francis Group, LLC 310 Kinematic Geometry of Surface Machining Regardless of whether the generation of the surface