Kinematic Geometry of Surface Machinin Episode Episode 11 pptx

Points O1.T and O2.T are the points of the cor-responding focal surfaces f1.T and f2.T for the surface T at point M: Examples of focal surfaces constructed for a local patch of the sculp

can be generated The form transition surface is machined instead This is

due to violation of the fifth 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 Representagenera-tion 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 hΣ 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 satisfied if

and only if the following condition is satisfied at every point of the nominal

sculptured surface P:

P

a P

n P

t P n

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.)

where the position vector of a point of a nominal sculptured surface P( ) n

is designated as rP( )n ; the position vector of the corresponding point of the

actual part surface P( ) a is designated as rP( )a, the position vector of a point of

the surface of tolerance P( )t is designated as rP( )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 satisfied, 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 nP( )n ⋅[ ]h

Fulfillment of the set of six conditions of proper part surface generation is

necessary and sufficient 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 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 infinitesimal vicinity of a cutter-contact-point (CC-point) K is a vital

link for verification of whether the third necessary condition of proper PSG

is globally satisfied 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 rP( )n and rP( )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 rP( )n.

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 specified by the unit tangent vector tP 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 detecEqua-tion 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 finding those lines on the part sursur-face 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 infinitesimal 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 defined 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 C1.P and C2.P through a point M of smooth, regular

sculptured surface P Principle surfaces C1.P and C2.P are passing through

the surface P unit normal vector n P, and through the directions specified by

the principal unit tangent vectors t1.P and t2.P Principal radii of curvature

R1.P and R2.P of the surface P are measured in the principal plane sections

C1.P and C2.P Centers of curvature O1.P and O2.P of the sculptured

sur-face at point M (Figure 7.27) are located within the straight line through the

unit normal vector nP 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 ArR( / )-indicatrix (see P T Chapter 4 ).

terms of unit normal vector nP to the surface P; and in terms of corresponding

radii of principal curvature, either R1.P or R2.P (Figure 7.27):

f1.P(U V P, P)=rP(U V P, P)−R1.P⋅nP (7.41)

f2.P(U V P, P)=rP(U V P, P)−R2.P⋅nP (7.42)

Elementary substitution R1.P=k1−.P and R2.P=k2−.P yields expression of the

focal surfaces f1.P, f2.P (Equation 7.41 and Equation 7.42) in terms of principal

curvatures:

f2.P(U V P, P)=rP(U V P, P)−k−2.P⋅nP (7.43)

f1.P(U V P, P)=rP(U V P, P)−k−1.P⋅nP (7.44)

Radii of principle curvatures R1.P and R2.P in Equation (7.41) and

Equa-tion (7.42) are computed using one of the equaEqua-tions represented in Chapter 1

(Equation 1.14 and Equation 1.19)

Radii of principle curvatures R1.P and R2.P can be expressed in terms of the

mean curvature %M P and of the Gaussian curvature %G P of the surface P:

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.)

Ultimately, equations for the focal surfaces f1.P(U V P, P) and f2.P(U V P, P)

can be represented in the form

The focal surfaces f1.P and f2.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

f1.T(U V T, T), and f2.T(U V T, T) for this surface T are depicted in Figure 7.29b

In Figure 7.29, the respective lines of curvature are designated as C(1.P, C(2.P

and C(1.T, C(2.T , correspondingly Points O1.T and O2.T are the points of the

cor-responding focal surfaces f1.T and f2.T for the surface T at point M:

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.)

Focal surfaces f1.P and f2.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 verification of whether or not the third

necessary condition of proper PSG is satisfied globally, it is necessary to plot

both of the focal surfaces f1.P and f2.P for the sculptured surface P and the

similar focal surfaces f1.T and f2.T for the generating surface T of the cutting

tool in a common coordinate system

An example of the relative configuration of the focal surfaces f1.P, f2.P and

f1.T, f2.T at the point K of contact of the given surfaces P and T is illustrated

in Figure 7.30 The saddle-like (GP<0 local patch of a sculptured surface P )

is machined with a convex patch ( GT>0, MT>0 ) of the generating surface

T of the cutting tool In the case under consideration, angle µ of the local

Configuration of the focal surfaces f1.P, f2.P for the sculptured surface P relative to the focal

surfaces f1.T and f2.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.)

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 first principle planes C1.P C1.T

(the identity is due to µ =0 ) intersect the surfaces P and T The lines of the

intersection C(1.P and C(1.T are convex lines ( k1.P>0; k1.T >0) Therefore, no

problem is observed to satisfy the third necessary condition of proper PSG

in this plane section The second principle planes C2.P C2.T (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(2.P of the intersection is a concave curve The

line C(2.T of the intersection is the convex curve Because the distance KO2.T

exceeds the distance KO2.P (i.e., KO2.T >KO2.P), the principle curvatures

k2.P and k2.T correspond to each other as |k2.P|>k2.T Because the

inequal-ity |k2.P|>k2.T is valid, the third necessary condition of proper PSG in the

second principal section of the surfaces P and T is not satisfied

Summariz-ing, one can conclude that the third necessary condition of proper PSG is not

satisfied in the infinitesimal vicinity of the CC-point K (Figure 7.30).

Analysis of Table 7.1 allows for analytical expression of the criterion for

verification of whether the third necessary condition of proper PSG is

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 nP

The third condition of proper PSG could be satisfied globally when each of

the focal surfaces f1.T and f2.T is entirely located between the convex surface

P and the corresponding focal surface f1.P or f2.P Focal surfaces f1.T and f2.T

can touch one or both focal surfaces f1.P or f2.P In a similar way, location of

the focal surfaces f1.T and f2.T, for concave and for saddle-like local patches

of surface P can be specified Focal surfaces f 1.T and f2.T must not intersect

the sculptured surface P and the corresponding focal surfaces f 1.P and f2.P

for the generating surface of the form-cutting tool Otherwise, the third

nec-essary condition of proper PSG would be violated

Focal surfaces f1.P and f2.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 f1.P

and f2.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 satisfied, then

certain constraints are imposed on the actual configuration of the focal

sur-faces For the purpose of verification of accessibility of the surface P by the

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 first focal surface f1.T of the generating

sur-face T of the cutting tool is located within the straight line along the unit

normal vector nP 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 f1.T is allowed within the

straight-line segment, as well as at its endpoints Therefore, without loss of generality,

instead of two focal surfaces f1.P and f1.T , just one

CT-dependent character-istic surface )f1(U V T, T) can be employed This surface features the summa

(R1.P+R1.T) of the first principal radii of curvature

The locus of points, determined in the above way, forms the first CT-dependent

characteristic surface )f1(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

first CT-dependent characteristic surface )f1 can be expressed in terms of the

parameters rP, nP , R1.P , and R1.T:

)

f1(U V T, T)=rP(U V T, T) (− R1.P+R1.T)⋅nP (7.52)

A similar analysis can be performed for the second focal surface f2.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 )f2 can be expressed in terms of the parameters rP, nP,

R2.P, and R2.T:

)

f2(U V T, T)=rP(U V T, T) (− R2.P+R2.T)⋅nP (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 )f1 and )f2 do not intersect

the sculptured surface P, then the third necessary condition of proper PSG

is satisfied 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 )f1 and )f2 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 satisfied 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 ( f1.P, f2.P and

f1.T, f2.T) to two CT-dependent characteristic surfaces ()f1 and )f2)

The 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 sursur-face by the corresponding

CT-dependent characteristic surfaces )f1 and )f2 Therefore, every point of the

boundary curve rbc satisfies the corresponding set of two equations:

solutions to Equation (7.56) and Equation (7.57) can be significantly simplified

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

rP−sgnk1.P⋅sgn⋅k1.T⋅sgn (k1.P+k1.T)⋅nP=rP(U P,,V P) (7.58)

rP−sgnk2.P⋅sgn⋅k2.T⋅sgn (k2.P+k2.T)⋅nP=rP(U P,,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 significantly 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 R1.P and R2.P of the

surface P (and the similar principal radii of curvature R1.T and R2.T of the

surface T) uniquely specify the torus surface.

The first principal radius of curvature R1.P is equal to the radius of the

generating circle of the torus surface, and the second principal radius of

cur-vature R2.P 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= 2.P−R1.P) 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 imExam-plemented

Example 7.2

Consider machining of a torus surface P with the flat-end milling cutter

(Figure 7.31) The radius r of the generating circle of the surface P is equal to

r=50mm , and the radius R of the directing circle of the surface P is equal to

R=90mm Gaussian (curvilinear) coordinates θP and ϕP of a point on the

surface P vary in the range of 0≤θP≤180and 0≤ϕP≤360 Using

Equa-tion (7.58) and EquaEqua-tion (7.59) in the commercial software MathCAD allows

the equation

rbc P

P P

( )

sincos

θ

θθ

for the position vector rbc( )θP of a point of the boundary curve for the

CT-dependent characteristic surface

When machining the torus surface P, the milling cutter rotates about its

axis with a certain angular velocity ωT The milling cutter is traveling with

respect to the work occupying various positions T1, T2, T3, and T4 relative to

the surface P The milling cutter contacts the surface P at the corresponding

CC-points K1, K2, K3, and K4 The boundary curve rbc( )θP subdivides the

surfaces P onto the cutting-tool-accessible and onto the

cutting-tool-not-acces-sible ℜ (shadowed) regions The boundary curve rbc( )θP (see Equation 7.60)

indicates that the positions T1 and T2 of the milling cutter are feasible The

cutter position T3 is also allowed, and it is limited in its position Due

to the CC-point, K4 is located within the cutting-tool-not-accessible region

ℜ—the position T4 of the milling cutter is not feasible In that position of the

milling cutter, the third necessary condition of proper PSG is not satisfied;

thus, the surface P cannot be machined properly.

Example 7.3

Consider machining of a torus surface P with the cylindrical milling cutter

(Figure 7.32) The same surface P as that in Example 7.2 could be machined

with a cylindrical milling cutter of the radius 50 mm When machining the

torus surface P, the cutter rotates about its axis with a certain angular

veloc-ity ωT The milling cutter is traveling with respect to the work

Using Equation (7.58) and Equation (7.59) in the commercial software

MathCAD, allows the equation

rbc P

P P

( )

sincos

θ

θθ

21 7931







(7.61)

for the position vector rbc( )θP of a point of the boundary curve for the

CT-dependent characteristic surface

When machining the torus surface P, the milling cutter occupies various

positions T1, T2, T3, and T4 relative to the surface P It is contacting the

sur-face P at the CC-points K1, K2, K3, and K4 correspondingly The boundary

curve rbc( )θP subdivides the surfaces P onto the cutting-tool-accessible and

onto the cutting-tool-not-accessible ℜ (shadowed) regions The boundary

curve rbc( )θP (see Equation 7.60) indicates that the positions T1 and T2 of

the milling cutter are allowed The position T3 of the cutting tool is also

allowed, and it is the limited cutter location Because the CC-point K4 is

located within the cutting-tool-not-accessible region ℜ, the position T4 of the

milling cutter is not allowed In that position of the milling cutter, the third

necessary condition of proper PSG is not satisfied; thus, the surface P cannot

be machined properly

7.3.1.4 Cases of Local-Extremal Tangency of the Surfaces P and T

Possible kinds of contact of the surfaces P and T are investigated in Chapter 4

In the theory of surface generation, pure local-extremal tangency of the

sur-faces is out of practical interest However, this kind of surface contact could

be observed in the form of quasi-kinds of surface contact when relative

dis-placements of the contacting surfaces are maximal

Local-extremal kinds of contact of the surfaces P and T are observed when

the equality k P= −k T is valid Under such a scenario, the focal surfaces f1.P, f2.P

and f1.T, f2.T (or the two CT-dependent characteristic surfaces )f1 and )f2) are

not helpful for solving the problem of verification of the global satisfaction of

the third necessary condition of proper PSG In the case under consideration,

another tool must be implemented

On the premises of the above analysis, it is recommended to use

deriva-tives of the corresponding functions In this way, the derivative-focal-surfaces

(DF-surfaces) are introduced [19] The DF-surfaces are analytically described

by the following equation:

1 2 , ( )

where n designates the smallest integer number under which any

uncer-tainty in global satisfaction of the third necessary condition of proper PSG

does not arise, and ∂n P T

P T n R dC

1 2

1 2 , ( ) , ( )

( designates the derivative of R1 2, ( )P T in the tion of C(n P T

direc-1 2 , ( )

In the cases under consideration, it is necessary to determine the DF-surfaces

for the sculptured surface P:

1

.

P P

n P P

R dC

2

.

P P

n P P

R dC

It is also necessary to determine the similar DF-surfaces for the generating

surface T of the cutting tool:

1

.

T T

n T T

R dC

2

.

n T T

R dC

In order to globally satisfy the third necessary condition of proper PSG,

the shape, the parameters, and the relative disposition of the DF-surfaces

f1.P, f2.

P, and f1.

T, f2.

T must be correlated with the shape, the parameters,

and the relative location of the surfaces P and T, say in the way similar to

that considered above

Similarly, the derivative-cutting-tool-dependent (DCT-dependent)

charac-teristic surfaces can be introduced:

n T T n

R dC

n T T n

R dC

R

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 for partitioning the sculptured surface P onto the

cutting-tool-accessible and onto the cutting-tool-not-accessible regions

7.3.2  implementation of R-Surfaces

A proper correspondence between the normal curvatures k P of the part

sur-face P and the corresponding normal curvatures k T of the generating surface

T of the cutting tool is one of the major prerequisites for proper generation of

the surface P in the differential vicinity of the CC-point.

7.3.2.1 Local Consideration

The geometry of contact of the surfaces P and T can be analytically described

by the indicatrices of conformity Cnf P T R ( / ) and Cnf P T k( / ) [16,17,20] and

by the AnR( / )-indicatrix,P T  or Ank( / ) -indicatrix [13] (see P T Chapter 4) For

the purpose of verification of global satisfaction of the third necessary

condi-tion of proper PSG implementacondi-tion, these characteristic curves have proven

to be convenient in CAD/CAM applications

It is critically important to stress that all of the characteristic curves

Cnf P T R ( / ) , Cnf P T k( / ) , AnR( / ) , and AnP T k( / ) specify the same direc-P T

tions of the extremal rate of conformity of the surfaces P and T at the current

CC-point This important property of the characteristic curves is illustrated by

an example of machining of a bicubic Bezier surface P (Figure 7.33)

The matrix equation for a bicubic Bezier patch P that is defined by a 4 × 4

array of points is as follows [6]:

where [ ] [PP pi j i,] , ,

j

= = 1 4

1K4, and position vectors of the control points are denoted

as pi j, In Equation (7.69), the bicubic patch is expressed in a form similar to

the Hermite bicubic patch [6]

The matrix [ ]P contains the position vectors for points that define the

charac-teristic polyhedron and, therefore, the Bezier surface patch In the Bezier

formu-lation, only four corner points p11, p41, p14, and p44 actually lie on the surface

patch The points p21, p31, p12, p13, p42, p43, p24, and p34 control the slope of

the boundary curves The remaining four points p22, p32, p23, and p33 control

the cross-slopes along the boundary curves in the same way as the twist

vec-tors of the bicubic patch The Bezier surface is completely defined by a net of

design points describing two families of Bezier curves on the surface

In Figure 7.33, the direction of the minimal diameter d cnfmin of the indicatrix

of conformity Cnf P T R( / ) aligns with the direction tmaxcnf at which the rate of

conformity of the surfaces P and T is maximal.

 The equation of the characteristic curves AnR( / ), and AnP T k( / ) is derived by Radzevich P T

[13] on the premises of the equation of the well-known surface—namely, of the surface of

Plücker’s conoid (see Chapter 4).

In order to satisfy the third necessary condition of proper PSG, all diameters

d cnf =2r cnf of this characteristic curve must be nonnegative — that is, in all

directions through the point K, the relationship r cnf ≥0 must be satisfied

The indicatrix of conformity Cnf P T R( / ) (see Equation 4.59) yields a

con-clusion on the actual kind of contact of the surfaces P and T at a current

CC-point (Figure 7.34a)

When the surfaces makes a regular point contact, then the minimal

diam-eter d cnfmin of the indicatrix of conformity Cnf P T R ( / ) is positive ( dmincnf >0), as

depicted in Figure 7.34b A CC-point of that kind cannot be a point of the

boundary curve rbc

0 30

60

90 120

( ) a

P

T K

60

90 120 150

Cnf R (P/T )

t2.P

t1.P t1.P

(b)

0 30

60

90 120

Examples of satisfaction and of violation of the third necessary condition of proper part

sur-face generation [The current cutter-contact-point K in (c) represents a point of the boundary

curve that subdivides the surface P into the cutting-tool-accessible and the

cutting-tool-not-accessible regions.]