Notation

Ank (P) Andrew’s indicatrix of normal curvature of the surface P Ank (P/ T) Andrew’s indicatrix of normal curvature of the surfaces

P and T Anlk (T) Andrew’s indicatrix of normal curvature of the generat-

ing surface T of the cutting tool AnlR (P) Andrew’s indicatrix of radii of normal curvature of the

surface P AnR (P/ T) Andrew’s indicatrix of normal radii of curvature of the

surfaces P and T AnlR (T) Andrew’s indicatrix of radii of normal curvature of the

generating surface T of the cutting tool CC-point Cutter contact point Cnfk(P/ T) Indicatrix of conformity of the part surface P and of the

generating surface T of the cutting tool at the current contact point K (normal curvatures)

CnfR(P/ T) Indicatrix of conformity of the part surface P and of the generating surface T of the cutting tool at the current contact point K (radii of normal curvatures)

Cpi[i a (i ± 1)] Couple of elementary coordinate system transformation Crv(P) Curvature indicatrix of the surface P Crv(T) Curvature indicatrix of the generating surface T of the

cutting tool C1.P , C2.P The first and the second principal plane sections of the

part surface P C1.T, C2.T The first and the second principal plane sections of the

generating surface T of the cutting tool Ds(P/ T) Matrix of the resultant displacement of the cutting tool

with respect to the part surface P Dup(P) Dupin’s indicatrix of the surface P Dup(P/ T) Dupin’s indicatrix of the surface of relative curvature R Dup(R ) Dupin’s indicatrix of the surface of relative curvature R Dup(P) Dupin’s indicatrix of the generating surface T of the

cutting tool E A characteristic line EP, FP , GP Fundamental magnitudes of the first order of the sur-

face P ET, FT, GT Fundamental magnitudes of the first order of the gener-

ating surface T of the cutting tool Eu(y, q, j) Operator of the Eulerian transformation( Ffr Feed rate per tooth of the cutting tool

[ ] ( Ffr Limit feed rate per tooth of the cutting tool

Ffr Vector of the feed-rate motion of the cutting tool( Fss Magnitude of the side step of the cutting tool [ ] ( Fss Limit magnitude of the side step of the cutting tool

Fss Vector of the side-step motion of the cutting tool F 1, F 2, F 3 The rate of degree of conformity functions Glnd(P) Gauss’ indicatrix of the surface P Glnd(T) Gauss’ indicatrix of the generating surface T of the

cutting tool GMap(P) Gauss’ map of the surface P GMap(T) Gauss’ map of the generating surface T of the cutting tool G P, G T Full (Gaussian) curvature of the surface P, and of the generating surface T of the cutting tool HP Discriminant of the first fundamental form of the sur-

face P HT Discriminant of the first fundamental form of the gener-

ating surface T of the cutting tool Jrk(P/ T) A planar curvature characteristic curve JrR(P/ T) A planar radii of curvature characteristic curve K Point of contact of the surfaces P and T (or a point within

the line of contact of the surfaces P and T) LP, MP, NP Fundamental magnitudes of the second order of the sur-

face P LT, MT, NT Fundamental magnitudes of the second order of the

generating surface T of the cutting tool Mch(P/ T) Indicatrix of machinability of the surface P with the cut-

ting tool T M P, M T Mean curvature of the surface P, and of the generating

surface T of the cutting tool NP Perpendicular to the surface P NT Perpendicular to the generating surface T of the cutting

tool P Part surface to be machined P mr Chip (material) removal output P sg Part surface generation output P lk(P) Plücker’s indicatrix of normal curvature of the surface P P lk(T) Plücker’s indicatrix of normal curvature of the generat-

ing surface T of the cutting tool P lR(P) Plücker’s indicatrix of normal radii of curvature of the

surface P P lR(T) Plücker’s indicatrix of radii of normal radii of curvature

of the generating surface T of the cutting tool R Surface of relative curvature Rfx(YiZi) Operator of reflection with respect to YiZi-coordinate

plane

Rfy(ZiXi) Operator of reflection with respect to YiZi-coordinate plane Rfz(XiYi) Operator of reflection with respect to XiYi-coordinate

plane RPY(jx, jy, jz) Operator of “Roll/Pitch/Yaw” transformation Rs(A a B) Operator of the resultant coordinate system transforma-

tion, say from the coordinate system A to the coordinate system B

Rt(jx, X) Operator of rotation through an angle jx about the X axis Rt(jy, Y) Operator of rotation through an angle jy about the Y axis Rt(jz, Z) Operator of rotation through an angle jz about the Z axis Rt(jA, A) Operator of rotation through an angle jA about the A 

axis not through the origin of the coordinate system Rt(jA, A0) Operator of rotation through an angle jA about the A0-

axis through the origin of the coordinate system Rtw(AaB) Operator of nonorthogonal coordinate system trans-

formation R1.P, R2.P The first and the second principal radii of curvature of

the surface P R1.T , R2.T The first and the second principal radii of curvature of

the generating surface T of the cutting tool T The generating surface of the cutting tool TP Discriminant of the second fundamental form of the

surface P TT Discriminant of the second fundamental form of the

generating surface T of the cutting tool TI(P/ T) Resultant matrix of tolerances on relative configuration

of the cutting tool with respect to the part surface P Tr(ax, X) Operator of translation at a distance ax along the X axis Tr(ay, Y) Operator of translation at a distance ay along the Y axis Tr(az, Z) Operator of translation at a distance ax along the Z axis T1.P, T2.P Tangent vectors of the principal directions on the sur-

face P T1.T, T2.T Tangent vectors of the principal directions on the gener-

ating surface T of the cutting tool UP, VP Curvilinear (Gaussian) coordinates of a point of the sur-

face P UT, VT Curvilinear (Gaussian) coordinates of a point of the gen-

erating surface T of the cutting tool UP, VP Tangent vectors to the curvilinear coordinate lines on

the surface P UT, VT Tangent vectors to the curvilinear coordinate lines on

the generating surface T of the cutting tool VΣ Vector of the resultant motion of the generating surface

T of the cutting tool with respect to the part surface P

XNC,YNC,ZNC Cartesian coordinates of a point in the coordinate system associated with the multi-axis numerical control machine

XP,YP,ZP Cartesian coordinates of a point of the surface P XT,YT,ZT Cartesian coordinates of a point of the generating sur-

face T of the cutting tool dsi[i a (i ± 1)] Matrix of an elementary i-th displacement of the cutting

tool with respect to the part surface P [h] Tolerance on accuracy of the machined part surface P hfr Height of the surface waviness hss Height of the surface cusps in the direction of vector Fss

of the side-step motion hΣ Resultant deviation of the machined surface from the

desired part surface k1.P, k2.P The first and second principal curvatures of the surface

P k1.T, k2.T The first and second principal curvatures of the generat-

ing surface T of the cutting tool nP Unit normal vector to the surface P nT Unit normal vector to the generating surface T of the

cutting tool rcnf Position vector of a point of the indicatrix of conformity

CnfR(P/ T) rP Position vector of a point of the surface P rT Position vector of a point of the generating surface T of

the cutting tool tli[i a (i ± 1)] Matrix of the i-th element of the resultant tolerance on configuration of the cutting tool with respect to the part

surface P t1.P, t2.P Unit tangent vectors of the principal directions on the

surface P t1.T, t2.T Unit tangent vectors of the principal directions on the

generating surface T of the cutting tool uP, vP Unit tangent vectors to the curvilinear coordinate lines

on the surface P uT, vT Unit tangent vectors to the curvilinear coordinate lines

on the generating surface T of the cutting tool xPyPzP Local Cartesian coordinate system with the origin at the

point of contact of the surfaces P and T

f1.P, f2.P The first and the second fundamental forms of the surface P f1.T, f2.T The first and the second fundamental forms of the generating sur-

face T of the cutting tool

Σ Crossed-axis angle a The clearance (flank) angle of the cutting tool b The tool wedge angle g The rake angle of the cutting tool gcf The rake angle of the cutting tool in the chip-flow direction d The cutting angle e The tool-tip (nose) angle j e The major cutting edge approach angle j e1 The minor cutting edge approach angle fn Normal pressure angle of a gear-cutting tool l The angle of inclination of the cutting edge m Angle of the local relative orientation of surfaces P and T tP Torsion of the surface P tT Torsion of the generating surface T of the cutting tool zT Setting angle of the gear finishing tool wP Coordinate angle on the part surface PwT Coordinate angle on the part-generating surface T of the cutting tool

R Surface of relative curvature cnf Conformity max Maximal min Minimal opt Optimal P Part surface being machined T Generating surface of the form-cutting tool