A possibility of application of Plücker's conoid for mathematical modeling of contact of two smooth regular surfaces in the first order of tangency

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Mathematical and Computer Modelling 42 (2005) 999-1022 www.elsevier.eom/locate/mcm

A Possibility of A p p l i c a t i o n of Pliicker's C o n o i d for M a t h e m a t i c a l M o d e l i n g of C o n t a c t of T w o S m o o t h R e g u l a r Surfaces in t h e First Order of T a n g e n c y S.

P. RADZEVICH E a t o n Corp. 31900 S h e r m a n Ave. M a d i s o n Heights, MI 48071, U.S.A. sradzevich©tractech,

com

st e p h e n _ r a d z e v i c h O h o t m a i l ,

com

(Received July 2004; revised and accepted January 2005) Abstract

The purpose of the paper is to investigate a possibility of application of Plficker's conoid in engineering geometry of surfaces. Ttle research reveals that Pliicker's conoid itself is a complex 3D characteristic image. On the premises of Pliicker's conoid, the 2D characteristic curves of novel kind are introduced. We refer to these characteristic curves as to the AuR(P)-indicatrix and Ank(P)-indicatrix of the surface P. The characteristic curves A u R ( P ) and A u k ( P ) in many cases are similar to the corresponding Dupin's indicatrices D u p R ( P ) and D u p k ( P ). For two interacting surfaces the developed approach yields introducing of the Pliicker's relative indicatrices PIR(P/T) and PI k (P/T). Finally, the 3D characteristic curves PtR(P/T ) and Plk (P/T) reduced to the planar characteristic curves ArtR(P/T) and Ar~k(P/T), i.e., to the so-called AuR(P/T )- and Auk(P/T )relative indicatrices. The characteristic curves AuR(P/T) and Auk(P/T) could be employed, (a) for analytical description of local topology of composite surfaces along the linkage curve; (b) for developing of the optimal tool-paths for sculptured surface machining on multiaxis NC machine, as well as for analytical description of conditions of proper sculptured surface machining; (c) for improving accuracy of solution to the problem of contact of two elastic bodies, etc. These characteristic curves reported in the paper in many cases are similar to the indicatrices of conformity CnfR (P/T) and Cnfk (P/T) of two smooth regular surfaces earlier developed by the author. The obtained results perfectly correlate with the results obtained by other researchers, as well as with the results earlier developed by the author. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Smooth of conformity.

surface, Plficker's conoid, Pliicker's indicatrix, Dupin's indicatrix, Indicatrix

1. I N T R O D U C T I O N Pliicker's conoid [1] is a well-known ruled surface. This surface is bearing name of the famous German mathematician and physicist Julius Plficker (1802-1868) known for his research in the It is a pleasure to thank the anonymous referees for their careful reading of the original manuscript and their very constructive remarks. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2005.01.033

Typeset by fl,~4S-TEX

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S . P . RADZEVICH

field of a new geometry of space [2]. A brief analysis reveals that Pliicker's conoid could be applied in engineering geometry of surfaces. It yields introducing of planar characteristic curves that are useful for C A D / C A M applications [3,4], for mechanical engineering [5,6], and for other applications. 2. L I T E R A R Y

SURVEY

Plficker's conoid is considering below from the prospective of application in mechanical engineering: (a) for designing of composite surfaces, (b) for developing of the optimal kinematics of sculptured surfaces machining on multiaxis NC machine, (c) for resolving problems that relate to stress analysis in contact of two elastic bodies, and (d) for visualization results of solution to the above problems. Some of the latest results in the field are reported in [7]. During past decades the Pliicker's conoid is investigated by many researchers (see [8-10] and others). Examples of practical application of Plficker's conoid for solving problems that relate to engineering geometry of surfaces could be found out in [11-14].

3. T H E P L U C K E R ' S

CONOID

Pliicker's conoid is a smooth regular ruled surface [15]. A ruled surface sometimes is also called the cyIindroid, which, from another hand is the inversion of the cross-cap [16].

4,

4,

(~)

(b)

5 4,

4

(c)

(d)

F i g u r e 1. E x a m p l e s of Pliicker's conoid w i t h various n u m b e r of folds.

Possibility of Application of Pliicker's Conoid

1001

Pliicker's conoid can also be considered as an example of right conoid [15]. 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 [17]. DEFINITION 1. The Pliicker's conoid is a ruled surface formed by rotating a straight line about the connecting axis, and by moving the straight line up-and-down the axis.

As with the cathenoid, another ruled surface, Pliicker's conoid must be reparameterized to see the rulings. The illustrative examples of various Pliicker's conoids are plotted in Figure 1. 3.1. B a s i c s Ruled surface can be swept out by moving a line in space and therefore, has a parameterization of the form, x(u, v) = b(u) + v 3(u), (1) where b is called the directrix (also called the base curve) and ~ is the director curve [15]. The straight lines themselves are called rulings. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled regular surface is everywhere nonpositive. 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 [18].

3.2. Analytical Representation For the Pliicker's conoid von Seggern [19] gives the general functional form as

(2)

ax 2+by ~ - z x 2 - z y 2 = 0 , whereas Fischer [20] and Gray [21] give

2xy z - x2 + y2'

(3)

Another form of Cartesian equation z a((x 2 _ y2)/(x2 + y2)) for two folds Pliicker's conoid is known as well [22]. The last equation yields the following matrix representation of nonpolar parameterization of the Pliieker's conoid [1], rpc(U, v) =

[u

v

2uv

u2+v2

1

lT

(4)

The Pliicker's conoid could be represented by the polar parameterization,

rp~(r,0) = J r

cos0

r sin0

2cos0sin0

I]T,

(5)

A more general form of the Pliicker's conoid is parameterized below, with "n" folds instead of just two [1]. A generalization of Pliieker's conoid to n folds is given by [21],

rpc(r,O)=[r eosO r s i n 0

sin(n0)

1] T

(6)

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 curvature with two undulations while the generalized form is more flexible with the number of undulations of the outputted curvature being determined by the value of "n". Cartesian parameterization of equation of the multifold Pliicker's eonoid (see equation (6)) therefore gives [22],

E

o
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S.P. RADZEVICH

T h e surface a p p e a r a n c e depends upon the actual n u m b e r of folds. T h e illustrative examples of the Pliicker's conoid (Figure 1) relate to the surfaces with n = 2, n = 3, n = 4, and n = 5. In order to represent the Pliicker's conoid as a ruled surface, it is sufficient to represent the above equation (6) in the form of equation (1),

Lljrrc°s°lI c°s°] L ° l Ic°i°] | r sin0 |

r sin0

rpc(r' 0 ) = |sin(nO)| =

0

2cos0sin0

=

si 0

2cos0sin0

1

+r

.

(8)

0

Taking the perpendicular plane as the xy-plane and take the line to be the x-axis gives the parametric e q u a t i o n [21],

rpc[Veosv(u)

vsinv(u) h(u)

1] T

(9)

E q u a t i o n in cylindrical coordinates [22]: z = a cos(n0), which simplifies to z = a cos 20 if n = 2. T h e interested reader is referred to [1,18,22] for details on analytical representation of the Plucker's conoid, as well as on visualization of the surface.

3.3. L o c a l P r o p e r t i e s B o n n e t was the first (1867) to prove [23] t h a t the specification of the first and of the second f u n d a m e n t a l forms determines a unique surface if the Gauss' equation (that is k n o w n as his theorema egregium) and the Codazzi-Mainardi's relations of compatibility are satisfied, and t h a t two surfaces t h a t have identical first and second f u n d a m e n t a l forms m u s t be congruent (or s y m m e t rical). This B o n n e t ' s t h e o r e m is essential justification for considering the differential g e o m e t r y of surfaces in connection with the six f u n d a m e n t a l magnitudes. Local properties of the Pliicker's conoid could be analytically described by the first and by the second f u n d a m e n t a l forms of the surface. For practical application some useful auxiliary formulas are also required. T h e first and the second f u n d a m e n t a l forms [22] of the Pliicker's conoid could be represented

as

~1 =~ ds2 = d P 2 + (p2 _t_ n2a2 sin2 (nO)) dO 2,

(10)

(11)

n a [sin (nO) dp - n p cos (nO) dO] dO.

A s y m p t o t e s are given by the equation p~ = k a ~ sin(n0). T h e y strictly correlate to Bernoulli's lenmiscates [22]. For the simplified case of the Pliicker's conoid n = 2, the first and the second fundamental forms reduce to [22] ~1 ~ d s 2 = d p2 _4_ (f12 _4_4a 2 cos 2 2 0 d 02, E = 1,

F = 0,

G = p2 + 4a 2 cos 2 20,

H = v/-G,

4a

i~2 ~ ----ff [sin2Odp - npcos2OdO] dO, L =0,

M-

2 a cos20 H

'

N -

4 a p sin20 H

(12) (13) (14) (15)

Due to the consideration below is limited to the case of n = 2, auxiliary formulas for references would be helpful.

Possibility of Application of Pliicker's Conoid

1003

3.4. A u x i l i a r y F o r m u l a s At u = u0, v = v0 the tangent to the surfaces is parameterized by [1]:

U+Uo

rpc(U, v) =

v +vo 2 ( - u u 2 v o 4- u v 3 4- u o v 2 ( - v 4- vo) 4- u3(v 4- vo))

(16)

~8 + vg 1 The surface normal is its double line (see Figure 1) [24]. The infinitesimal area of a patch on the surface is given by [1],

~ ds=

~1+ 4 ( u -(u2v)2(u + v2) + v2)3 du dr.

(17)

Gaussian curvature of the Pliicker's conoid could be computed from [1], G (11, v) =

(18)

4 (u 4 - v4) 2 (u 6 -/- V4 (4 + v ~) + u2v 2 ( - 8 + 3v 2) + l/,4 (4 + 3v2)) ~'

Mean curvature of the Pliicker's conoid is equal to [1], 4uv

M (u, v) =

(19)

(u 2 4-v2) 2 (1 + 4 ( u - v ) 2 (u + v)2/(u 2 4-v2)3) a/2.

4. O N A N A L Y T I C A L DESCRIPTION OF LOCAL TOPOLOGY OF THE SMOOTH REGULAR SURFACE

P

For the needs of engineering application, the following characteristics of smooth regular sur-

face P (a) (b) (c)

are of prime importance: tangent plane to the surface P, unit normal n p to the surface P, and the surface P principal curvatures kl.p and k2.p, as well as the surface P normal curvature kp at the prespecified direction.

The Pliicker's conoid could be used for visualization of distribution of the surface P normal curvature at a given point. The corresponding Plficker's conoid can be determined at every point of smooth regular surface P. The surface P normal unit vector n p can be employed as the axis of the Pliicker's conoid. The rulings are the straight lines that intersect z-axis at right angle. The generating straight-line segments of the Pliicker's conoid are always parallel to the tangent plane to the surface P at the point, at which the Pliicker's conoid is erected. In consideration below, other applications of the tangent plane to the surface P are of importance as well. Consequently, the above consideration yields the natural way for connecting the Plticker's conoid to the surface P itself.

4.1. Preliminary R e m a r k s An example of application of Pliicker's conoid is given by Struik [24]. He considers a cylindroid, which is represented by the locus of the curvature vectors at a point P of a surface belonging to all curves passing through P, Z (X 2 4- y 2 ) = [gl.pX 2 _~ ]g2.py2.

(20)

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S . P . RADZEVICH

Here, in equation (20), is designated k l . g and k 2 . p a r e the principal curvatures of the surface P. It is assumed by convention 1 that kl.p > k2.p. The curvature vector is defined in the following way (See [24, p. 131). According to [24], we can introduce a proportionality factor kp such that kp -

dtp

d~ - kpnp.

(21)

The vector k p = ~ expresses the rate of change of the tangent when we proceed along the curve. It is called the curvature vector. The factor kp is called the curvature; ]kp] is the length of the curvature vector. Although the sense of n p may be arbitrary chosen, that of ~ is perfectly determined by the curve, independent of its orientation; when S changes sign, t p also changes sign. When n p (as it often done) is taken in the sense of S, then ~p is always positive, but we shall note adhere to this convention. 4.2. T h e P l i i c k e r ' s C o n o i d In order to develop an appropriate graphical interpretation of the Plficker's conoid P1R(P) of the surface P, let's consider a smooth regular patch of the part surface P that is given by vector equation, r. = r.(Up, v.), (22) where r p position vector of a point of the surface P, Up and Vp curvilinear (Gaussian) coordinates on P. From prospective of natural connecting of the Pliicker's conoid to the surface P itself, axis of the Pliicker's conoid P I R ( P ) aligns to the unit normal vector n g to the surface P at K. The unit normal vector is equal to n p = u p . v p . Here is designated u p = U p / I U p I , v g = V p / I V p I. The vectors U p and V p are tangent to corresponding coordinate lines Up and Vp through K. Orp They are equal to U p = areaup and V p = b--¢7, respectively. For further consideration, the normal radii of curvature Rp = kp 1 of the surface P at K it is required to be computed. The formula Rp = f f 2 1 . P / ~ 2 , g for computation the current value of Rp is of the simplest structure. In order to simplify computations, this formula could be reduced to the Euler's formula for normal radii of curvature, Rp(9~) = (R;.% cos z ~ ÷ R~.~ sin 2 (/9)-1,

(23)

where R1.p and

/~I.P

the principal radii of curvature of the surface P at K , angle that the normal plane section Rp(99 ) makes with the first principal direction t l . p of the surface P

Point C1 coincides with tile center of curvature of the surface P in the first principal plane section of P at K. It locates on the axis of the Pliicker's conoid P I R ( P ) . The straight-line segment of the length R 1.P extends from C1 in the direction of tl.p. The unit tangent vector tl.p indicates the first principal direction of the surface P at K. It makes right angle with the axis of the surface P I R ( P ) . The straight-line segment of the same length R 1.P extends from C1 in the opposite direction - - t l . P . 1Reminder: t h e inequality k l . v > k2.p is always satisfied [25]. U s u a l l y t h i s inequality is r e p r e s e n t i n g in t h e form kl. P > k2.p, w h i c h is incorrect. If t h e equality kl. P = k2. P occurs, t h e n all n o r m a l c u r v a t u r e s of t h e surface P at K are of t h e s a m e value, as well as of t h e s a m e sign. ( T h e equality kl. P = k2. P observes for umbilics, a n d for t h e p l a n e surface.) For t h i s reason, at an umbilical point, t h e principal directions t l . p a n d t2. P are undefined. Therefore, t h e principal c u r v a t u r e s are u n d e f i n e d as well. T h i s results in t h a t t h e correct r c p r e s e n t a t i o n of t h e above inequality is a s k l . P > k2. P .

Possibility of Application of Pliicker's Conoid

1005

Point C2 coincides with the center of curvature of the surface P in the second principal plane section of P at K. It is remote from C1 at a distance (R1.p - Re.p). (Remember, that normal radius of curvature R e , as well as the principal radii of curvature RI.p and R z p , are the algebraic values in nature). The straight-line segment of the length R2.p extends from C2 in the direction of t2.p. The unit tangent vector t2.p indicates the second principal direction of the surface P at K. It also makes right angle with the axis of the P1R(P). The straight-line segment of the same length R~.p extends from C2 in the direction - t 2 . p . The applied above unit tangent vectors h . P and t2.p are equal to

tl'p-

T1.p ]Wl.p[

W2.p

and

tl. p

--iT2.p[

correspondingly. Here, the principal directions T1.p and T2.p are determined by the ratio OUp Jr,2, which in fact is equal to [7] OVp ~

-(GpLp

- EpNp)

O~P ] 1,2 =

4- V / ( G p L p

- EpNp)

2 - 4(GpMp

- FpNp)

(FpLp

2 (CpMp - FpNp)

- EpMp)

(24)

A certain point C coincides with the center of curvature of the surface P in the normal plane section of P at K in an arbitrary direction that is specified by the corresponding value of central angle 4. The point C is located on the axis of the surface P1R(P). The normal radius of curvature Rp(~) corresponds to the principal radii of curvature R1.p and R2.p in the manner R1. P < R p ( ~ )

< R2. P.

According to the above-discussed way, a straight-line segment of the length R p = Rp(~) rotates about and travels up-and-down the axis of the Pliicker's conoid P1R(P). In such way, the Pliicker's conoid could be represented as a locus of consecutive positions of the straight-line s e g m e n t R p : Rp(~9).

Figure 2 reveals 2 that the Pl/icker's conoid perfectly reflects topology of the surface P in differential vicinity of K. Therefore, the surface P1R(P) could be used as a tool for visualization of appearance of a smooth regular surface P, as well as for visualization of change of it local parameters. In order to plot the Pliicker's conoid Rr = R p + R T together with the surface P itself (Figure 2), it is required to represent equation of both of the surfaces in a common coordinate system, for example in the coordinate system X s Y s Z s . In order to represent the surfaces P and P1R(P) in a common coordinate system, the operator of resultant coordinate system transformation R s (S ~ P) is required to be composed. The operator R s (S ~ P ) could be computed using the following generalized equation [13], Operator of Resultant Coordinate System Transformations ~ R s (Si --* Si±l) nt

nt

= l-I Cpi[i--* (i-4-1)] = r I [Trk @ ~ , x ) . i=l

i=l •

Rtk (~3~,X)] . Y

Cp~[,i----*(i4-1)]

In equation (25), Cpi[i --~ (i ± 1)] denotes the local "Translation~Rotation" couple, I~nax denotes number of elementary coordinate system transformations in a given series of consequent transformations, and nt denotes total number of consequent coordinate system transformations in that same series of elementary transformations. 2It is of importance to point out here, that for the reader's convenience, the Pliicker's conoid (Figure 2) is scaled along the axes of the local coordinate system (with the solely goal for better visualization of the surface P local geometrical properties).

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S.P.

-t ~ o

RADZEVICH

I/ R 2 . p

yR.

Xp Rs (S --* P)

Ys ~

Figure 2. The Pliicker's conoid PIR(P), and the Pliicker's curvature indicatrix P[R(P), naturally connected to the concave patch of smooth regular surface P. Formally, couple Cpi[i --~ (i + 1)] possesses the following property. One component of the couple Cpi[i --~ (i ± 1)] is always equal to unity matrix E4x4 while another component is not equal to E4x4. So, if for example Tr (as, X) ¢ E4x4, then the operator R t (p~, X) = E4x4 and vice versa. The similar is true with respect of all others operators of elementary coordinate system transformations. The couples Cp~[i --* (i 4-1)] of "Translation~Rotation" proved to be useful in CAD/CAM applications. Here, Trk(5~, X) denotes the operator of transition, and Rtk(~e, X) denotes the operator of rotation of consequent coordinate systems transformations. The interested reader is referred to Appendix B in [26] for details and for examples of application. After been erected at the point of the smooth regular surface P, the characteristic surface P1R(P) clear indicates: (a) the actual values of principal radii of curvature R1.p and R2.p, (b) location of the centers Ox.p and 02.p of the principal radii of curvature, (c) orientation of the principal plane sections C1.p and C2.p (i.e., the principal directions tl.p and t2.p), as well as the current value of normal radii of curvature R(p), location of it center Op for any prespecified section by normal plane Cp through the given direction Therefore, in the case under consideration, i.e., from prospective of application of the surface P1R(P) for analysis of geometry of part surfaces, the Plficker's conoid could be considered as an example of a characteristic surface. In addition to the Plficker's conoid PIR(P) of the described above kind (Figure 2), a eharacteristic sm'face P l k ( P ) of another kind could be introduced as well. While generating the Plficker's conoid PIR(P) (Figure 2), instead of straight-line segment of the length Rp(~), the straight-line segment of length kp(p) can be used. (Here, kp(p) = Rpl(~) is normal curvature of P at /4 in the given plane section through /(.) This results in that a characteristic surface Plk(P) of novel kind could be introduced. In many aspects, the surfaces P1R(P) and Plk(P) resemble each other. They also appear similarly, except of the cases, when Rp(~) and/or kp(q0) is equal to zero (0), or to infinity (oc).

Possibility of Application of Pliicker's Conoid

1007

We refer to the characteristic surface P1R(P) as to the Pliicker's conoid of the first kind, and we refer to the characteristic surface P l k ( P ) as to the Pliicker's conoid of the second kind. The conoids P1R(P) and P l k ( P ) are inverse to each other P1R(P) = P l ikn v (P), and vice versa). As it is following from the above, the Pliicker's conoid, e.g., P1R(P) and P l k ( P ) clear reflects change of parameters of local topology in the differential vicinity of a point of smooth regular surface P. Therefore, it sounds promising to employ the characteristic surfaces P1R(P) and P l k ( P ) for resolving problems t h a t relate to engineering geometry of surfaces. Moreover, it really is. Surface of the Pliicker's conoid P1R(P), as well as the surface P l k ( P ) could be used for engineering purposes. 4.3. T h e Pliicker~s C u r v a t u r e

Indicatrix

The distribution of normal curvatures of the surface P in differential vicinity of K is the only issue that is of interest for the current research. The b o u n d a r y curve of the Plficker's conoid contains all the required information, while the surface P1R(P) itself represents additional information about local topology of the surface P, which is out of our current interest. Therefore, this is the proper point to remind and to follow the rule called Ockham's razor 3. Thus, without loss of accuracy of the approach t h a t is under developing in the current paper, the Pliicker's conoid itself could be replaced with the b o u n d a r y curve of the surface P1R(P). We refer to the boundary curve P i n ( P ) of the characteristic surface P1R(P) as to the Pliicker's curvature indicatrix of the first kind of the surface P at K . DEFINITION 2. The Pliicker's curvature indicatrix P I R ( P ) of the first kind of the surface P is a smooth regular 3D curve that bounds the surface of the corresponding Pliicker's conoid P I R ( P ) . The Pliicker's curvature indicatrix is represented therefore by the end-points of the position vector of the length of R p ( ~ ) that is rotating about and travels up-and-down the axis of the surface P1R(P). This immediately leads to equation of this characteristic curve,

PlR(P)~rR(~)=[Rp(~)eosqa

Rp(~)sin~

Rp(~)

1] T,

(26)

where R p ( ~ ) is given by the Euler's formula Rp(~a) = (R1.1 cos s ~ + R~.~ sin s ~a)-1. A brief analysis of equation (24) is done. The analysis reveals that for the most kinds of local patches 4 of smooth regular surface P [13,27] the Pl/icker's curvature indicatrix P I R ( P ) of the first kind is a closed regular 3D curve. For the surface local patches of parabolic and of saddle-like type, the Plficker's indicatrix P I R ( P ) separates onto two, and onto four branches correspondingly. In the degenerated eases, it could be reduced even to a planar c u r v e - - t o a circie, e.g., for umbilic local patches of the surface P. Equation, similar to t h a t above equation (26) is valid for the Pliicker's curvature indicatrix P l k ( P ) of the second kind of the surface P, Ptk(P)~rk(~)

[kp(~)cos~

kp(~)sin~

kp(~)

1] T ,

(27)

where k p ( ~ ) = kl.p cos 2 ~ + k2.p sin s ~. DEFINITION 3. The P1iicker's curvature indicatrix Ptk (P) of the second kind of a smooth regular surface P is a smooth regular 3D curve that bounds the surface of the corresponding P1iicker's conoid Plk (P). Usually, the Plficker's curvature indicatrix P l k ( P ) is represented with a closed curve. Further possible simplification of analytical description of local topology of smooth regular surface P could be based on the following consideration. a William of Ockham, also spelled Occam (b.c. 1285, Ockham, Surrey, England~d. 1347/49, Munich, Bavaria (now in Germany)). We remember him mostly because he developed the tools of logic. He insisted that we should always look for the simplest explanation that fits all the facts, instead of inventing complicated theories. The rule, which said "plurality should not be assumed without necessity" is coalle "Ockham's razor". 4 From the prospective of engineering application, 10 different kinds of local patches of smooth regular surface P are distinguished [27], (1988) and [13], (2001). The distinction depends upon sign and the actual value of mean Mp and full (Gauss') Cp curvature of the surface P at K. The interested reader is referred to [13,27] for details.

1008

S . P . RADZEVICH

4.4. A n R ( P ) - I n d i c a t r i x o f t h e S u r f a c e P Aiming further simplification of analytical description of local topology of smooth regular surface, the Pliicker's curvature indicatrix could be replaced with the corresponding planar characteristic curve. As it is following from equation (26), the first two elements Rp(qo)cos ~ and Rp(~) sin p at the right-hand side portion of the above equation contain all the required information on distribution of normal radii of curvature of PIR(P) at K. The third element R(~) is out of our current interest. Hence, instead of consideration of Pliicker's curvature indicatrix P l k ( P ) (see equation (26)) this yields consideration of the characteristic curve that could be analytically described by the equation, ArtR(P)~r~R(~)=[Rp(~)cos~

Rp(~)sin~

0

1] T.

(28)

We refer to this planar characteristic curve as to the A n R ( P ) indicatrix of the first kind of P atK. DEFINITION 4. The ArtR(P)-indieatrix of smooth regular surface P I R ( P ) is a planar characteristic curve, which could be represented as the projection of the Plficker's curvature indicatrix PIR(P) onto the tangent plane surface to the surface P. The characteristic curve A n R ( P ) is simpler rather than the Pliieker's indicatrix PIk(P). The AnR(P)-indicatrix is always a planar curve, while the indicatrix PIR(P) is a 3D curve. The distribution of normal curvature of the surface P at K could be given by another planar characteristic curve, A n k ( P ) ~ rik(~) = [kp(9~) cosD~ kp(~) cos99

0

1] T .

(29)

We refer to the characteristic curve (see equation (29)) as to the Ank(P)-indicatrix of the second kind of P at K. DEFINITION 5. The Ank(P)-indicatrix of smooth regular surface PIR(P) is a planar characteristic curve, which can be represented as the projection of the Pliicker's curvature indicatrix Ptk (P) onto the tangent plane surface to the surface P. Consideration of the similar kind immediately leads to the well-known Dupin's indicatrix [13,28]. Euler's formula kl.p cos e p + kzp sin 2 9~ = kp yields representation in the form, kl.p

kp

cos 2

+ ~ p P sin S ~ = 1.

(30)

Transition from polar coordinates to Cartesian coordinates could be performed using formulas x p = p cos~ and yp = p sin~. These formulas yield: cosep = X2p/p2 and sine~ = y2p/pe. After substituting the last formulas into the above equation equation (30), one could obtain k l . p . X2 ke.P. kp -/9 - ~ -}- k p

y2 D--ff = 1.

(31)

It is convenient to designate p = V / ~ 1. This immediately leads to equation for the Dupin's indicatrix Dup (P) that is represented in the Darboux trihedron, kl.p X2p + k z p y~ = 1. The radius of the indicatrix Dup (P) in any direction is equal to square root of the radius of curvature in that direction a. 5The same equation of t h e D u p i n ' s indicatrix could be derived in a n o t h e r way. Coxeter [28] considers a pair of conics obtained by expanding an equation in Monge's form z = z(x, y) in a Maclauren series z = z(0, 0) + zlx + z2y + (1/2)(zllx~ + 2zl~xy 4- z22Y 2) 4- . . . . (1/2)(bllx 2 4- 2bl2xy 4- b22y2). This gives t h e equation (bll z2 4- 2b12xy 4- b2~y2) - 4-1 of t h e Dupin's indicatrix.

Possibility of Application of Pl~icker's Conoid

1009

18

260

q~

280

Figure 3. The AnR(P)-indicatrix of the surface P (at K: R1. P = 3 ram, R 2 . p 15 mm) plotted together with the corresponding Dupin's indicatrix Dup (P).

=

An example of the ArtR(P)-indicatrix is shown in Figure 3. The characteristic curve A n R ( P ) is computed for local patch of the surface P at which principal radii of curvature are equal to R1.p = 3mm, and R2.p = 15ram. It is of importance to point out here that the direction of the minimal diameter dR aligns with the first principal direction tl.p, and the direction of the maximal diameter DR aligns with the second principal direction t2.p on the surface P at K. Therefore, the directions of dn and DR are always orthogonal to each other. For analytical description of local properties of the surface P at K the Dupin's indicatrix is in wide use. It is of interest to compare the ArtR(P)-indicatrix with the corresponding Dupin's indicatrix Dup (P). In order to do that, the characteristic curve Dup (P) is computed for the same point K of the surface P (R1.p = 3mm, R2. P = 15ram). The characteristic curve Dup (P) is also plotted in Figure 3. For the reader's convenience, the characteristic curve Dup (P) is magnified by ten times with respect to it original (computed) parameters. The direction of the minimal diameter dDup aligns with the first principal direction t l . p , and the direction of the maximal diameter DDup aligns with the second principal direction t2.p on the surface P at K. As it follows from Figure 3, both characteristic curves, i.e., the ArtR(P) and Dup (P) indicate the same directions for the first R1.p as well as for the second R2.p radii of curvature of P at K. However, the difference in shape of the characteristic curves ArtR(P) and Dup (P) observes. The Dupin's indicatrix is a planar smooth regular curve of the second order. In the case under consideration, it is always convex with uniform change of curvature. The Artn(P)-indicatrix is also a planar smooth regular curve. However, points of inflection are inherited to this curve in nature. This is due to the ArtR(P)-indicatrix is a curve of the fourth order. Higher order enhances capabilities of the characteristic curve ArtR(P). Because of the higher order, the Artn(P)-indicatrix reflects the distribution of normal radii of curvature, while the Dupin's indicatrix reflects the distribution of square root of normal radii of

1010

s.P. IR.ADZEVICH

curvature of P at K. In order to make the difference clear, it is sufficient to represent equation of the Dupin's indicatrix in the form similar to that for AnR(P)-indicatrix (see equation (28)), Dup (P)::> rDup(~)= [ f p p ((/)) cos ))

fpp(~)sin(/P

0 1] T

(32)

It is evident fl'om the above that equation (28) and equation (32) are similar to each other. 5.

RELATIVE

CHARACTERISTIC

CURVES

The consideration above is aiming resolving problems in various fields of engineering: (a) in CAGD: for analytical description and/or visualization of local topology of regular and of composite surfaces, especially at points along the linkage curve; (b) in CAD: for designing composite surfaces with prescribed requirements for curvature continuity between surfaces (along and across the linkage curve); (c) in CAM: (a) for developing the optimal trajectories of the cutter motion relative to the work while machining a sculptured surface on multiaxis NC machine, and (b) to satisfy the necessary conditions of proper sculptured surface machining on multiaxis NC machine; (d) in mechanical engineering (e.g., in theory of elasticity): for compressive stresses analysis, and for improving accuracy of solution to the problem of contact of two smooth regular surfaces in the first order of tangency. Of course, the potential area of application of the obtained results is not limited to the listed above areas of engineering. Many other geometrical problems of the similar nature could be also resolved with application the developed characteristic curves. 5.1. A P o s s i b i l i t y of A p p l i c a t i o n of T w o P l i i c k e r ' s C o n o i d s In order to solve any of the above-listed problems, it is required to develop an appropriate analytical description of the distribution of normal curvature of the two smooth regular surfaces, which join each other along linkage curve ((a) and (b)), or contact each other at a point or along a certain curve ((c) and (d)). In order to resolve the problem of analytical description of geometry of contact of two smooth regular surfaces application of two Pliicker's conoids at first glimpse sounds promising. (Below, while developing an analytical method for resolving geometrical problem of the nature like the above-listed Problems (a)-(d), we refer to the problem under consideration as to the problem of contact of two smooth regular surfaces, regardless of further application of the obtained results.) In order to develop an appropriate solution to the problem of analytical description of geometry of contact of two smooth regular surfaces that interact with each other, the characteristic surface P I R ( P / T ) that reflects the difference between the corresponding normal radii of curvature of two surfaces P and T could be introduced. Equation (8) yields the following matrix representation of the equation of the surface P1R(P/T),

(Rp - RT) cos ~] P1R(P/T) ~ R*R(~) =

(Rp

-

RT) sin ~ [

We refer to the characteristic surface P1R(P/T) as to the Plucker's relative conoid.

(33)

Possibility of Application of Pliicker's Conoid

1011

DEFINITION 6. The Plucker's relative conoid P1R(P/T) is a ruled surface formed by rotating a strMght-line segment of the length (Rp-RT) about the common perpendicular to the surfaces P and T, and by moving the segment up-and-down the axis of rotation according to location of the corresponding centers of principal curvature. Because the centers of principal curvature of the surfaces P and T, i.e., the centers C1.p and C2.p, as well as the centers C1.T and CZT in general case do not coincide with each other, the actual reciprocation of the straight-line segment of the length (Rp-RT) could be restricted by different pairs of the limiting points CI.p, Czp, C1.T, C2.T. Various location of the limiting points on the axis of rotation results in deformation of the surface Pln(P/T) in it axial direction. The deformation of such kind does not affect the surface appearance in the direction of (Rp-RT) that is of prime importance for the current research. Figure 4 illustrates an example of the P1/icker's relative conoid for a certain point K at which the principal radii of curvature of the surfaces P and T are equal to R1.p = 2 mm, R2.p = 3 ram, RI.T = --2 mm, R2.T = --5 ram, respectively. It is of importance to point out here that for the example (see Figure 4) the angle # of the faces P and T local relative orientation is equal to # = 45 °. Note that the apparent gap in the center of the surface is an artifact of the plotting, and not a feature of the surface. The characteristic surface Pln(P/T) could be analytically described by equation (33). This indicates that the Pliicker's relative conoid properly reflects the rate of conformity of the surfaces P and T at K. However, the above example (see Figure 4) reveals that the surface P1R(P/T) itself is inconvenient for application in engineering geometry of surfaces. In order to fix this undesirable problem, one may decide to go that same way as the way above (see Section 5.3), and to introduce the Pliicker's relative indicatrix PIn(P/T). Equation of this 3D characteristic curve (Figure 4) immediately follows from the above equation (33),

PIR(P/T)~RR(q~)=[(Rp-RT)COSW

(Rp--RT)sinw

(Rp--RT)

1] -F.

(34)

Further, the characteristic curve Pt•(P/T) could be reduced to a corresponding planar characteristic curve. In order to shorten the paper, we omit the intermediate consideration, and go directly to the AnR(P/T)-relative indicatrix of the surfaces P and T at K.

zp

3.5

?.5

5 \ y, Figure 4. The Pliicker's relative conoid PIR(P/T) and the Plficker's relative indicatrix ArtR(P/T) for the surfaces P and T at (R1.P = 2ram, R2.p = 3mm, R1.T --2 mm, R2.T = --5 ram, and # = 45°).

1012

S . P . RADZEVICH

t 2 " P 9 0 ~ ~

¢4 tZ.T

} AnR (P

Q.P tl.p 0T

180 :0

An/~

270

,., N

cp

Figure 5. The ArtR(P/T)-indicatrix of the surfaces P and T at K (R1. P = 2mm, R2.p = 3mm, R1.T = 2mm, R2.T -= --5mm, lz = 45 ° ) plotted together with the corresponding AnR(P)-indicatrix and AnR(T)-indicatrix.

5.2. A n R ( P / T ) - R e l a t i v e I n d i c a t r i x o f t h e S u r f a c e s P a n d T Aiming simplification of analytical description of the geometry of contact of the surfaces P and T, the Pliicker's relative indicatrix kr = kp kT could be replaced with the corresponding planar characteristic curve. Equation of the 2D AnR(P/T)-relative indicatrix of the surfaces P and T at K could be obtained from equation (34),

A n n ( P / T ) ~]Ril~(~) = [ ( R p - - R T ) c o s ~

(Rp-RT)sin~

0

1]-

(35)

We refer to this planar characteristic curve as to the Artn(P/T)-relative indicatrix of the first kind of the surfaces P and T at K. The ArtR (P/T)-relative indicatrix of the first kind analytically describes the distribution of the difference of normal radii of curvature of the surfaces P and T at/(. DEFINITION 7. The ArtR(P/T)-relative indicatriz of the first kind of the interacting surfaces P

and T is a planar characteristic curve, which could be represented as the projection of the Plficker's relative indicatrix PIR( P / T ) onto the common tangent plane to the surface P and T. An example of the AnR(P/T)-relative indicatrix of the surfaces P and T at K is shown in Figure 5. The characteristic curve A r t R ( P / T ) is computed for the case of contact of convex local patch of the surface P (R1. P = 3 ram, and R2.p = 15 ram) with concave local patch of the surface T (R1. T = - 2 ram, and R2.T = --5 ram). The surfaces P and T are turned through the angle # = 45 ° relative to each other around the common perpendicular. The corresponding A v t n ( P )indicatrix, as well as ArtR(T)-indicatrix are also plotted in Figure 5. It is of importance to point out here t h a t the direction of the minimal diameter dim~~ and the direction of the maximal diameter dp~ x of the characteristic curve A r t R ( P / T ) do not align neither with the principal directions t l . p and t2.p on the surface P, nor with the principal directions tl.T and t2.T on the surface T. The extremal directions of the Anu(P/T)-relative indicatrix are not orthogonal to each other. In general case they make a certain angle ~ ¢ 90 °. The above immediately yields the following conclusion. The directions of the e x t r e m u m (i.e., of the m a x i m u m and of the minimum) rate of conformity of the surface P and T at K could be orthogonal to each other only in particular cases. In general case, these directions are not orthogonal to each other.

Possibility of Application of Pliicker's Conoid

1013

The shape and parameters of the ArtR(P/T)-relative indicatrix depend upon algebraic value of the principal radii of curvature k~ = kp - kT, R2.p and R1.T, R2.T of the surfaces P and T at K, as well as upon the actual value of the angle # of the surfaces local relative orientation. The Dupin's indicatrix Dup (P/T) of the surface of relative curvature indicates that the directions of the extremum rate of conformity of the surfaces P and T at K are always orthogonal to each other. The above proves, that this conclusion is not correct and results in errors in calculations. The characteristic curve ArtR(P/T) is simpler rather than the Pliieker's relative indicatrix PIR(P/T) itself. The AnR(P/T)-relative indicatrix is always a planar curve, while the Pliicker's relative indicatrix PIR(P/T) is a 3D curve. The distribution of the difference of normal curvature of the surfaces P and T at K could be analytically described by another planar characteristic curve,

Ank(P/T) ~ Rik(~) = [(kp - kT)COS~

(kp -- leT) sin ~

0

1] 7

(36)

We refer to the characteristic curve (see equation (36)) as to the 3knk (P/T)-relative indicatrix of the second kind of the surfaces P and T at K. DEFINITION 8. The Ank (P/T)-relative indieatrix of the second kind of the interacting surfaces P and T at K is a planar characteristic curve, which could be considered as the projection of the Pliicker's relative indicatrix Ptk(P/T) onto the common tangent plane to the surface P and T. Extremums of the characteristic curves AnR(P/T) and Artk (P/T) can be computed following known methods (see Appendix A). The reported in the paper planar characteristic curves A n R ( P ) and AnR(P/T), as well as the characteristic curves A n k ( P ) and Ank(P/T) originate from Pliicker's conoid. Equations (28), (29), (35), and (36) of the corresponding indicatrices A n R ( P ) , AnR(P/T), and A n k ( P ) , Ank(P/T) directly follow from equation (8) of the surface of Pliicker's conoid (see [3; 13, Section 4.9, pp. 257 260]). The characteristic curves similar to that above could be also derived if the Pliicker's conoid would be replaced with corresponding Dupin's indieatrix. Following that last way one could come up with the planar characteristic curves of another sort known as the indicatrix of conformity CnfR(P/T) of smooth regular surfaces (see Appendix B). The interested reader is referred to [3,7,13,29] for details. It is worth pointing out here about results of research in the field of surfaces contact done by Favard [30]. He completed a profound investigation of conditions of contact of two smooth regular surface in various order of tangency that exceed the first order. Besides these results are certainly important, they are too far from the needs of engineering geometry of surfaces, and therefore, it is incorrect to compare the results reported in the paper with the results earlier obtained by Favard [30]. 6. I L L U S T R A T I V E

EXAMPLES

OF

APPLICATION

The illustrative examples below are arranged in that same order that corresponds to the order the problems to be resolved are listed at the beginning of Section 5. 6.1. In order to make visible the distribution of normal radii of curvature R(p) of smooth regular surface P in differential vicinity of the point K, the Pliicker's conoid P1R(P) itself could be applied (Figure 6). Aiming simplification of equation, it is recommended to represent the characteristic curve P1R(P) in Darboux' trihedron xpypzp. Axes xp and yp of the local coordinate system xpypzp are pointed along the principal directions t l . p and t2.p on the surface P at K , and zp is pointed along the unit normal vector k~ = h i . P c o s 2 ~ - k2. P sin 2 ~a - k l . r cos2(qO -t- /z) -t-

1014

S.P.

I{ADZEVICH

Xp

Figure 6. Distribution of normal curvature of a smooth regular surface P in the differential vicinity of the point /(. 90

/j

j'\

/ ;'>2_J A%(PIT),,(\, .... ~ \ r

~

~.... 2 ~ n !

2~

.....

2.P

""

270 q~ (a)

(b)

Figure 7. Distribution of normal curvature of a composite surface P [7] in the differential vicinity of the point K on the linkage curve.

k2.Tsin2(~

+ #). Along with the Pliicker's conoid itself, in the case under consideration, the A r t R ( P ) and A n k ( P ) characteristic curves could be applied as well. For visualization of the distribution of normal radii of curvature of composite surface along the linkage curve application of the Pliicker's conoid P1R(P), as well as of the Pliicker's indicatrix P l R ( P ) is inconvenient. In such case, it is preferred to employ the ArtR(P)-indicatrix (see equation (28)) or A n k ( P ) - i n d i c a t r i x (see equation (29)). Figure 7 presents an example of composite surface, which is composed of two patches of smooth regular surfaces P and T. The characteristic curves A n R ( P ) and A n R ( T ) are tangent to each other along the linkage curve. Because of this, the diameter of the relative characteristic curve ArtR(P/T) at that same direction is equal to zero. The charactcristic curves ArtR(P), ArtR(T), and AnR(P/T) could be computed at every point of the linkage curve. Figure 7 provides a clear illustration of how normal radii of curvature of the composite surface are distributed in differential vicinity of every point of the linkage curve.

Possibility of Application of Pliicker's Conoid

1015

6.2. Fitting surfaces with the desired kind of continuity across the linkage curve is a critical issue when designing composite surfaces [12,14]. In the event of tangent continuity, the surfaces have the same normal curvature in the direction that is tangent to the linkage curve. Thus, in case of tangent continuity, it is required the A n R ( P ) - i n d i c a t r i x and the A n R ( T ) - i n d i e a t r i x be tangent to each other or, the s a m e - - t h e minimal diameter of the AnR(P/T)-relative indicatrix in the direction of the linkage curve be equal to zero. Figure 7 is a perfect illustration of how this requirement could be satisfied. More interesting occurs when the curvature continuity of the composite surface is required. In order to satisfy the curvature continuity condition, the surfaces P and T have to have the same value of normal curvature not only along the linkage curve, but in the transverse direction as well. The A n R ( P ) - i n d i c a t r i x and the A n R ( T ) - i n d i c a t r i x have to be congruent to each other in order to satisfy the condition of curvature continuity of the composite surface. The same requirement to composite surface would be satisfied, if the AnR(P/T)-relative indicatrix degenerates into point at every point along the linkage curve (Figure 8). Application of the characteristic curves ArtR(P) and A n R ( T ) , as well as the the second order of smoothness of the composite surface.

AttR(P/T) yield

90

___

Xp (a)

_ _ _ .

,o

270 q~ (b)

Figure 8. The second order of smoothness of the composite surface.

6.3. As an example of application of the characteristic curves A n R ( P ) , A n R ( T ) , and AnR(P/T) in CAM, let's consider the following two issues, which are critical in sculptured surface machining (SSM) on multiaxis NC machine. The first issue relates to development of the optimal tool-paths for SSM. The second issue covers the topic of conditions of proper SSM.

6.3.1.

It is proven [3,13] t h a t a closed-form solution to the problem of the optimal tool-paths generation for SSM on multiaxis NC machine can be represented in an integral form,

1016

S. P, RADZEVICH

f[2 /OXp [-0-57 (t)cos =

_

OXp_(t)_sinp(Opt)(t) )

dt

0 Yp (t3 JQ t~ ( O0- ~Yp p (t) c°s ~(°Pt) (t) -- -O-@-p " ' sin p(opt) ( t ) ) •dt /

t~ 0 Zp

(37)

_ 0 Zp (t) sin ~(opt)(t)~ • dt / 1

Here, in equation (37) ~(opt) designates the angle, which the optimal tool-path makes with the first principal direction t l . p at every point along a certain tool-path 6 [31,32]. The optimal tool-path is always directed orthogonally to the direction on P at CC-point K , at which the rate of conformity of the generating surface T of the cutter to the part surface P is the highest possible [3,13,29]. The AnR(P/T)-relative indicatrix yields identification of this extremal direction. The highest rate of conformity of the surface T to the surface P at the CC-point K observes in the direction, at which diameter of the characteristic curve AnR(P/T) is equal to zero. In order to compute the optimal value of angle ~(opt) it is required to solve the following equation: p(opt) ~ ~ R ~ R ( p ) = 0 (see equation (35)). After substituting the computed value of ~(opt) into equation (37), equation of the optimal tool-path for SSM could be derived. The example clarifies a way in which the AnR(P/T)-relative indicatrix could be applied for solving geometrical problems in CAD. 6.3.2. Another issue that is of critical importance for CAD is satisfaction of the necessary conditions of SSM [3,13,29]. The AnR(P/T)-relative indicatrix could be applied in order to verify either the third necessary condition of proper SSM satisfied or it doesn't. In order to satisfy the third necessary condition of proper SSM, it is required to eliminate interference between the generating surface T of the cutter and between the sculptured surface P being machined [3,13,29]. The third necessary condition of proper SSM could be satisfied if and only if all diameters of the ArtR(P/T)-relativc indicatrix are always nonnegative RiR(~) _> 0 (see equation (35)). Figure 9 presents an example of machining of the sculptured surface P in the differential vicinity of the CC-point K. The principal radii of curvature of the saddle-like surface P at K are equal to RI.p = 2 mm, R2.p = - 3 ram. The principal radii of curvature of the convex machining surface T of the cutter at K are equal to RI.T = 2 ram, RZT = 5 mm. At the initial position of the cutter relative to the surface P the angIe of the surfaces P and T local relative orientation is equal to p* = 0°. At this position of the cutter relative to the work the An*R(P/T)-relative indicatrix intersects itself and, thus negative values of the diameter of the characteristic curve An*R(P/T ) observe (for the readers convenience, one brunch of the characteristic curve An*R(P/T ) is shown in solid line, while another brunch is shown in dash line). Due to negative diameters observe in the Art*R(P/T)-relative indicatrix, the third necessary condition of proper SSM is violated, and, thus, the surface P could not be machined in full accordance to the requirements specified in the blueprint. In order to machine that same sculptured surface P with that same cutter T, the cutter could be turned through a certain angle about the common perpendicular n p ( n r ) . If under such a scenario the angle of the surfaces P and T local relative orientation is equal to p** = 30 °, no negative diameters of the An~*(P/T)-relative indicatrix observe. Therefore, the sculptured surface P could be machined with that same cutter T, but with another value of the angle ft. 6This equation is derived on the premises of two patents: Pat. No. 1185749, USSR, [31], Filed: October 24, 1983, and Pat. No. 1249787, USSR, [32] Filed: December 27, 1984.

Possibility of A p p l i c a t i o n of Pliicker's C o n o i d

1017 90

1

30

IP/T) !

180 i

2

330

270 q~

P (a)

(b)

F i g u r e 9. Satisfaction a n d violation of t h e t h i r d n e c e s s a r y c o n d i t i o n of proper SSM.

The example (Figure 9) clear shows how the AnR(P/T)-relative indicatrix could be applied for solving important geometrical problems in CAM. 6.4. Analytical description of the geometry of contact of two smooth regular surfaces is a critical issue in mechanical engineering, e.g., for the theory of elasticity. Many engineering applications require the analysis of compressive stresses be performed, and contact stresses be computed. Usually, computation of contact stresses is performing on the premises of the Dupin's indicatrix of the surface of relative curvature. The surface of relative curvature of the two contacting bodies is defined by the equation kT = kp - kT. (An approach similar to that below could be developed based on equation RT = Rp-I-R T instead of equation k~ = kp-kT.) Here is designated k~-normal curvature of the surface of relative curvature, kp and ]CT a r e the algebraic values of curvature of the contacting surfaces P and T measured at the same plane section of the P and T at K, at which k~ is measured. The equation k~ = kp - kT could be easily transformed to the equation of the surface of relative curvature itself,

kr = kl.p cos 2 ~ -- k2.p sin 2 ~

- kl.T

COS2(~fl-t- ~t) -~- k2.T sin2(~ + #).

(38)

Equation (38) yields computation of the Dupin's indicatrix Dup~ (P/T) of the surface of relative curvature for P and T at K. The principal radii of curvature of the surfaces P and T at K are equal to R1.p 2mm, R2.p 3mm, RI.T = --5ram, and R2.T = --15mm correspondingly (Figure 10). The contacting surfaces are turned around common perpendicular through the angle of local relative orientation # = 45 °. This characteristic curve is represented with the ellipse DupT(P/T ). The principal directions tl.~ and t2.~ of the surface of relative curvature are always orthogonal to each other. For the readers convenience, the characteristic curve Dup (S~) is magnified by six times. Shown in Figure 10, the AnR(P/T)-relative indieatrix of the surfaces P and T is computed for that same values of normal radii of curvature, and the angle of the surfaces local orientation as the Dupin's indicatrix Dup (S~) is shown. This characteristic curve reflects the actual distribution of the rate of conformity of the surface T to the surface P at K. Figure 10 yields making two important conclusions: (a) the principal directions tl.~ and t2.~ of the surface of relative curvature do not align with the directions t m i n and t m a x , and

S. P. RADZEVICH

1018

90 120

~

60

150

180 ......

'r(P/T) 210

3

240

"'"

~ 300

270 Figure 10. On comparison of the AnR(P/T)-relative indicatrix with the Dupin's indicatrix Dup~(P/T) of the surface of relative curvature of the surfaces P and T.

IJ O

7.8

©

7.6 o

7.4

~6 L~

7.2

O q~ O

6.8

6.6 L 0

-t-

0.5

#(opt)

~----~ 1.5 /4 rad

Figure 11. Impact of value of the angle tt of the surfaces local relative orientation onto the rate of conformity of the surface T to the surface P at K. (b) tile d i r e c t i o n s train and t . . . . of t h e e x t r e m a l r a t e of c o n f o r m i t y of t h e surfaces P an d T are n o t always o r t h o g o n a l t o each other. T h e s e two conclusions clear i n d ic a t e s t h a t a p p l i c a t i o n of t h e D u p i n ' s i n d i c a t r i x D u p (S~) for a n a l y t i c a l d e s c r i p t i o n of t h e g e o m e t r y of c o n t a c t of two s m o o t h r e g u l a r surfaces in general case results in inaccuracies. cases, for w h i c h

T h e c h a r a c t e r i s t i c curve D u p ( S . ) could be ap p l i ed only in p a r t i c u l a r

AnR(P/T)-relative

i n d i c a t r i x d e g e n e r a t e s to c o r r e s p o n d i n g D u p i n ' s i n d i cat r i x .

Possibility of Application of Pliicker's Conoid

1019

The AnR(P/T)-relative indicatrix reflects the geometry of contact of two surfaces precisely. This advantage is important, and is directly following from that the characteristic curve AnR(P/T) is a curve of higher (of the fourth) order rather t h a n the Dupin's indicatrix, which is a planar curve of the second order 7. Equation (35) indicates t h a t position vector of a point of the AnR(P/T)-relative indicatrix of the first kind is a function of the angle # of the surfaces P and T local relative orientation at K . For every given value of # the corresponding value of the angle ~(opt) could be computed. Figure 11 illustrates the impact of the angle # on the optimal value of the angle qO(°pt). This curve is computed for the surfaces P and T with R1.p = 2 ram, R2.p = 3 mm, R1.T = --2 mm, and R2.T = --5 mm. It satisfies the condition 8%R~R(F) = 0. Figure 11 illustrates t h a t for any two smooth regular surfaces P and T exists the o p t i m u m value of the angle #. The similar is valid for the relative indicatrix of the second kind (see equation (36)). Application of the AnR(P/T)-relative indicatrix does not resolve the problem of computation of compressive stresses of two elastic bodies [33-35]. It is helpful only for more detailed understanding of the problem, and for it proper formulation. However, it clear indicates that application of the Dupin's indicatrix of the surface of relative curvature could not provide an exact solution to the problem. Therefore, in order to improve the accuracy of solution to the problem of contact of two smooth regular surfaces in the first order of tangency an appropriate approach to be develop must incorporate analytical description of the geometry of contact of two surfaces. The approach could be based on application of the AnR(P/T)-relative indicatrix of the contacting surfaces. 7.

CONCLUSION

A possibility of application of Pliicker's conoid in engineering geometry of surfaces is investigated in the paper. At first glimpse, application of Plficker's conoid sounds promising for solving problems in (a) CAGD, (b) C A D / C A M , (c) mechanical engineering (gearing), etc. The research reveals t h a t Plticker's conoid itself is a complex 3D characteristic image. Without loss of accuracy, it could be replaced with 2D characteristic curves. The novel 3D characteristic curves P1.R(P) and P t k ( P ) could be considered as a kind of extraction from the surface of Pliicker's conoid. Further, these characteristic curves are reduced to 2D characteristic curves, i.e., to the so-called AnR(P)-indicatrix and A n k ( P ) - i n d i c a t r i x of the surface P. The characteristic curves A r t R ( P ) and A r t k ( P ) in m a n y cases are similar to the corresponding Dupin's indicatrices D u p e ( P ) and D u p k ( P ) , and thus could be employed for anal:gcical description of the local topology of smooth regular surfaces. Definitions of the Pliieker's relative conoid P1R(P/T) and Plk (P/T) are introduced, and corresponding equations of the characteristic surfaces are derived. This yields introducing the Pliieker's relative indicatrices PIR(P/T) and PIk(P/T). Finally, the 3D characteristic curves PIR(P/T) and PIk(P/T) are reduced to the planar characteristic curves AnR(P/T) and Ank(P/T), i.e., to the so-called AnR(P/T)- and Ank(P/T)-relative indicatrices. The characteristic curve AnR(P/T), as well as the A n k (P/T) could be employed for analytical description of the local topology of composite surfaces along the linkage curve; for developing the optimal tool-paths for sculptured surface machining on multiaxis NC machine, as well as for analytical description of conditions of proper sculptured surface machining; for improving accuracy of solution to the problem of contact of two elastic bodies. 7In order to derive equation of the characteristic curve author.

AnR(P/T) another

approach [7] has been used by the

1020

S.P. RADZEVICH

Consideration based on the Pliicker's conoid yields the indicatrices of novel kind that are reported in the paper while consideration based on the Dupin's indieatrices yields indicatriees of conformity CnfR(P/T) earlier developed by the author. These characteristic curves of novel kind are similar to the indieatrices of conformity CnfR(P/T) and Cnfk(P/T) of two smooth regular surfaces earlier developed by the author. Application of the developed planar characteristic curves A n n ( P ) , Ank(P) and AnR(P/T), Ank(P/T) is illustrated by examples. Reported in the paper results perfectly correlate with the results obtained by other researchers [3,7,11-14,24,27,36] as well as with the results earlier developed by the author [37]. The obtained results complement the earlier developed methods of investigation of conditions of contact of smooth regular surfaces in the first order of tangency [37]. APPENDIX

A

Conditions for the min/max of the characteristic curve represented in the following way.

AnR(P/T)

(see equation (35)) can be

(a)

90

90

180

o 210 ~ ~ - - - ~ i R

....... (t] T ) ( ~3 0/ 0 0

300

270 (b)

27O (c)

Figure A1. Characteristic curves CnfR(P/T) and A n R ( P / T ) identically determine the direction t m~x of maximum rate of conformity of the surfaces P and T at a cn[ current CC-point K.

Possibility of Application of Plticker's Conoid

1021

First, the necessary condition for the m i n / m a x of the characteristic curve AnR(P/T) is determined by the expression, The necessary condition ORiR(~) _ O. for m i n / m a x of the A.rtR(P/T) ~ O~a

(A-l)

Second, the sufficient conditions for the m i n / m a x of the characteristic curve AnR(P/T) yield the following analytical representation, The sufficient condition 0 RiR(qa) > 0. for min of the AnR(P/T) =~ O~

(A-2)

The sufficient condition ORiR(p) < O. for max of the AnR(P/T) =* O~

(A-3)

The similar equations can be employed for computation of extremums of equation (36).

APPENDIX

B

It is proven analytically that both planar characteristic curves, say the reported in the paper characteristic curve AnR(P/T), as well as the earlier developed by the author indieatrix of conformity CnfR(P/T) of two smooth regular surfaces P and T specify that same direction tom}x at which the rate of conformity of the surfaces P and T is maximum. Consider machining of a Bezier patch with a filleted-end milling cutter (Figure A-la). The surfaces P and T make contact at a CC-point K. Figure A-1 reveals that the direction tcm~fx of the surfaces P and T maximum rate of conformity, which is determined using CnfR(P/T) (Figure Alb) is identical to the direction tcm~)X that is computed using AnR(P/T) characteristic curve (Figure A-lc). Therefore, from the standpoint of the reported research, both the characteristic curves AnR(P/T) and CnfR(P/T) are equivalent to each other.

REFERENCES 1. h t t p : / / w w w . m a t h . h m c . e d u / f a c u l t y / g u / c u r v e s _ a n d _ s u r f a c e s / s u r f a c e s / p l u c k e r . h t m l . 2. J. Pliicker, On a new geometry of space, Philosophical Transactions of the Royal Society of London 155, 725-791, (1865). 3. S.P. Radzevich, Machining of Sculptured Surface on Multi-Axis NC Machine Monograph, (in Russian), Copy of the monograph is available from The Library of Congress, Vishcha Shkola Publishing House, Kiev, (1991). 4. S.P. Raxtzevich and E.D. Goodman, Efficiency of multi-axis N C machining of sculptured part surfaces, In Machining Impossible Shapes, IFIP TC5 International Conference on Sculptured Surface Machining (SSM'98), November 9-11, 1998, Chrysler Technology Center, Michigan, USA, (Edited by G.J. Olling, B.K. Choi and R.B. Jerrard), pp. 42-58, Kluwer Academic Publishers, Boston, MA, (1998). 5. M. Shilor, Editor, Recent advances in contact mechanics, Mathl. Comput. Modelling 28 (4-8), 1-543, (1998). 6. D.A. Hills, D. Nowell and A. Sackfield, Mechanics of Elastic Contact, ISBN 0 7506 0540 5, pp. 496, Butterworth-Heinemann, Ltd., Linacre Hause, Jordan Hill, Oxford, (1993). 7. S.P. Radzavich, Geometric features of contact of two surfaces in first order of tangency, Mathl. Comput. Modelling 39 (9/10), 1083-1112, (2004). 8. R. Dixon, Mathographics, p. 20, Dover, New York, (1991). 9. J. von Hoschek, Linien-Ceometrie, p. 263, Hochschultaschenbficher-Verlag, (1971). 10. W. Wunderlich, Darstellende Ceometria II, 133/133a, Hochshultaschenbiicher-Verlag, Mannheim, (1967). 11. D.B. Dooner and A.A. Seireg, The Kinematic Geometry of Gearing. A Concurrent Engineering Approach, p. 450, John Wiley and Sons, Inc., New York, (1995). 12. J. Pegna and F.-E. Wolter, Geometrical criteria to guarantee curvature continuity of blend surfaces, ASME Journal Mechanical Design 114, 201-210, (March 1992). 13. S.P. Radzevich, Fundamentals of Surface Machining, Monograph, (in Russian), p. 592, Rastan, Kiev, (2001); Copy of the monograph is available from The Library of Congress, and online at: h t t p ://www. c s e . b u f f a l o • e d u / ~ v a r 2 / ; and h t t p ://www. t r a c t e c h , c o m / d o c s / R a d z e v i c h - S P - N o n o g r a p h - 2 0 0 1 , pdf. 14. X. Ye, The Gaussian and mean curvature criteria for curvature continuity between surfaces, Computer-Aided Geometric Design 13 (16), 549 567, (August 1996). 15. E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman and Hall/CRC, New York, (2000). 16. U. Pinkall, Mathematical Models from the Collections of Universities and Museums, (Edited by G. Fischer), p. 64, Vieweg, Braunschweig, Germany, (1986).

1022

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

34. 35. 36. 37.

S.P.

RADZEVICH

E. Kreyszig, Differential Geometry, p. 352, Dover, New York, (1991). http ://www. btinternet, com/~krys 1/gallery/vrml_geol. html. D. yon Seggern, CRC Standard Curves and Surfaces, p. 288, CRC Press, Boca Raton, FL, (1993). G. Fischer, Editor, Mathematical Models from the Collections of Universities and Museums, pp. 4-5, Vieweg, Braunschweig, Germany, (1986). A. Gray, Pliicker's conoid, In Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, p. 435-437, CRC Press, Boca Raton, FL, (1997). h t t p ://www. mathcurve, c o m / s u r f a c e s / p l u c k e r / p l u c k e r , shtml. Bonnet, Journ. Ec. Polytech xiii, 31, (1867). D.J. Struik, Lectures on Classical Differential Geometry, Second Edition, p. 232, Addison-Wesley Publishing Company, Inc., Massachusetts, (1961). A.W. Nutbourn and R.R. Martin, Differential Geometry Applied to Curve and Surface Design, Volume 1, p. 282, Ellis IIorwood, Ltd., Chichester, West Sussex, England, (1988). S.P. Radzevich, Design of shaving cutter for plunge shaving a topologically modified involute pinion, ASME J. of Mechanical Design 125, 632-639, (September 2003). S.P. Radzevich, Classification of Surfaces, Monograph, (in Russian), UkrNIINTI No. 1440-Uk88, p. 185, Kiev, (1988). H.S.M. Coxeter, Dupin's indicatrix~ In Introduction to Geometry~ Second Edition, pp. 363 365, Willey, New York, (1969). S.P. Radzevich, Conditions of proper sculptured surface machining, Computer-Aided Design 34 (10), 727740, (September 2002). J. Favard, Coures de G~om~trie Diff~rentialle Locale, viii p. 553, Gauthier-Villars, Paris, (1957). S.P. Radzevich, A Method of Sculptured Surface Machining on Multi-Axis NC Machine, Pat. No. 1185749, B23C 3/16, Filed: October 24, 1983, USSR. S.P. Radzevich, A Method of Sculptured Surface Machining on Multi-Axis NC Machine, Pat. No. 1249787, B23C 3/16, Filed: December 27, 1984, USSR. H. Hertz, Uber die Beruhrung Fester Elastischer Korper, Journal fiir die Reine und Angewandte Mathematik, 156-171, (1881); "Uber die Ber~uhrung Pester Elastischer Korper und Uber die Harte, Berlin, (1892);H. Hertz, Gesammelte Werke, Volume 1, Reprinted, pp. 155-196, Leipzig, (18955); (Translated by D.E. Jones and G.A. Schott), pp. 146-183, McMillan and Co., London, (1896). A.I. Luri'e, Three-Dimensional Problems of the Theory o/Elasticity, p. 493, John Wiley and Sons, New York, (1964). S.P. Timoshenko and J.N. Gudier, Theory of Elasticity, Third Edition, McGraw-Hill, New York, (1972). S.P. Radzevich and E.D. Goodman, Computation of optimal workpiece orientation for multi-axis N C machining of sculptured part surfaces, ASME J. of Mechanical Design 124 (12), 201-212, (June 2002). S.P. Radzevich, Methods for Investigation of Conditions of Contact of Surfaces, Monograph, (in Russian), UkrNIINTI No. 759-Uk88, p. 104, Kiev, (1987).