Kinematic Geometry of Surface Machinin Episode Episode 5 ppsx

The Geometry of Contact of Two Smooth, Regular Surfaces 117Equation 4.59 of the indicatrix of conformity Cnf P T R / yields an equation of one more characteristic curve.. Advantages of

106 Kinematic Geometry of Surface Machining

Crv(P) of a smooth, regular surface P are listed along with the corresponding

sign of the mean MP and of the Gaussian GP curvatures (Figure 4.6):

Convex elliptic (MP > 0, GP > 0) in Figure 4.6a

Concave elliptic (MP < 0, GP < 0) in Figure 4.6b

Convex umbilic (MP > 0, GP > 0) in Figure 4.6c

Concave umbilic (MP < 0, GP < 0) in Figure 4.6d

Convex parabolic (MP > 0, GP = 0) in Figure 4.6e

Concave parabolic (MP < 0, GP = 0) in Figure 4.6f

Quasi-convex hyperbolic (MP > 0, GP < 0 ) in Figure 4.6g

Quasi-concave hyperbolic (MP < 0, GP < 0) in Figure 4.6h

Minimal hyperbolic (MP = 0, GP < 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 infinity

4.3.8  introduction of the Jrk (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 = dVP/

dU P at a given point of a sculptured surface P, one can rewrite the equation

E dU

P P P

2 1

2

22.

. FF dU dV P P P+G dV P P2 (4.46)for normal curvature in the form of

2 2

In the particular case when L M N P: P: P=E F G P: P: P, the normal curvature

k Pis independent of h Surface points with this property are known as

umbilic points and flatten points

In general cases when k P changes as h changes, the function kP =k P( )η is

a rational quadratic form, as illustrated in Figure 4.7 The extreme values k 1.P

and k2.P of k P =k P( )η occur at the roots η1 and η2 of

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) definition 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 first 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 nP 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 nP 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 concon-vex

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

first example ( R T( ) 2 >R T( ) 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 flattened section The radius of normal

curva-ture R T( )3 of the flattened plane section T( ) 3 approaches infinity ( R T( ) 3 → ∞)

Thus, the inequality R T( )3 >R T( )2 >R T( )1 is valid Therefore, the rate of conformity

of the surface T to the surface P in Figure 4.8c is also getting higher.

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 nP

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 P( )ϕP = R P( )ϕP (4.51)

where r P is the position vector of a point of the Dupin’s indicatrix Dup(P) of

the surface P, and ϕP is the polar angle of the indicatrix Dup(P).

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.)

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 cnf( , )ϕ µ = R P( ) sgnϕ R P( )ϕ + R T( , ) sgϕ µ nnR T( , )ϕ µ

=r P( )sgnϕ R P( )ϕ +r T( , )sgnϕ µ R T( , )ϕ µ (4.55)

where r P = | | is the position vector of a point of the Dupin’s indicatrix of the R P

surface P and r T= |R 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 sgnR T( , )ϕ µ are assigned to each of the functions r P( )ϕ = R P( ) ϕ and

r T( , )ϕ µ = R T( , ) 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 defines location of a point a P of the

Dupin’s indicatrix Dup(P), and the position vector r T( , )ϕ µ defines location of

a point a T of the Dupin’s indicatrix Dup(T), then the position vector r cnf( , )ϕ µ

defines 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 cnf( , )ϕ µ =Ka C

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) = f1.P/f2.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) =

j1.T/f2.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 φ1.P≥0 and φ1.T≥0 are always

valid Therefore, Equation (4.55) can be rewritten in the following form:

r cnf =r P( )sgnϕ φ2 -.P+r T( , )sgnϕ µ φ2 -.T (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):

The Geometry of Contact of Two Smooth, Regular Surfaces 113

Here, the radii of principal curvature R1.P and R2.P are the roots of the

Recall that the inequality R1.P<R2.P 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 first 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 first 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

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].

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 ωPcan be calculated by the formula

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 ωP 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 first 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 sursur-face 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 defined 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) simplifies to

L x P P2+2M x y P P P+N y P P2 = ±1 (4.60)

L x T T2+2M x y T T T +N y T T2 = ±1 (4.61)

 The diameter of a centro-symmetrical curve can be defined as a distance between two points

of the curve, measured along the corresponding straight line through the center of symmetry

of the curve.

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 cnf(min)=r cnf(min)( )µ For the derivation of the equation of the

character-istic curve r cnf(min)=r cnf(min)( )µ , the following method can be employed

A given relative orientation of the surfaces P and T is specified 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

ϕ1 of equation ∂ϕr cnf( , ) 0 The additional condition ϕ µ = ∂

ϕ r cnf( , )ϕ µ must

be satisfied as well In order to determine the necessary value of the angle

ϕ1, the equation ∂ ϕr cnf ( , ) 0 must be solved with respect to m After sub-ϕ µ =

stituting the obtained solution µ(min) to Equation (4.48) of the indicatrix of

conformity Cnf P T R ( / ), the equation r cnf(min)=r cnf(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 cnf(max) =r cnf(max)( )ϕ , can be derived The last characteristic curve reflects 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

Two angles ϕmin and ϕmax 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:

∂

∂ϕr cnf( , )ϕ µ =0 (4.64)

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 ϕmin and ϕmax is not equal to 0 5 π This

means the equality ϕmin-ϕmax= ±0 5 n is not observed, and in most cases, the π

relationship ϕmin-ϕmax≠ ±0 5 n is valid (Here n is an integer number.) The π

condition ϕmin=ϕmax±0 5 n is satisfied only in cases when the angle μ of the π

surfaces P and T local relative orientation is equal to µ= ±0 5 n, and thus the π

principal directions t1.P and t2.P of the surface P, and the principal directions

t1.T and t2.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 ϕmin and

ϕmax=ϕmin+ °90 Equation (4.64) allows for two solutions ϕmin and ϕmax

Therefore, it is easy to compute the extremal difference ∆ min =ϕmin-ϕmin,

as well as the extremal difference ∆ max=ϕmax-ϕmax

Generally speaking, neither the extremal difference ∆ min nor the extremal

difference ∆ 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

satisfies the relationship µ= ±0 5 πn.

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 configuration 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: k1.P =4mm- 1 and k2.P =0

Prin-cipal curvatures of the surface T are equal: k1.T=1mm- 1 and k2.T=0 The angle

m of the surfaces P and T local relative orientation is equal toµ = °45

Two approaches can be implemented for the analytical description of the

geometry of contact of the surfaces P and T The first 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

kR =k1 Pcos2ϕ-k1 Tcos (2ϕ µ+ ) (4.65)Therefore, the equality

120 Kinematic Geometry of Surface Machining

half of π Therefore, the relationship ϕmax-ϕmin 90° between the

extre-mal angles ϕmin and ϕmax 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 configuration 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 infinitely

close to the curve as the distance from the origin increases to infinity are

referred to as the asymptotes This definition 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 cnf =r cnf( , )ϕ µ

Derivation of the equation of the asymptotes of the characteristic curve

r cnf =r cnf( , )ϕ µ can be accomplished in just a few steps:

For a given indicatrix of conformity r cnf =r cnf( , )ϕ µ, compose a function

Solve the equation r cnf ( , )ϕ µ =0 with respect to j The solution ϕ0 to this

equation specifies the direction of the asymptote

Calculate the value of the parameter m0 The value of the parameter m0 is

The Geometry of Contact of Two Smooth, Regular Surfaces 121

The asymptote is the line through point (m0,ϕ0+0 5 )π , and with the

direc-tion ϕ0 Its equation is

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 finally

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 first 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 first 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 reflects 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 scientific 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 specified 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 specifies 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

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 more for proper analytical description

of the geometry of contact of two surfaces in the first order of tangency

Principal features of the equation of this characteristic curve cause a

principal disadvantage of Dupin’s indicatrix Dup(P/T) The disadvantage

above is inherent to Dupin’s indicatrix, and it cannot be eliminated

4.4.6  important Properties of the indicatrix of Conformity Cnf R  (P/T)

Analysis of Equation (4.59) of the indicatrix of conformity Cnf P T R( / ) reveals

that this characteristic curve possesses the following important properties:

The indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T is a

pla-nar characteristic curve of the fourth order It possesses the property

of central symmetry, and in particular cases it also possesses the

property of mirror symmetry

The indicatrix of conformity Cnf P T R ( / ) is closely related to the surfaces’ P

and T second fundamental forms φ 2.P and φ2.T This characteristic curve

is invariant with respect to the kind of parameterization of the surfaces P

and T, but its equation does The last is similar in much to an indicatrix

of conformity Cnf R (P/T) is represented in different reference systems A

change in the surfaces’ P and T parameterization leads to changes in the

equation of the indicatrix of conformity Cnf P T R( / ), while the shape and

parameters of this characteristic curve remain unchanged

The characteristic curve Cnf P T R( / ) is independent of the actual value of

the coordinate angle ωP that makes the coordinate lines U P and V P

on the part surface P It is also independent of the actual value of the

coordinate angle ωT that makes the coordinate lines U T and V T on

the generating surface T of the cutting tool However, parameters of

the indicatrix of conformity Cnf P T R( / ) are dependent upon the angle

m of the surfaces P and T local relative orientation Therefore, for the

given pair of surfaces P and T, the rate of conformity of the surface

varies correspondingly to variation of the angle m, while the surface T

is spinning around the unit vector of the common perpendicular

4.4.7  The Converse indicatrix of Conformity of the Surfaces P and T 

in the First Order of Tangency

For Dupin’s indicatrix Dup(P/T) of the surface of relative curvature, a

cor-responding inverse Dupin’s indicatrix Dup P T k( / ) exists Similarly, for the

indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T, a corresponding

converse indicatrix of conformity Cnf P T k( / ) exists This characteristic curve

The Geometry of Contact of Two Smooth, Regular Surfaces 123

can be expressed directly in terms of the surfaces’ P and T normal

curva-tures k P and k T:

Cnf P T k( / )⇒r cnf cnv( , )ϕ µ = | ( )| sgnk P ϕ ⋅ Φ- 2 P- |k T( , )| sgnϕ µ ⋅ Φ- 2 T (4.69)

For derivation of the equation of the converse indicatrix of conformity

Cnf P T k( / ), the Euler’s formula for a surface normal curvature is used in the

following representation:

k P( )ϕ =k1.Pcos2ϕ+k2.Psin2ϕ (4.70)

k T( , )ϕ µ =k1.Tcos (2ϕ µ+ )+k2.Tsin (2 ϕ µ+ ) (4.71)

In Equation (4.70) and Equation (4.71), the principal curvatures of the part

surface P are designated as k1.P and k2.P , and k1.T and k2.T designate the

principal curvatures of the generating surface T of the cutting tool.

After substitution of Equation (4.70) and Equation (4.71) into Equation (4.69),

one can come up with the equation for the converse indicatrix of conformity

Cnf P T k ( / ) of the surfaces P and T in the first order of tangency:

where principal curvatures k1.P , k2.P and k1.T , k2.T can be expressed in terms

of the corresponding fundamental magnitudes E P , F P , G P of the first order

and L P , M P , N P of the second order of the part surface P, and in terms of the

corresponding fundamental magnitudes E T , F T , G T of the first order and

L T , M T , N T of the second order of the generating surface T of the cutting

tool Following this, Equation (4.72) of the inverse indicatrix of conformity

Cnf P T k( / ) can be cast into the form similar to Equation (4.59) of the ordinary

indicatrix of conformity Cnf P T R ( / ) of the surfaces P and T.

Similar to the indicatrix of conformity Cnf P T R( / ) , the characteristic curve

Cnf P T k( / ) also possesses the property of central symmetry In particular

cases of the surface contact, it also possesses the property of mirror

sym-metry Directions of the extremal rate conformity of the surfaces P and T are

orthogonal to one another only in degenerated cases of the surfaces contact

Equation (4.72) of the converse indicatrix of conformity Cnf P T k( / ) is

con-venient for implementation when the surface P, or the surface T, or both

have points or lines of inflection In the points or lines of inflection, radii of

normal curvature R P T( ) of the surface P(T) are equal to infinity This causes

indefiniteness when computing the position vector r cnf( , )ϕ µ of the

charac-teristic curve Cnf P T R( / ) Equation (4.72) of the converse indicatrix of

confor-mity Cnf P T k( / ) is free of these disadvantages and is therefore recommended

for practical applications

124 Kinematic Geometry of Surface Machining

4.5  Plücker’s Conoid: More Characteristic Curves

More characteristic curves for the analytical description of the geometry of

contact of two smooth, regular surfaces in the first order of tangency can be

derived on the premises of Plücker’s conoid [9]

4.5.1  Plücker’s Conoid

Several definitions for Plücker’s conoid are known First, Plücker’s conoid is

a smooth, regular, ruled surface A ruled surface is sometimes also called the

cylindroid, which is the inversion of the cross-cap.

Plücker’s conoid can also be considered as an example of a right conoid

A ruled surface is called a right conoid if it can be generated by moving a

straight line intersecting a fixed straight line such that the lines are always

perpendicular

As with the cathenoid, another ruled surface, Plücker’s conoid must be

reparameterized to see the rulings Illustrative examples of various Plücker’s

conoids are considered in [10]

4.5.1.1  Basics

The ruled surface can be swept out by moving a line in space; therefore, it has

a parameterization of the following form:

x( , )u v =b( )u +v u( ) (4.73)

where b is the directrix (also referred to as the base curve) and v is the director

curve The straight lines are the rulings The rulings of a ruled surface are

asymp-totic curves Furthermore, the Gaussian curvature on a ruled, regular surface is

nonpositive at all points The surface is known for the presence of two or more

folds formed by the application of a cylindrical equation to the line during this

rotation This equation defines the path of the line along the axis of rotation

4.5.1.2  Analytical Representation

For the Plücker’s conoid, von Seggern [20] gives the general functional form as

a x2+b y2-z x2-z y2=0 (4.74)whereas Fischer [3] and Gray [4] give

=+

2

 Plücker’s conoid is a ruled surface, bearing the name of famous German mathematician and

physicist Julius Plücker (1802–1868), who is known for his research in the field of a new

geom-etry of space [9].

The Geometry of Contact of Two Smooth, Regular Surfaces 125

Another form of Cartesian equation for a two-folds Plücker’s conoid is

The equation above yields the following matrix representation of nonpolar

parameterization of the Plücker’s conoid:

The Plücker’s conoid could be represented by the polar parameterization:

rpc( , ) [ cosrθ = r θ rsinθ 2cos sinθ θ 0 ]T (4.77)

A more general form of the Plücker’s conoid is parameterized below, with

“n” folds instead of just two A generalization of Plücker’s conoid to n folds

is given by the following [4]:

rpc( , ) [ cosr θ = r θ rsinθ sin( ) ]nθ 0 T (4.78)

The difference between these two forms is the function in the z axis The

polar form is a specialized function that outputs only one type of

curva-ture with two undulations; the generalized form is more flexible with the

number of undulations of the output curvature being determined by the

value of n.

Cartesian parameterization of the equation of the multifold Plücker’s

conoid (see Equation 4.78) therefore gives [21]

1+

-≤ -≤

(4.79)

The surface appearance depends upon the actual number of folds [10]

In order to represent the Plücker’s conoid as a ruled surface, it is sufficient

to represent Equation (4.78) in the form of Equation (4.79):

rpc r

r r n

r

( , )

cossinsin( )

coθ

θθθ

sssincos sin cos sin

θθ

r

20

0020

θθ00

(4.80)

126 Kinematic Geometry of Surface Machining

Taking the perpendicular plane as the xy plane and taking the line to be

the x-axis gives the parametric equation [4]:

(4.81)

The equation in cylindrical coordinates [21] is z a= cos( )nθ , which

simpli-fies to z a= cos 2θ if n = 2.

4.5.1.3  Local Properties

Following Bonnet’s theorem (see Chapter 1), local properties of Plücker’s

conoid could be analytically expressed in terms of the first and of the second

fundamental forms of the surface For practical application, some useful

aux-iliary formulas are also required

The first and the second fundamental forms [21] of Plücker’s conoid could

be represented as

φ1⇒d s2=dρ2+(ρ2+n a2 2sin ( ))2 nθ dθ2 (4.82)

φ2⇒n a θ ρ- ρ θ θ θ

H[sin( )n d n cos( ) ]n d d (4.83)

Asymptotes are given by the equation ρn=k a nsin(nθ) They strictly

cor-relate to Bernoulli’s lemniscates [21]

For the simplified case of Plücker’s conoid n = 2, the first and the second

fundamental forms reduce to [21]

=0, = -2 cos2θ, = -4 ρsin2θ (4.87)

Because the discussion of auxiliary formulas that follows is limited to the

case of n = 2, auxiliary formulas for further reference would be helpful.