106 Kinematic Geometry of Surface Machining Crv(P) of a smooth, regular surface P are listed along with the corresponding sign of the mean M P and of the Gaussian G P curvatures (Figure 4.6): Convex elliptic (M P > 0, G P > 0) in Figure 4.6a Concave elliptic (M P < 0, G P < 0) in Figure 4.6b Convex umbilic (M P > 0, G P > 0) in Figure 4.6c Concave umbilic (M P < 0, G P < 0) in Figure 4.6d Convex parabolic (M P > 0, G P = 0) in Figure 4.6e Concave parabolic (M P < 0, G P = 0) in Figure 4.6f Quasi-convex hyperbolic (M P > 0, G P < 0 ) in Figure 4.6g Quasi-concave hyperbolic (M P < 0, G P < 0) in Figure 4.6h Minimal hyperbolic (M P = 0, G P < 0 ) in Figure 4.6i Phantom branches of the characteristic curve in Figure 4.6g through Figure 4.6i are indicated by dashed lines. For a plane local patch of a surface P, the curvature indicatrix does not exist. All points of this characteristic curve are remote to innity. 4.3.8 Introduction of the Jr k (P/T ) Characteristic Curve For the purpose of analytical description of the distribution of normal curvature in differential vicinity of a point on a smooth, regular surface, Böhm recom- mends [1] that the following characteristic curve be employed. Setting h = dV P / dU P at a given point of a sculptured surface P, one can rewrite the equation k L dU M dU dV N dV E dU P P P P P P P P P P P P = = + + + Φ Φ 2 1 2 2 2 2 2 . . FF dU dV G dV P P P P P + 2 (4.46) for normal curvature in the form of k L M N E F G P P P P P P P = + + + + 2 2 2 2 η η η η (4.47) In the particular case when L M N E F G P P P P P P : : : := , the normal curvature k P is independent of h. Surface points with this property are known as umbilic points and atten points. In general cases when k P changes as h changes, the function k k P P = ( ) η is a rational quadratic form, as illustrated in Figure 4.7. The extreme values k P1. and k P2. of k k P P = ( ) η occur at the roots η 1 and η 2 of η η 2 1 0 - =E F G L M N P P P P P P (4.48) • • • • • • • • • © 2008 by Taylor & Francis Group, LLC 108 Kinematic Geometry of Surface Machining The discussed methods of higher-order analysis target the development of an analytical description of the rate of conformity of the generating surface T of the cutting tool to the part surface P at the current point K of their contact. The higher the rate of conformity of the surfaces P and T, the closer are these surfaces to each other in differential vicinity of the point K. This qualitative (intuitive) denition of the rate of conformity of two smooth, regular surfaces needs a corresponding quantitative measure. 4.4.1 Preliminary Remarks Consider two surfaces P and T in the rst order of tangency that make contact at a point K. The rate of conformity of the surfaces P and T can be interpreted as a function of radii of normal curvature R P and R T of the surfaces. The radii of normal curvature R P and R T are taken in a common normal plane section through point K. For a given radius of normal curva- ture R P of the surface P, the rate of conformity of the surfaces depends on the corresponding value of radius of normal curvature R T of the generat- ing surface T. In most cases of part surface generation, the rate of conformity of the surfaces P and T is not constant. It depends on orientation of the normal plane section through the point K and changes as the normal plane section is turning about the common perpendicular n P . This statement immedi- ately follows from the above conclusion that the rate of conformity of the surfaces P and T yields interpretation in terms of radii of normal curvature R P and R T . Illustrated in Figure 4.8 is the change of the rate of conformity of the surfaces P and T due to the turning of the normal plane section about the common perpendicular n P . In Figure 4.8, only two-dimensional examples are shown, for which that same normal plane section of the surface P makes contact with different plane sections T i( ) of the generating surface T. In the example shown in Figure 4.8a, the radius of normal curvature R T ( )1 of the convex plane section T ( )1 of the surface T is positive ( R T ( )1 0> ). The con- vex normal plane section of the surface T makes contact with the convex normal plane section ( R P > 0 ) of the surface P. The rate of conformity of the generating surface T to the part surface P in Figure 4.8a is relatively low. Another example is shown in Figure 4.8b. The radius of normal curvature R T ( )2 of the convex plane section T ( )2 of the surface T is also positive ( R T ( )2 0> ). However, its value exceeds the value R T ( )1 of radius of normal curvature in the rst example ( R R T T ( ) ( )2 1 > ). This results in the rate of conformity of the surface T to the surface P (Figure 4.8a) being higher compared to what is shown in Figure 4.8b. In the next example (Figure 4.8c), the normal plane section T ( )3 of the surface T is represented with a locally attened section. The radius of normal curva- ture R T ( )3 of the attened plane section T ( )3 approaches innity ( R T ( )3 → ∞ ). Thus, the inequality R R R T T T ( ) ( ) ( )3 2 1 > > is valid. Therefore, the rate of conformity of the surface T to the surface P in Figure 4.8c is also getting higher. © 2008 by Taylor & Francis Group, LLC 110 Kinematic Geometry of Surface Machining 4.4.2 Indicatrix of Conformity of the Surfaces P and T Introduced in this section is a quantitative measure of the rate of conformity of two surfaces. The rate of conformity of two surfaces P and T indicates how the surface T is close to the surface P in differential vicinity of the point K of their contact, say how much the surface T is congruent to the surface P in differential vicinity of the point K. Quantitatively, the rate of conformity of a surface T to another surface P can be expressed in terms of the difference between the corresponding radii of normal curvature of the surfaces. In order to develop a quantitative measure of the rate of conformity of the surfaces P and T, it is convenient to implement Dupin’s indicatrices Dup(P) and Dup(T) of the surfaces P and T, respectively. It is natural to assume that the higher rate of conformity of the surfaces P and T is due to the smaller difference between the normal curvatures of the surfaces P and T in a common cross-section by a plane through the common normal vector n P . Dupin’s indicatrix Dup(P) indicates the distribution of radii of normal curvature of the surface P as it had been shown, for example, for a concave elliptic patch of the surface P (Figure 4.10). The equation of this characteristic curve for surface P (see Equation 4.37) in polar coordinates can be repre- sented in the following form: Dup P r R P P P P ( ) ( ) ( )⇒ = ϕ ϕ (4.51) where r P is the position vector of a point of the Dupin’s indicatrix Dup(P) of the surface P, and ϕ P is the polar angle of the indicatrix Dup(P). (a) n P T P K (b) P T R P R T K FIGURE 4.9 Analytical description of the geometry of contact of the surface P being machined and of the generating surface T of the cutting tool. (From Radzevich, S.P., Mathematical and Computer Mod- eling, 39 (9–10), 1083–1112, 2004. With permission.) © 2008 by Taylor & Francis Group, LLC 112 Kinematic Geometry of Surface Machining can be employed for indication of the rate of conformity of the surfaces P and T at point K. The equation of indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T is postulated of the following structure: Cnf P T r R R R R cnf P P T ( / ) ( , ) ( ) sgn ( ) ( , ) sg⇒ = + ϕ µ ϕ ϕ ϕ µ nn ( , )R T ϕ µ = +r R r R P P T T ( )sgn ( ) ( , )sgn ( , ) ϕ ϕ ϕ µ ϕ µ (4.55) where r R P P = | | is the position vector of a point of the Dupin’s indicatrix of the surface P and r R T T = | is a position vector of a corresponding point of the Dupin’s indicatrix of the surface T. Here, in Equation (4.55), the multipliers sgn ( )R P ϕ and sgn ( , )R T ϕ µ are assigned to each of the functions r P R P ( ) ( ) ϕ ϕ = and r T R T ( , ) ( , ) ϕ µ ϕ µ = just for the purpose of remaining the corresponding sign of the functions — that is, that same sign that the radii of normal curvature R P ( ) ϕ and R T ( , ) ϕ µ have. Because the position vector r P ( ) ϕ denes location of a point a P of the Dupin’s indicatrix Dup(P), and the position vector r T ( , ) ϕ µ denes location of a point a T of the Dupin’s indicatrix Dup(T), then the position vector r cnf ( , ) ϕ µ denes location of a point a C (see Figure 4.10) of the indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T. Therefore, the equality r Ka cnf C ( , ) ϕ µ = is observed, and the length of the straight-line segment Ka C is equal to the distance a a P T . Ultimately one can conclude that position vector r cnf of a point of the indi- catrix of conformity Cnf P T R ( / ) can be expressed in terms of position vectors r P and r T of the Dupin’s indicatrices Dup(P) and Dup(T). For the computation of current value of the radius of normal curvature R P ( ), ϕ the equality R P (j) = f 1.P /f 2.P can be used. Similarly, for the computation of cur- rent value of the radius of normal curvature R T ( , ) ϕ µ , the equality R T (j, m) = j 1.T /f 2.T can be employed. Use of the angle m of the surfaces P and T local rela- tive orientation indicates that the radii of normal curvature R P ( ) ϕ and R T ( , ) ϕ µ are taken in a common normal plane section through the point K. Further, it is well known that the inequalities φ 1 0 .P ≥ and φ 1 0 .T ≥ are always valid. Therefore, Equation (4.55) can be rewritten in the following form: r r r cnf P .P T .T = + - - ( )sgn ( , )sgn ϕ φ ϕ µ φ 2 1 2 1 (4.56) For the derivation of equation of the indicatrix of conformity Cnf P T R ( / ) , it is convenient to use Euler’s equation for R P ( ) ϕ (see Equation 1.31): R R R R R P P P P P ( ) sin cos . . . . ϕ ϕ ϕ = ⋅ ⋅ + ⋅ 1 2 1 2 2 2 (4.57) © 2008 by Taylor & Francis Group, LLC The Geometry of Contact of Two Smooth, Regular Surfaces 113 Here, the radii of principal curvature R P1. and R P2. are the roots of the qua- dratic equation: L R E M R F M R F N R G P P P P P P P P P P P P ⋅ - ⋅ - ⋅ - ⋅ - = 0 (4.58) Recall that the inequality R R P P1 2. . < is always observed. Equation (4.57) and Equation (4.58) allow expression of the radius of normal curvature R P ( ) ϕ of the surface P in terms of the fundamental magnitudes of the rst order E P , F P , and G P , and of the fundamental magnitudes of the second order L P , M P , and N P . A similar consideration is applicable for the generating surface T of the cutting tool. Omitting routing analysis, one can conclude that the radius of normal curvature R T ( , ) ϕ µ of the surface T can be expressed in terms of the fundamental magnitudes of the rst order E T , F T , and G T , and of the funda- mental magnitudes of the second order L T , M T , and N T . Finally, on the premises of the above-performed analysis, the following equation for the indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T can be derived: r E G L G M E G N E cnf P P P P P P P P P ( , ) cos sin sin ϕ µ ϕ ϕ = - + 2 2 22 2 1 ϕ φ sgn .P - + + - + + E G L G M E G N E T T T T T T T T T cos ( ) sin ( ) sin 2 2 2 ϕ µ ϕ µ (( ) sgn ϕ µ φ + - 2 1 .T (4.59) Equation (4.59) of the characteristic curve* Cnf P T R ( / ) is published in [7] and (in a hidden form) in [8]. Analysis of Equation (4.59) reveals that the indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T at the point K is represented with a pla- nar centro-symmetrical curve of the fourth order. In particular cases, this characteristic curve also possesses a property of mirror symmetry. Mirror symmetry of the indicatrix of conformity observes, for example, when the angle m of the local relative orientation of surfaces P and T is equal m = ±p∙ n/2, where n designates an integer number. It is important to note that even for the most general case of surface gen- eration, position vector r cnf ( , ) ϕ µ of the indicatrix of conformity Cnf P T R ( / ) is not dependent on the fundamental magnitudes F P and F T . Independence of * An equation of this characteristic curve is also known from [7] and (in a hidden form) from [8]. © 2008 by Taylor & Francis Group, LLC 114 Kinematic Geometry of Surface Machining the characteristic curve Cnf P T R ( / ) of the fundamental magnitudes F P and F T is due to the following. The coordinate angle ω P can be calculated by the formula ω P P P P F E G = arccos The position vector r cnf ( , ) ϕ µ of a point of the indicatrix of conformity Cnf P T R ( / ) is not a function of the coordinate angle ω P . Although the position vector r cnf ( , ) ϕ µ depends on the fundamental magnitudes E P , G P and E T , G T , the above analysis makes it clear why r cnf ( , ) ϕ µ is not dependent on the fun- damental magnitudes F P and F T . Two illustrative examples of the indicatrix of conformity Cnf P T R ( / ) are shown in Figure 4.11. The rst example (Figure 4.11a) relates to the cases of contact of a saddle-like local patch of the part surface P and of a convex elliptic- like local patch of the generating surface T. The second one (Figure 4.11b) is for the case of contact of a convex parabolic-like local patch of the part sur- face P and of a convex, elliptic-like local patch of the generating surface T. For both cases (see Figure 4.11), the corresponding curvature indicatrices Crv(P) and Crv(T) of the surfaces P and T are depicted as well. The imaginary (phan- tom) branches of the Dupin’s indicatrix Dup(P) for the saddle-like local patch of the part surface P are represented by dashed lines (see Figure 4.11a). Surfaces P and T can make contact geometrically but physical conditions of their contact could be violated. Violation of the physical condition of con- tact results in the surfaces P and T interfering with one another. Implemen- tation of the indicatrix of conformity Cnf P T R ( / ) immediately uncovers the interference of the surfaces, if there is any. Three illustrative examples of the violation of physical condition of contact are depicted in Figure 4.12. When the correspondence between radii of normal curvature is inappropriate, then the indicatrix of conformity Cnf P T R ( / ) either intersects itself (Figure 4.12a), or all of its diameters become negative (Figure 4.12b and Figure 4.12c). The value of the current diameter* d cnf of the indicatrix of conformity Cnf P T R ( / ) indicates the rate of conformity of the surfaces P and T in the cor- responding cross-section of the surfaces by normal plane through the com- mon perpendicular. Orientation of the normal plane sections with respect to the surfaces P and T is dened by the corresponding central angle j. For the orthogonally parameterized surfaces P and T, the equation of Dupin’s indicatrices Dup(P) and Dup(T) simplies to L x M x y N y P P P P P P P 2 2 2 1+ + = ± (4.60) L x M x y N y T T T T T T T 2 2 2 1+ + = ± (4.61) * The diameter of a centro-symmetrical curve can be dened as a distance between two points of the curve, measured along the corresponding straight line through the center of symmetry of the curve. © 2008 by Taylor & Francis Group, LLC The Geometry of Contact of Two Smooth, Regular Surfaces 117 Equation (4.59) of the indicatrix of conformity Cnf P T R ( / ) yields an equation of one more characteristic curve. This characteristic curve is referred to as the curve of minimal values of the position vector r cnf , which is expressed in terms of j. In the general case, the equation of this characteristic curve can be represented in the form r r cnf cnf (min) (min) ( )= µ . For the derivation of the equation of the character- istic curve r r cnf cnf (min) (min) ( )= µ , the following method can be employed. A given relative orientation of the surfaces P and T is specied by the value of the angle m of the surfaces P and T local relative orientation. The minimal value of r cnf (min) is observed when the angular parameter j is equal to the root ϕ 1 of equation ∂ ∂ = ϕ ϕ µ r cnf ( , ) 0 . The additional condition ∂ ∂ > 2 2 0 ϕ ϕ µ r cnf ( , ) must be satised as well. In order to determine the necessary value of the angle ϕ 1 , the equation ∂ ∂ = ϕ ϕ µ r cnf ( , ) 0 must be solved with respect to m. After sub- stituting the obtained solution µ (min) to Equation (4.48) of the indicatrix of conformity Cnf P T R ( / ) , the equation r r cnf cnf (min) (min) ( )= ϕ of the curve of minimal diameters of the characteristic curve Cnf P T R ( / ) can be derived. In a similar way, one more characteristic curve, say the characteristic curve r r cnf cnf (max) (max) ( )= ϕ , can be derived. The last characteristic curve reects the dis- tribution of the maximal values of the position vector r cnf in terms of j. 4.4.3 Directions of the Extremum Rate of Conformity of the Surfaces P and T Directions, along which the rate of conformity of the surfaces P and T is extremum (that is, it reaches either its maximum or its minimum value), are of prime importance for many engineering applications. This issue is espe- cially important when designing blend surfaces, for computation of param- eters of optimal tool-paths for the machining of sculptured surfaces on a multi-axis NC machine, for improving the accuracy of the solution to the problem of two elastic bodies in contact, and for many other applications in applied science and in engineering. Directions of the extremal rate of conformity of the surfaces P and T (i.e., the directions pointed along the extremal diameters d cnf (min) and d cnf (max) ) can be determined from the equation of the indicatrix of conformity Cnf P T R ( / ) . For convenience, Equation (4.48) of this characteristic curve is transformed and is represented in the form r r r r cnf P P .P ( , ) | cos sin |sgn | . . ϕ µ ϕ ϕ φ = + + - 1 2 2 2 2 1 11 2 2 2 2 1 . . cos ( ) sin ( )|sgn T T .T r ϕ µ ϕ µ φ + + + - (4.63) Two angles ϕ min and ϕ max specify two directions within the common tangent plane, along which the rate of conformity of the surface T to the surface P reaches its extremal values. These angles are the roots of the following equation: ∂ ∂ = ϕ ϕ µ r cnf ( , ) .0 (4.64) © 2008 by Taylor & Francis Group, LLC 118 Kinematic Geometry of Surface Machining It is easy to prove that in the general case of two sculptured surfaces in contact, the difference between the angles ϕ min and ϕ max is not equal to 0 5. π . This means the equality ϕ ϕ π min max = ±0 5 n is not observed, and in most cases, the relationship ϕ ϕ π min max ≠ ±0 5 n is valid. (Here n is an integer number.) The condition ϕ ϕ π min max .= ± 0 5 n is satised only in cases when the angle μ of the surfaces P and T local relative orientation is equal to µ π = ±0 5. n , and thus the principal directions t 1.P and t 2.P of the surface P, and the principal directions t 1.T and t 2.T of the surface T are either aligned or are directed oppositely. This enables one to make the following statement: In the general case of two sculptured surfaces in contact, directions along which the rate of conformity of two smooth, regular surfaces P and T is extremal are not orthogonal to each other. This conclusion is important for engineering applications. The solution to Equation (4.28) returns two extremal angles ϕ min and ϕ ϕ max min = + °90 . Equation (4.64) allows for two solutions ϕ min * and ϕ max * . Therefore, it is easy to compute the extremal difference ∆ ϕ ϕ ϕ min min min * = - , as well as the extremal difference ∆ ϕ ϕ ϕ max max max * = - . Generally speaking, neither the extremal difference ∆ ϕ min nor the extremal difference ∆ ϕ max is equal to zero. They are equal to zero only in particular cases, say when the angle μ of the surfaces P and T local relative orientation satises the relationship µ π = ±0 5. n . Example 1 As an illustrative example, let us describe analytically the geometry of contact of two convex parabolic patches of the surfaces P and T (Figure 4.13). In the exam- ple under consideration, the design parameters of the gear and of the shaving cutter together with the given gear and the cutter conguration yield the follow- ing numerical data for the computation. At the point K of the surfaces contact, principal curvatures of the surface P are equal: k mm P1 1 4 . = - and k P2 0 . = . Prin- cipal curvatures of the surface T are equal: k mm T1 1 1 . = - and k T2 0 . = . The angle m of the surfaces P and T local relative orientation is equal to µ = °45 . Two approaches can be implemented for the analytical description of the geometry of contact of the surfaces P and T. The rst one is based on imple- mentation of Dupin’s indicatrix of the surface of relative curvature. Another is based on application of the indicatrix of conformity Cnf P T R ( / ) of the sur- faces P and T at point K. The First Approach For the case under consideration, Equation (4.28) reduces to k k k P T R = - + 1 2 1 2 . . cos cos ( ) ϕ ϕ µ (4.65) Therefore, the equality ∂ ∂ = - + + + = k k k P T R ϕ ϕ ϕ ϕ µ ϕ µ 2 2 1 1. . sin cos sin( )cos( ) 00 (4.66) is valid. © 2008 by Taylor & Francis Group, LLC 120 Kinematic Geometry of Surface Machining half of π. Therefore, the relationship ϕ ϕ max * min * - 90° between the extre- mal angles ϕ min * and ϕ max * is observed. In the general case, directions of the extremal rate of conformity of the surfaces P and T are not orthogonal to one another. The example reveals that in general cases of two smooth, regular sculp- tured surfaces in contact, the indicatrix of conformity Cnf P T R ( / ) can be implemented for the purpose of accurate analytical description of the geom- etry of contact of the surfaces. Dupin’s indicatrix of the surface of relative normal curvature can be implemented for this purpose only in particular cases of the surface’s conguration. Application of Dupin’s indicatrix of the surface of relative curvature enables only approximate analytical descrip- tion of the geometry of contact of the surfaces. Dupin’s indicatrix of the surface of relative curvature could be equivalent to the indicatrix of con- formity only in degenerated cases of contact of two surfaces. Advantages of the indicatrix of conformity over Dupin’s indicatrix of the surface of relative curvature are that this characteristic curve is a curve of the fourth order. 4 . 4.4 Asymptotes of the Indicatrix of Conformity Cnf R (P/T) In the theory of surface generation, asymptotes of the indicatrix of confor- mity Cnf P T R ( / ) play an important role. The indicatrix of conformity could have asymptotes when a certain combination of parameters of shape of the surfaces P and T is observed. Straight lines that possess the property of becoming and staying innitely close to the curve as the distance from the origin increases to innity are referred to as the asymptotes. This denition of the asymptotes is helpful for derivation of the equation of asymptotes of the indicatrix of conformity of the surfaces P and T. In polar coordinates, the indicatrix of conformity Cnf P T R ( / ) is analyti- cally described by Equation (4.59). For convenience, the equation of this char- acteristic curve is represented below in the form of r r cnf cnf = ( , ) ϕ µ . Derivation of the equation of the asymptotes of the characteristic curve r r cnf cnf = ( , ) ϕ µ can be accomplished in just a few steps: For a given indicatrix of conformity r r cnf cnf = ( , ) ϕ µ , compose a function r cnf * ( , ) ϕ µ that is equal: r r cnf cnf * ( , ) ( , ) ϕ µ ϕ µ = 1 (4.67) Solve the equation r cnf * ( , ) ϕ µ = 0 with respect to j. The solution ϕ 0 to this equation species the direction of the asymptote. Calculate the value of the parameter m 0 . The value of the parameter m 0 is equal m g 0 1 = ∂ ∂ - ( ) ( , ) ϕ µ ϕ under the condition ϕ ϕ = 0 . © 2008 by Taylor & Francis Group, LLC The Geometry of Contact of Two Smooth, Regular Surfaces 121 The asymptote is the line through point ( , . )m 0 0 0 5 ϕ π + , and with the direc- tion ϕ 0 . Its equation is r m ( ) sin( ) ϕ ϕ ϕ = - 0 0 (4.68) In particular cases, asymptotes of the indicatrix of conformity Cnf P T R ( / ) can coincide either with the asymptotes of the Dupin’s indicatrix Dup(P) of the surface P, or of the Dupin’s indicatrix Dup(T) of the surface T, or nally with Dupin’s indicatrix Dup(P/T) of the surface of relative curvature R . 4.4.5 Comparison of Capabilities of the Indicatrix of Conformity Cnf R (P/T) and of Dupin’s Indicatrix of the Surface of Relative Curvature Both characteristic curves — that is, the indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T, and Dupin’s indicatrix Dup(P/T) of the surface of relative curvature can be used with the same goal of analytical description of the geom- etry of contact of the surfaces P and T in the rst order of tangency. Therefore, it is important to compare the capabilities of these characteristic curves. A detailed analysis of capabilities of the indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T (see Equation 4.59) and of Dupin’s indicatrix of the surface of relative curvature Dup(P/T) (see Equation 4.37) is performed. This analysis allows the following conclusions to be made. From the viewpoint of completeness and effectiveness of analytical description of the geometry of contact of two surfaces in the rst order of tangency, the indicatrix of conformity Cnf P T R ( / ) is more informative com- pared to Dupin’s indicatrix Dup(P/T) of the surface of relative curvature. It more accurately reects important features of the geometry of contact in dif- ferential vicinity of the point K. Thus, implementation of the indicatrix of conformity Cnf P T R ( / ) for scientic and engineering purposes has advan- tages over Dupin’s indicatrix of the surface of relative curvature Dup(P/T). This conclusion is directly drawn from the following: Directions of the extremal rate of conformity of the surfaces P and T that are specied by Dupin’s indicatrix Dup(P/T) are always orthogonal to one another. Actually, in the general case of contact of two sculp- tured surfaces, these directions are not orthogonal to each other. They could be orthogonal only in particular cases of the surfaces’ contact. The indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T properly species the actual directions of the extremal rate of conformity of the surfaces P and T. This is particularly (but not only) due to the fact that the characteristic curve Cnf P T R ( / ) is a curve of the fourth order, while the Dupin’s indicatrix Dup(P/T) of the surface of relative curvature is a curve of the second order. © 2008 by Taylor & Francis Group, LLC [...]... ∂2ϕ 144 Kinematic Geometry of Surface Machining Δj 0. 15 2 0.1 Δj 2 15 3 0. 05 1 ρr.cnf 1 0 .5 –10 5 0 0 0. 05 1 5 2 0.1 3 0. 15 0.2 ρr.cnf 10 –0 .5 Figure 4.20 The impact of the errors of the computations on the direction of the maximal conformity of the generating surface T of the cutting tool to the part surface P The following statement: If a surface P to be machined and the machining surface of a cutting... 4 .5. 3 Relative Characteristic Curves The considered properties of Plücker’s conoid can be employed for derivation of the equation of a planar characteristic curve for analytical description of the geometry of contact of two surfaces for the needs of the theory of surfaces generation 4 .5. 3.1 On a Possibility of Implementation of Two of Plücker’s Conoids At a first glimpse, the implementation of two of. .. the surfaces P and T make contact within the infinitely small area This kind of surfaces contact is referred to as the local -surface contact of the surfaces of the first kind As long as the second (and not higher) derivatives are considered, then the local -surface contact of surfaces of the first kind is identical to the true -surface contact of the surfaces 2 Consider line kind of contact of the surfaces...122 Kinematic Geometry of Surface Machining An accounting of the members of higher order in the equation of Dupin’s indicatrix Dup(P/T) of the surface of relative curvature does not enhance the capabilities of this characteristic curve and is useless An accounting of the members of higher order in Taylor’s expansion of the equation of Dupin’s indicatrix gives nothing... this point: In the general case of surface contact, directions of the extremal (i.e., of the maximal and of the minimal) © 2008 by Taylor & Francis Group, LLC 137 The Geometry of Contact of Two Smooth, Regular Surfaces rate of conformity of the surfaces P and T at the point K are not orthogonal to one another The directions of the extremal rate of conformity of the surfaces P and T could be orthogonal... surfaces P and T make contact within an infinitely small area This kind of surfaces contact is referred to as the local -surface contact of surfaces of the second kind As long as the second (and not higher) derivatives are considered, then the local -surface contact of the surfaces of the second kind is identical to the true -surface contact of the surfaces In differential vicinity of point K, the surfaces... normal radii of curvature of the part surface P and of the generating surface T of the cutting tool at point K © 2008 by Taylor & Francis Group, LLC 136 Kinematic Geometry of Surface Machining t2.P 90 120 C2.P 30 ϑ R(P/T) C1.P max dind 210 R(T) t2.T 60 C1.T 150 180 μ min 0 t1.P dind < 0 C2.T 330 μ R(P) 240 270 300 t1.T Figure 4.16 An example of the An R ( P / T )-relative indicatrix of the surfaces P... purpose of solving the problem of analytical description of the geometry of contact of two smooth, regular surfaces In order to develop an appropriate solution to the problem under consideration, the characteristic surface Pl R ( P/T ) that reflects summa of the corresponding normal radii of curvature of the surfaces P and T could be introduced The following matrix representation of the equation of the surface. .. single plane section of the surfaces P and T through the common unit normal vector n P Torsion of the surfaces P and T along the arc of contact is identical — that is, geodesic (relative) torsion τ g P ≡τ g T This kind © 2008 by Taylor & Francis Group, LLC 146 Kinematic Geometry of Surface Machining of surfaces contact is referred to as the true-line contact of the surfaces When two surfaces are in... the indicatrix of conformity (of both kinds) or the relative indicatrix (of both kinds), and not of any other characteristic curve, enables correct results of computation of the geometry of contact of two surfaces to be obtained 4.6.2 Impact of Accuracy of the Computations on the Desired Parameters of the Indicatrices of Conformity CnfR(P/T) All computations performed by the NC system of a numerically . Group, LLC 112 Kinematic Geometry of Surface Machining can be employed for indication of the rate of conformity of the surfaces P and T at point K. The equation of indicatrix of conformity Cnf. cases of the surface s conguration. Application of Dupin’s indicatrix of the surface of relative curvature enables only approximate analytical descrip- tion of the geometry of contact of the surfaces indicatrix Dup(P/T) of the surface of relative curvature R . 4.4 .5 Comparison of Capabilities of the Indicatrix of Conformity Cnf R (P/T) and of Dupin’s Indicatrix of the Surface of Relative Curvature Both