GENERALIZAWQNS C35" Ch‘RVATURiS ASSQCEAKD WSTE‘E A CURVE GE? A SURFACE Thesis for Hm Degree 05 pk. D. MICHIGAN STATE UNIVERSITY Joseph Donald Novak 1959 This is to certify that the thesis entitled GENERALIZATIONS OF CURVATURES ASSOCIATED WITH A CURVE ON A SURFACE presented by Joseph Donald Novak has been accepted towards fulfillment of the requirements for Ph.D. degree in Mathematics QM é Zen, Major professor Date June 3, 1959 0-169 LIBRARY Michigan State University .. _.__._.....__...-..———._.4 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE NEWS 51 NF; UH! . SITY !__L_______i _J MSU Is An Affirmative Action/Equal Opportunity Institution emu»: “AW/”firth? «m.— M. ; ....." . ~ .fi : .. : ~«.n‘l”.‘.“Qg! M 1W GENERALIZATIONS 0F CURVATURES ASSOCIATED WITH A.CURVE*ON A SURFACE by Joseph Donald Novak A THESIS Submitted to the School for Advanced Graduate Simdies of Michigan State University of Agriculture and,Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1959 9/5/82. 9/20/49: 'me author wishes to- express his sincere thanks and appreciation to his major professor, Dr. V. 6. Grave, for the helpful advice and encouragement given during the writing of this thesis. ii GENERALIZATIONS OF CURYATURES ASSOCIATED WITH.A CURVE ON A SURFACE by Joseph Donald Novak AN ABSTRACT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and.Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1959 Approved 45”,. ABSTRACT Toe main problem of this thesis concerns itself with the gener- alization of the geodesic curvature, normal curvature and geodesic tor- sion of a curve on a surface in ordinary space. The method of general- ization depends upon introducing at a point of the curve a trihedral which is determined by an arbitrary unit vector V associated with the curve. has generalized curvatures are defined to be the elements of a curvature matrix associated with the trihedral. When the arbitrary vector V is chosen to be the unit vector tangent to the curve, the gen- eralized curvatures simplify to the ordinary geodesic curvature, normal curvature and geodesic torsion of the curve. The vector V is first limited to be a unit vector in the tangent plane of the surface and the generalized curvatures turn out to be the angular spread of Graustein and the normal curvature and the indicatrix torsion of a vector field studied by Pan. However, the point of view taken in this thesis is different in that the emphasis is placed on the curve with a variable vector V rather than upon a fixed vector field. The notion, introduced by Grove and Pan, of a first curvature of a curve relative to a congruence of unit vectors associated with the surface is extended by considering the derived vector of any unit vector in the tangent plane as compared to the derived vector of the unit vec- tor tangent to the curve. A relative parallelism is defined which is somewhat different from that obtained by Pan. When the vector V associated with the curve is allowed to be any unit vector not necessarily tangent to the surface, the generalized iv curvatures are obtained in terms of the curvatures previously studied for the limited case. A new type of parallelism, called.normal paral- lelism, is defined for vectors which need not lie in the tangent planes of the surface. the asymptotic curves prove to be autoeparallel curves with respect to this normal parallelism. The theory previously developed is applied to the study of con- gruences of lines associated with a surface. Conditions are obtained for certain nets of curves on.the surface. A.method developed'by V. G. Grove for the study of ruled surfaces proves to be inportant in the above generalizations. Therefore, in the firstrpart of the thesis this method is introduced.and extended to char- acterize ruled surfaces and curves on a ruled surface. TABLE OF CONTENTS CHAPTER PAGE 1.1monucrIoN................ 1 2. mECURVAWREMATRIXOFARULEDSURFACB. . . . . . .14 3. CLASSIFICATION or RULED SURFACES AND CURVES on RULEDSURFACES............. 9 h. CLASSICAL CASE: VECTOR v TANCENT TO THE CURVE . . . . 26 S. VECTOR v TANGENT TO THE SURFACE s . . . . . . . . 32 6. ms RELATIVE 'r-CECDESIC CURVATURE . . . .1 . . . . 56 7. THE 0mm CASE: vac'ma v ANY UNIT VECTOR . . . . . 65 8. consumer or LINES ASSOCIATED mm A SURFACE s . . . 76 BIBLIOGRAPHI............... 1. INTRODUCTION The main problem of this thesis concerns itself with the gener- alization of the geodesic curvature, normal curvature and geodesic tor- sion of a curve on a surface in ordinary space. The method of general- ization depends upon introducing at a point of the curve a trihedral which is determined by an arbitrary unit vector V associated with the curve. The generalized curvatures are defined to be the elements of a curvature matrix associated with the trihedral. When the arbitrary vector V is chosen to be the unit vector tangent to the curve, the gen- eralized curvatures simplify to the ordinary geodesic curvature, normal curvature and geodesic torsion of the curve. After demonstrating in Chapter 1; the application of the curva- ture matrix method to the classical case when V is the unit vector tan- gent to the curve, we take up in Chapter 5 the first generalization ' when V is limited to be a unit vector in the tangent plane of the sur- face. This case is treated separately because the later generalization to any unit vector V utilizes the curvatures obtained in this limited case. Then, too, we obtain some properties and theorems which do not hold for the more general case. The curvatures associated with a unit vector T1 in the tangent plane turn out to be the angular spread (or associated curvature) of Graustein [3] and the normal curvature [8] and the indicatrix torsion [10] of a vector field studied by Pan. How- ever, the point of view taken here is different in that the emphasis is placed on the curve with a variable vector V rather than upon a fixed vector field. As a consequence, the generalized lines of curvature l differ from the lines of curvature of a vector field defined by Pan, 'with the advantage that certain preperties of ordinary lines of curva- ture can'be generalized. Before we take up the most general case for the unit vector V associated.with the curve, a generalization of the relative first curva- ture of a curve studied by Grove [6] and by Pan [9] is introduced in Chapter 6. As in the cases mentioned, there is defined a curvature of a curve on a surface relative to a congruence of unit vectors F associ- ated with the surface, but the definition is extended.by considering the derived vector of any unit vector in the tangent plane as compared to the derived vector of the unit vector tangent to the curve. A.relative parallelism is defined.which is somewhat different from.that obtained by Pan, although all his theorems concerning the relative associate curvature of a vector and relative parallel displacement along a curve remain'valid. The most general case for the unit vector V, not necessarily tans gent to the surface, associated with the curve is considered.in Chapter 7. In this case the generalized curvatures are Obtained.in.terms of the curvatures previously studied in Chapters 5 and 6. A new type of paral- lelism, called normal parallelism, is defined for vectors which need not lie in the tangent plane of the surface. The asymptotic curves prove to be autoaparallel curves with respect to this normal parallelism. In Chapter 8 the theory developed in the previous chapters is applied to the study of congruences of lines associated with a surface. Conditions are obtained for certain nets of curves on the surface which are related to the congruence of lines, particularly for those curves which are lines of striction for ruled surfaces of the congruence. A.method developed.by Grove [5, pp. 213-219] for the study of ruled surfaces, which also utilizes a curvature matrix, proves to be important and.helpful in the abovewmentioned generalizations. Therefore in Chapter 2‘we present the essentials of Grave's method and.define his curvatures t and k of a ruled surface relative to a curve on the ruled surface. In terms of these curvatures certain classes of ruled surfaces and curves on ruled surfaces are characterized in Chapter 3. Thus a class of surfaces, called constangle ruled surfaces, is defined by the condition that t/k - constant, just as in the study of curves cylindri- cal helices are defined by'a similar condition. Some of the results of Chapter 3 are applied in the theory of the generalizations for curva- tures of a curve on a surface. 2. THE CURVATURE MATRIX OF A RULED SURFACE In a Euclidean space of three dimensions we shall consider a fixed left-handed coordinate trihedral of reference with origin 0 and 11-, x2-, xB-axes denoted by 0-(11,12,23). Referred to this coordinate system a point X has coordinates (x1312 ,1:3 ) and the position vector 1 - 6% has couponents (xl,xz,x3). In general, capital letters shall denote a vector with components given by lower-case letters; thus the expressions (2.1) ‘ v - (v1,v2,v3), v - v1 shall mean the vector V with components (vl,v2,v3) or v1. We shall now consider a general theory of real ruled surfaces; i.e., ruled surfaces containing real generators. Let 01(u1) be the unit vector in the direction of the generating line g1(u1) passing through a point A(u1) of an arbitrary nonisotropic curve C lying on the surface, of which the equation is I - !(u1), where ul is the arc length along the curve. A generic point P(X) of the ruled surface R is given by the vec- tor equation , (2.2) X - f(u1) o u201(u1), u: 3 111 1 11%.. We shall assume that the real functions yi(u1) and gi(u1) possess con- tinuous derivatives up to and including at least the fourth order. The directed distance AP is given by the parameter u2, and the generators g1(u1) of the ruled surface are the paranetric curves u1 - constant. We shall call 61 Egg-51mm of R and the curve C the directrix. Since the scalar product 01.01 - 1, it follows that 61.6; 8 0. (The mark ' will always indicate differentiation with respect to uh h 5 Hence, either 61 is a constant vector, in which case R is a cylinder, or the vector G; is perpendicular to 61. He assume that R is not a cylinder and write. (2.3) G; - 1:02, 62.02 - 1, k > o. ‘lhe function 1: is the courbure incline/e of C relative to 61 as defined by Aoust. Let G be defined as the vector product of GI and 62; i.e., 3 G3 - 61 x 62. Then the determinant lGlG2GBI - l, and the vectors 01,62, G3. in that order, form a left-handed system. The three vectors satisfy the equations ! _ 3 where the repeated index indicates the usual summation convention. By differentiating (2.5) 61.63 - 513, where 613 is the Kronecker delta, we obtain ' t (2.6) 01063 + Gd. Gi - 0. Substitution of (2.1;) into (2.6) gives the condition 1 3 _ k3 + 1:1 0, which states that the matrix (kg) is skew—symmetric. We call it the 325- vature matrix 9_f R relative to C. But from (2.3), 1% - 0, It: - 1:. Hence (2.1;) asst-es the form I I l (2.7) G]. - RG2, 02 I 4:61 - tGB, G3 I 1‘82. Eluations (2.7) will be called t_hg henet-Serret equations of R relative to C. For a non-cylindrical ruled surface there is just one common nor- mal n to two generators g(u1) and §(u1 +4 111). The vector normal to g and 'g' is given by (2.8) 01(u1) x 01(u1m1) - 01(u1) x [01(u1)m;(u1)an1+ G]. .mllm31461xa, tannin-O. _ Au +0 Au]- The limit of this vector as g approaches g over R is therefore 1:63. The limiting position of the foot of the common normal to g and g is called _t_l_l_e_ central pgint of g, and the locus of the central point is called 213 line 91 striction on R. The vector G is a tangent vector at 3 the central point, and hence 02 is noraal to the sm'face at the central point. Let the point 1 given by (2.2) be the central point. It follows therefore that at the central point I I (2.9) x .G - I -G . uzk - o. 2 2 Let 51,92,33 denote the cosines of the angles determined by the tangent vector I. of the curve C and the vectors 61,02,63,respectively. Hence ' 2 2 2 (2'10) I ' p1°'1 ’ $202 ‘ B363’ '31 ‘ B2 ‘ B3 ' 1’ From (2.9) it follows that the central point of g has the position vec- tor given by (2 11) ml) - ml) - ra (uh - 1/1: ° ‘ 261 ' r ' The directrix C will be the line of striction on R if and only if 32- O. The first and second partial derivatives of I are found to be 11 - x' - I’ o uZGi - 9101 + (52 4 name2 o 3363, I2 ‘ 91’ (2.12) 111- (Bi-kfizfu2k2)61 + (kBl+t53¢fl;*u2k.)Gz + (BQ-tBQ-uzkt)6 , xl2' kcz’ 122' 0’ where subscripts l and 2 indicate differentiation with respect to u1 and u2, respectively. The unit normal vector N to R at a point X on R is given by (2.13) N - (X1 2: 12)/L, L - 40:1 1: lJ-(x1 SE72). Substitution of (2.12) in (2.13) leads to (1' . u2k62) x “1 . 5:62 - (32 + nzkm2 ’ L 2 £3 + ($2 + uzk) Now let. C be the line of striction so that [32 - 0. The angle 6 between (2.114) N - the normal vector G at the central point of the generator g through it 2 and the normal vector N at the point X is given by cos 9 . 62-}! - 5;: _ r32 {83 + 0121:)? 41,2332 * (112)2 Hence (2015) tan 6 ‘ ‘12 . F5; The expression (2.16) p - rfl3 is called 1h: parameter 3f distribution 93 the ggnerator 3. Since p is independent of u2, 213 t_g_nge_n_t g}; 35. 222 between the; tangent p_l_a_£e_ at alum Xflébmramrflflmmfikbwufl mile 313 se-called central p_l_a_n_e_, 5.3 Emrtional 3.393 distance 2; 133 the central L. _A ruled surface with p-constant will be called an isoEential Q33 surface because of the property that, at points along each generator which are at the same distance 112 from the corresponding central points, the tangent planes to the ruled surface make equal angles with the corresponding central planes. The expression r33 is indepen- dent of the curve C used in its definition. By use of (2.12) and (2.114), oonponents of the first and second fundamental quadratic tensors of a [2, pp.121;, 215] are found to be gn . 5.15 .. pi + (32 c 1121:)?- 4 pg - 1 + (112102 + 21121432. g12 " x1'x2 " 31’ g22 ' J‘2'1‘2 " 1' (2.17) . . 2 . 2k . 2 . fidll'fl°x11 ' [fl3(kfll*t53*32+u k ) " (52"‘1 )(BB'tflz'u 1:13)]: qulz-qgn-xlz " k‘33’ V3d22 WEN-122 " 0’ where 2 3 " g11322 " g12"” When 0 is the line of striction, the Gaussian curvatm'e is given by 2 2 2 (2.18) a . ‘51“22 1‘13— . " r 33 . e (1.23;. (112m? Hence the given ruled surface is a developable if and only if the para- meter of distribution p is zero. Since 53 - 0 for a developable surface, the line of striction is the edge of regression. 3. CLASSIFICATION OF RULED SURFACES. AND CURVE ON RULED SURFACE ‘Ihe method of the previous chapter shall now be used to character- ise ruled surfaces and curves on ruled am‘faces in terms of the curva- A tures k and t and the direction cosines 51,92,53. Unless stated other- wise, we shall assume that the ruled surface is a 3.239;; (not a deve10p- able surface) and therefore [33 K 0. The asymptotic lines of a ruled surface R are the integral curves of the differential equation (3.1) daflduadup - o, (a - 1,2; 5 . 1,2). (In what follows Latin indices take the values 1,2,3 and Greek indices the values 1,2.) Substitution of (2.17) in (3.1) results in (3.2) [33(kalot53+aé+u2k')-(92cu2k)(B;-t52-u2kt)Jtdu112+2k53duldu2 - 0. Hence, the generators, u:l - const. , of the ruled surface form one family of asyluptotics. The other family is given by 2 ' 2 ' 2 ' 2 ' e ' “t "’ "’ e (3 3) 2k$3(u ) (B2*u k)(B3 $2 u kt) 53(k31*t$3+32+u k ) 'fine line of striction will be an asymptotic curve if and only if u2 - -r$2 satisfies equation (3.3), which condition results in , t t (3.1» (113,) - real . tsp - 91 . 3533. men the line of striction is the directrix C, it will be an asynptotic curve if and only if (305) RB]. 9 tBB ‘ oe ihe tensor s“8 is defined [2, p. 135] by the cowonents (3.6) ‘11 - 522 - 0, 512 . l/ag, e21 -= - l/ag. The lines of curvature of R are the integral curves of the differential equation 10 (3.7) caBgaYdflbdnYdua - 0. Substituting (2.17) and (3.6) in (3.7). we obtain I I (kB3[1+(u2k)2+2u2kB2] - alla3(kal+ts3+szw2k ) 2 ' 2 2 I 2 I (3.8) - (82m kaB-taz-u munch} - [53(kpl+t33+22+u k ) I - (BzouszBB-tBZ-uzkt)]du1du2 - k53(du2)2 - o. ‘Ihe line of striction will be a line of curvature if and only if u2 - -rB2 satisfies equation (3.8), which condition gives ' k (309) (r32) . (tfil ' k83)/t . B1 "EBB’ When the line of striction is the directrix C, it will be a line of cur- vature if and only if (3.10) tel 4:53 - o. A necessary and sufficient condition that a curve be a geodesic on R is that its principal normal at each point of the curve be normal to R at the point. From (2.10) and (2.111), the directrix C will be a geodesic if and only if the two vectors II I k ) ( I k ( I Y . (Bl - $2 01 I 52 I 31 . t33)02 O 33 ' t52)633 N " B362 ' £32‘33 are proportional, which condition results in B1 " “'32 " 0' I I Since a; . B3 4 pg - 1, the second equation of (3.12) simplifies to I B1(B1 - kflz) . 0' Hence, the directrix 0 will be a geodesic on R if and only if (3013) 3].. - kflz - 00 (3.11) (3.12) 11 As an application of Grove's method for the study of ruled sur- faces, it is very easy to prove the following 11mm 3.1 (Theorem _o_f_ £29933). _I_f_ _a_ m 313 a @391 surface satisfies fl 9_f_ the 3.13333 conditions, (a) g; m a geodesic, (b) of 2.21.9.3 3 L195. 93 striction, (c) 9_i_‘_ intersecting the m- mfléwcmmtw mgmfim conditions 3&2: i122 sari- MI Let the curve be the directrix 0. Then the three condi- tions are (a) a; - ks, - o. (b) a, - o. (c) a; - 0. Since k>0, it is obvious that am two of these conditions imply the third. When the line of striction is the directrix 0, condition (3.13) that it be a geodesic reduces to (3.11:) 51' - 0. We now seek the condition that the line of striction be a geodesic when an arbitrary curve On It is the directrix. Differentiation of (2.11) twice results in (3.15) x" - (a; - (r32>"161 . [k5, + ta, - k_ .99. isotgggential M surface (p = 399333.22 l 0). M. 'lhe generalized helicoid whose generators intersect the x3-axis at a constant angle a is given by x1 - ' u2 sin a. cos 9(u1), (3.22) x2 - u2 sin a sin 0(u1), x3-u14u2cosa, where u1 is the distance along the xB-axis from the origin to the point of intersection of the generator, u2 is the distance along the generator from the x3-axis, and 9(u1) is a function of 111 which determines the angle made by the plane through the generator and the xB-axis with the x11:3 -p1ane in the positive direction of rotation. It follows that 01 - (sin a cos 9, sin a sin 9, cos a), (3.23) G2 s' (-sin 6, cos 9, 0), (l3 - (-cos (t cos 9, -cos a sin 6, sin a), and (3.2h) where k} 0 since 0(a< fl and the directed angle 9 is chosen to in- crease as u1 increases. From (3.21;) and the relations (3.25) 91 - cos a, 52 c 0, $3 - sin a, we find that 1 (3°26) t/k ' '°°t “’ kBl ‘ tB3 ‘ 0: p = r53 “%§' . l6 ‘lherefore, the parameter of distribution is a non-zero constant if and only if $1 is a non-zero constant, in which case the generalized heli- coid is an ordinary ruled helicoid [ l, p. 11:6] with a constant 223- meter g_f helicoidal mo___t_i__on (211—91. comma! 3.1.. i riggt conoid will 9.9. 3 riggt helicoid g ;a_1_1_d £11 _i_f_ it _i_s also gisothgential ruled surface (p s const.}(o). Proof. In the details of the proof of theorem 3.h we now assume sin a - l and cos a - 0. Equations (3.26) then become - 1 (3'27) t-O, Bl'pa P'gg—e 1 Hence, if fig- is constant, the right conoid will be a right helicoid [l , p. 1116]. We summarize'the preceding results as follows: t/k - const. / 0, - constangle surface; t/k - const. )1 0, 32-0, kfllatBB-Opu-generalized helicoid; t/k - const. K 0, 52-0, ksl+t53-o, p-const. «um-helicoid; (3.28) t - 0, — — - ———--— ——right constangle surface; to - 0’ $2 . 0, Bl . 0, ‘ —r1ght comid; t - 0, B2 - 0, $1 a O, p - const.- right helicoid. Consider a point I near the point I on the line of striction of R, and consider also the corresponding generating vector 51 at I. From the Taylor developments [11, p. 3] I'I’Ybu1§1"(fiu1)2/2*Im(A111°)3/6*Y,lifiti3o, “90(411)3 51- 61+ ol'aml + 61"(su1)2/2 4» 0;" (4u1)3/6 + Go, limit 3L0 =0 oul-fioku )3’ l7 and equations (2.7) and (2.10), we find that any point on the neighbor- ing generator El is given by i-iebal-Ieglclegzozez363, where E’- hoflla u1+( Bi-hkz) (m1 )2/2+( 8.: -k291-ktBB-3hkk ' ) (A11 )3/6401, 5?. ma au1+(kelet53+nk')(au1)2/2 + [Zkfli4k'pl+2t3;+t'$3 (3.29) eb(k"-k3-kt2)][Au1]3/6 + 02. 53- 23 eu1e(a;-hkt)(eu1)2/2+w; Mal-3:33 .b(-2k't-kt')][nu1]3/6 e 03. he terms 01,02, and 03 are infinitesimals of order greater than three. Equations (3.29) represent the local coordinates of the point I referred to the trihedral x-(GIGZGB).with the striction line as directrix ((32-0). Demanding that the general equation of a quadric surface, a(€1)2+b(52)2»(£3)20d£1£2+eE1E3+f£2E34aE1+1E2&3": - o. be satisfied by the expressions (3.29) for €1,E2,£3 identically in h and Au1 as far as and including the terms in (Au1)2, we obtain the conditions a - g - n c 0, dk'O, OBB’H“O’ ”3'0, I 3 2cp§ . 2e5133 . 1(kp1 . tBB) - 0. he solution of (3.30) under the condition that 1:93 I 0 leads to the equation of the osculaggg guadric ks - tB (3.31) t(£2)2 - --1—£,3—-2(r.3)2 . 21:85.3 - (lnlplfezci‘ - 2535.2 - o. (3.30) 2bk2-ekt-0, eB +2fk534lk'I-O, 18 manna 3.5 1h: osculaEg guadric pf _a_ real scroll, _ngt _a rimt const_a_n_gle ruled surface, ig _a Moloid 93 935 sheet. Egg 3 riggt constgggge ruled surface _thg osculatigg guadric _i_s. _a_ m- bolic paraboloi . Proof. me invariants [7 , p. 191] of the osculating quadric (3.31) are 45- 1.23;) o, D c -k2t, 1 - 2t - kBl/BB, a - t2 -(ktal/p3)- :3 - [1/t1[(1nlpl)'12. If t I 0, then D f 0. be characteristic equation A3 . 1A2 4 JA - D ' 0 (3.32) will have three roots of the same sign if (3.33) DI>0, .00. But if m - fungal/53) - 2t21> o -(ktBI/33)(-2t2 and J - tz-(ktBl/B3)-k2-[1/h][(lnlpI)']2 < -t2-k2-(1/u1[(1nlpl )‘120. Dt - -k2t2('o, l9 and I and D differ in sign. ‘lherefore, the characteristic roots are not of the same sign. Since A) 0, D I 0, and the characteristic roots do not have the same sign, the osculating quadric must be a hyperboloid of one sheet [7 a P- 192]- Hhen t - 0, then D - 0 (the matrix corresponding to the deter- minant D is of rank two), and A> 0. Since J - - k2 - [1/M[(1n|pl)')2< 0. the two non-zero characteristic roots differ in sign. 'nlerefore, the osculating quadric is a hyperbolic paraboloid [ 7, p. 192], and the theorem is proved. We next determine the flecnode points on the generator g1 of R. Demanding that equation (3.31) be satisfied by the expressions (3.29) for 9,22,? conditionally in h and identically in au1 as far as and including the terms in (Au1)3, we obtain the condition I 2 3H3 31! ) k(—-:-3- k t - 2kt')b2 4 [hi2 (tel-1:33) e 2395'- “3 (3.3h) 333(k )2 1:519 k - 253k" 1h +1: 51- 1353*20‘51't 83)+{-—;—Bl- 5153;720 0, or, when t ,l o, Waltz/k |)' 112+ [2(1nlpl)" - h(kB3-tfl1)/p - (1nlpl)'(1nlks3l)')h (3.35) " 2[(k814’t$3)/k]' 4' (493(51/33). " (kBl‘tBB)(1nlk53|)./k " 00 If h is a root of (3.3M or (3.35), then the point I 4 ht}1 is a flecnode point on the generator g1, and a flecnode curve is determined by (3.36) ' m 1) - 2011) + h(u1)01(u1). 20 In general, there will be two distinct flecnode cm'ves, which may be 11088111817. Equation (3. 35) will have a zero root and the line of striction will be a flecnode curve if and only if (3.37) (kgot33)(1nlk33l)'/k - 2{(k31+t33)/k)' + hB3(B1/)33)' - 0. THERE! 3.6. 9.191 935 finite flecnode curve exists 93 _a_ rigt constgggle surface 2}; 93 _ag isoEggenfial const_g_nge surface. Proof. men t - 0, in (3.31.) the coefficient of 112 vanishes mile the other coefficients are different from zero. Similarly, for 2 in (3.35) vanishes p .. constant and t/k - constant, the coefficient of h while the other coefficients do not. 'lhese constants must be different from zero. THEDRD! 3.7. g 13 isotflgential M surface _th_g _tgg flecnode m 1% 133 guidistant gm _t_&_1_e_ ling 2f striction if and 9-1le if _t_h_e_ Ling 9_f_ striction _i_s_ _a. Ling 2f curvature. mg. If p -= constant I O, the coefficient of h in equation (3.35) will vanish and the too roots will be equal and opposite in sign it and only if 1:53 - tfll - 0, which by (3.10) is the condition that the line of striction be a line of curvature. THEOREM 3.8. 1123 pm, u2 = 32953., 333 constant distance meannn___smouoniinie mean. mission): if _t_h_g £13.33 surface 33 _a_ £1.23: helicoid. _l:r_9_9_f_. when the directrix C is the line of striction ((32 - 0), the differential equation (3.3) for the curved asymptotics on R becomes (3.38) 2(u2)' - -(t/p)(u2)2 9 (lnhol)'u2 - (1:31 . tB3)/k. This equation is satisfied by all constants if and only if 21 t-0, p-constantfo, 01-0, which by (3.28) are the conditions for a right helicoid. Associated Bertrand curves have the property that.their princi- pal normals coincide. In a somewhat analagous manner, we define two 93- noranal _ru_l_g_d surfaces to be two ruled surfaces which possess common nor- mals to the surfaces at corresponding points of their lines of striction. First we study the case in which the lines of striction coincide. THFDRDI 3.9. lb: .1523 general _r_u_l_..c_d surface R M _i_s_ conormal 3,529,332 surface Rflngjggf striction_i_s_9_b- moon—name mmmamamoz 332:5 593 Egg g striction m _a. constant 2g}: 9. _1_h_e_ £3;- vatures _o_§ 3:93 associated conormal surface _a_r_e 51.11.33 by k- kcose otsine, (3.39) _ t - ~k sin 0 e t cos 6. 23993. Let the directrix C be the line of striction on R. Since the normal vector '62 of ii along c must coincide with the normal vector G2 of R along C, the vectors 51,52,53 of R must satisfy the relations (11- GlcoseoGBsine, (3.1m) 5 - 02, 2 5 3 Where 0(u1) is the directed angle through which 01 is rotated about 02. By use of (3.140) and (2.7), it follows that -I - " ol-(kcoseetsinemzee'GB-EG I- - a(-ksin00tcose)02-GGl-t02. --Glsine+G3cos6, 2: I 3 The normals 62 and 52 will coincide if and only if 9 is a constant. The 5 22 cux'Vatures E and t of R are then given by (3.39 )3 and -I _- I 02"k°1‘fi3"k°1'w3'62° THEOREM 3.10. 213 93.9.9.3 general _nggg surface it conormal 33 _a_ 5.1192 _rt_1_l_gd surface R _i_s_ obtained _by W _t_h_g generamg 129.29.? 01 parallel lg _i__t_s_;_el_.f_ g constant distance 1193 62, _t_h_g surface m 2.2 _t_h_g 29.111112 9_n_ _thg _li_n_g gt; striction, _a_n_d_ Egg by rotatigg _th__i_g digglaced mtg; throfl _a_ constant angle _i_n the; 21355 perpendicular lg 02. 3313 923 exception 9£c_1_1_r_s_ 3.11.92. _tlr_1_g _li_n_g 31; striction 25 R is. 3 _li_r_1_g 93 curvature 3133 the constant distance is; r31. _I_n_ _th_i_._s_ gag _th_e__li_g_e_ pf striction 5 25 R 92' generates 11.1.29 3 fixed oint, 2n_d_ _t;h_e_ developable generated 3 ‘_t_h_e_ surface normals 319% £1.13 _1_i_r_1_§ 3f striction _ig 3 3939.. M. Let the line of striction 1(u1) be the directrix C on R. men any point on the, surface nomal 02 along C is given by (3-111) 3 - I at £202. If '31 is the arc length along the locus of I, it follows that - 1 d? d! d 2 2 ' 2 (3.1.2) 351 - 3:1 3‘31 - W11 - In: )61 + (z ) e, + ((33 - ts. )G3J/A. where (3.1.3) A - [(31 - no.2)2 + [(8)32 + (53 - a???” a 0. To find the most general ruled surface R which is conormal to R, it will b. no loss of generality to assume that the generator El is parallel to 81 since, by ‘lheorea 3.9, all other conormal surfaces with the same line or striction must be obtained by rotating g1 through a constant angle ' fibeut the normal 62. Hence we seem 51 - G . The curve Y(\'i1)will be 1 the line of striction on R if 23 (3.1.1.) 32 - if e2 - (52W; - 0. Hence £2 must be constant. The excluded case A - 0 implies that (3.1.5) 51 - kzz - 33 - v.2 - 0. (zzf - 0. thence (3.1;6) 1:93 - tfil - o, :2 - r51 - const. Conversely, for a scroll, conditions (3.h6) imply (3.16). 'Ihe first equation of (3.!46) is the necessary and sufficient condition that the line of striction be a line of curvature. mm 3.11. 133 distinct yes; 93 striction 93 2'2 commal wwmfwesakmd mgflmgmfim of curvature an the respective surfaces. as 13.92 g}: striction flailingsiwcmatmeaeshmmogsnmgfle ling g; striction 3g _tgl_e_ associated conomal surface is also a l_.’_t__n_e_ _o_f_ curvature. M. From 'lheorem 3.10, corresponding points I and T of the We lines of striction c and '5 are equally distant. merefore, we need to examine only the condition that the tangents at I and Y be parallel. By use of (2.10) and (3.1;2) with :32 = (52)' - o, the two tangents will be parallel if and only if (3.1.7) 33(31 - tr?) - 51(53 - v.2). Or (3.h8) 80433 - tfil) - o. The two curves coincide if 252 - 0, while 1:133 - 1431 -= O is the necessary and sufficient condition that C be a line of curvature. Conversely, if E2 ,l 0, condition (3.148) implies (3.1;?) and the tangents to c and ‘6 will be Parallel. For the proof of the second statement of the theorems, consider first the case of 'lheorean 3.9 in which 01 is rotated through a constant angle 6 about 62. From (3.!i0) it follows that '51 - y'. '51 - fllcos 6 + szin 6, .— I (3.119) Bz-Y'Ez-O, F It '5 sin 9 o e 3 - . 3 -$1 33008 . Hence. (3.39) and (3.13) result in (3.50) I? 93 - £31 - 1:33 - tfll. For the case in which 01 is moved parallel to itself a constant distance £2 along the normal 62’ we obtain for the ruled surface Ti 1? - VB, 3‘. - t/B, 3]. ' 61. £31 " (31 " 1‘52)/Ba du - G - -- - o, .52 2 (11-1-1 '53 - 03- £51 - (:33 - tat/B, mere (3.52) B - [(fil - 1:652 + (93 - “2)211/2 )4 0. Hence, (3.53) E ‘63 - “i 31 = (1:53 - tB1)/Ba. As a result of (3.50) and (3.53), it follows that (3.53) holds for any tum conormal surfaces, and the second part of the theorem is proved. HERE! 3.12. If a line of striction is cannon to two conornal ruled surfaces, _i_t will be _a_n mtotic curve _og both surfaces 25 95 93 neither surface. On conornal surfaces with distinct lines 9_f striction, _thg E _11933 of striction can m be asmtotic sun's: “mammal;- £1033. lhe first statement follows imediately because (3.39) and (3.19) imply that (3.51:) 1? Fl * 15.53 - kfll + tBB. 'Ihe relations (3.51) give (3.55) E 151 + 1‘53 - [tel + an - (k2 . t2)£2]/Bz. Hence, by the same argument as in meoreln 3.11, the relation (3.55) holds for any two conorlnal surfaces. For the case of real scrolls with distinct lines of curvature, (k2 " 1‘2 )EZ i‘ 09 and therefore (E '51 + '1": EB) and (143:l 4» +433) cannot vanish simultaneously. h. CLASSICAL CASE: VETOR V TANGEWT 'm M CURVE In order to duonstrate the motivation for the later generaliza- tions, we introduce the geodesic curvature, normal curvature, and geo- desic torsion of a curve on a surface as elements of a curvature matrix associated with a trihedral at a point of the curve. Graustein calls attention to this trihech'al and the curvature matrix[ It , p. 163 l, but he presents it as an after thought rather than as a basic tool in the study of these curva‘hlres and the associated curves on the surface. By comparing the trihedral with the one associated with the Frenet-Serret formulas, one is easily able to derive relationships between curvatures and to prove certain theorems about curves on a surface. In particuhr, in place of the somewhat artificial method often used, the geodesic tor- sion is introduced as a couponent of a vector just as are the geodesic curvature and the normal curvature. Let 8: xi - x1(u1,u2) denote a real proper surface of class not less than two in three-dimensional Euclidean space referred to a left- handed rectangular cartesian‘ coordinate system. Lot 8‘19 and (lap denote, respectively, the first and the second fundamental tensors of S such that 3 " 81l322 " giz>° and d " dudzz " “:2 f 0' '5‘" man” 11(u1,u2) satisfy the (lanes and the Weingarten differential equations ' 11’“? = dafi n1, (h.l) 1 n ta = "dangYxisY 3 where the functions n1 are the direction cosines of the normal line to to S at the. point x, and the coma indicates covariant differentiation 26 27 with respect to the metric tensor g 05' Consider a curve 6 on the surface S throng: the point 1. Assume that the curve 0 is Wood in a one-parameter family of curves satis- fying the differential equation (l..2) cal/cm2 - {IL/:2, z“ - 6011.112). where (ll-03) gap EOEB " 1 and the {C(ulmz) are of class not less than two. New introduce at a point of the curve 0 the trihedral composed of the following unit vectors: a The tangent to the curve, T1: ti - 3%- xi,a ' Kali,“ .9 The tangential normal,‘1’ : t: -= 110:1,“ " GBGEWEYxi’a s The normal to the surface, N: n1 =- jk/VE', — " .2... eijk( 3:315: " @232), {E— 11 au bu ?u where i,j,k take the values 1,2,3 cyclically, and s is the arc length of the curve. By use of the relations £an - £2711 - 612 - I/JE, (AM)2 + (A23)2 + (.31)? -- s. it can be proved that the determinant |T1T2N| 8 Maxi,“ naxi,a Ajk/y/E— I - 1. 'Dierefore, T1, T2 and N form a left-handed coordinate system. 28 The derived vector of T1 along 0 at a point P is given by 1 ' due 1 dup g} " (ti) " {unfiaa'a's' " €05! ’aBF a which, by use of (h.l), becomes 1 l (Itch) (ti). 3 gagpsz ’6 4’ dapgaEB 0 (Unless specified otherwise, the mark ' will indicate differentiation with respect to the arc lengths of the curve C.) with the aid of the relation 11%! ' gati- . not: ’ (lull) becomes (2.5) (55' - nattgzfité + dupetflni . By the same process and the fact that tapas - matam . it follows that (the) «if - mattflzflti + «21%.,naz‘3n’L . Since the curvatmre matrix must be skew-symmetric, we can write issue- diately (14.7) (n1). - mafia“: - daflnazflté . Eluations (h.5), (11.6) and (11.?) shall be written in vector form as 11;. " 181‘2 + an , I (h.8) .T2 - 4(ng - th , 29 where by definition a. the geodesic curvature x8 a uni ,BEB , the normal curvature In - daBEaKB : . _ _ l3 _ «B Y 5 and the geodesic torsion cg dag-nag a do 731mg E DFINITIONS 9E CURVES 9E 53 {33:1 . *Fa dndu . a dsB mtotic curve, xn- 0; dapdu du - 0. I Line 2.11 curvature, "g' o; ea‘edaYgflfiqudua - o. ‘Ihe Frenet-Serret formulas for the curve C in vector form are 1’1' " ”0'2 ’ (“-9) G; ' "“1 " “’3 ' G; - 1:02 , in which n: 0 and the unit vectors T1, 62 and 03 form a trihedral. for which [110203] - 1. Let Vdenote the (Elected anglemdebythe normal vectoerith the principal normal 62, whose components are denoted by g3. It follows that GzaTzsintthosv, (h.lo) --Tzcos\y+Nsinw. G3 From the first equations in 04.8) and (h.9), we obtain (hen) X02 ‘ 181,2 " ano The first equation of 04.10) ‘and equation (14.11) imply that (14.12) 18' 1 sin v, In a n cos v. Differentiation of the second equation of (h.10) with respect to 5 gives u; = N'sin v s N v'cos v - Técos v + T2 v'sin 1,, which by (14.8). (h.lo) and (2.42) simplifies to I I (b.13) G3 - (v + Tg)G2. Comparing (h.13) with the last equation of (h.9) leads to the result (hen) T ' *' ‘5 1'8. I lhe first equations in (h.8) and (14.9) give the relation (ham x-(§+xfivi As an interesting exercise, theorems can be proved merely by com- paring the curvature matrices in (the) and (h.9). we illustrate with one example. mm h.1. A gene geodesic line, 933 _a_ strai t line, 3:9. 2: line g_f_ curvature. Since 1' - x8 IO, and k f 0, equations 04.8) and (11.9) Proof. I T1 - an , (14.16) n' - - anl - tg(-T2), (4‘2), - th , and I ll - :02, man Gé-«H, .. G; ' 00 V In (h.16) the vectors T1, N and -T2 form a left-handed trihedral with 31 [TIN -'1‘2| s 1. Comparison of (14.16) and (lid?) shows that tg must va- nish. Incidentally, )(n - Z x, the sign mending upon mother the prin- cipal normal 62 has the same or opposite direction as the surface normal N. 5. moron v mm in ms SURFACE 3 Consider a unit vector ‘1:1 in the plane tangent to a surface S at P such that i i a tl-eax,a ,gafleefiel, ushers the e“ are functions of u“ of at least class two. As in chapter It, a curve C on S is imersed in a one-parameter family of curves satis- fying (14.2) and (14.3). Then to each point P of C there is associated a unit vector 1'1, and the a“ are functions of the arc length s along the curve 0. who angle which 11(9“) makes with the tangent r.“ to the curve oshallbedsnotedbye. with the use of (h.1), the derived vector of 1‘ along C at P is l (5.1) (tif - 9a,B(uB)'x1,a . d¢9a(up')'n1. The tangent vector T2 orthogonal to 1'1 is given by (5.2) T2: t2 - 0‘51,“ -= «Baefixi,(I - eBagBYGYxi,a . Since 11’ a . eat: ‘ 45°21” equation (5.1) can be written (5.3) (Hi). - ¢a0a,pipté 4 daaeafiflni. Similarly, with the additional relation that 96¢“, p - - M9“: '3, the de- rived vector of T2 along C at P can be shown to be i 1 - (5.1:) (132) - - he“, fiat: c dafidazfini. Since the curvature matrix must be skew-symmetric, it follows that 32 33 (5.5) (n1). = «lage‘tpti - dafit‘tfltg. Equations (5.3), (5.1.) and (5.5) shall be written in vector torn as Tl " t‘gTz ” tan’ (5.6) g .. - txg’l‘l - t‘tgfl, N' - - tanl + t‘thz , where tug - 9x8 .. ¢a6a,$£fi - eYaeYe“,BzB, (5.7) t‘n .. e“n . defleagfi’ t‘tg I 61:8 II - daBOGEB - sapdwgflaeazv. DEFINITIONSQE CURVA‘IUREBQEE WITH REPECTEQ Tl! tug (6:8) is defined to be the tl-geodeeic curvatmre of the curve C with respect to the vector T1 (ea) in the tangent plane. It is the angular spread studied by Graustein [ 3, p. 559]. tun (exn) shall be called the Tl-nornal curvatm'e of C with respect to the unit vector r1 (9“) in the tangent plans. It is the same as the nor-a1 curvature 9g _t_h__e_ Vector field 1‘1 with re- SPOCt to the curve C [ 8, De 955]e 1{cg (61:8) is defined to be the Tl-ggdesic torsion of C with respect to the unit vector 1‘1 (0“) in the tangent plane. In Pan's ternnnology [10, p. 1453]: this is his indicatric torsion 93 a vector field in a direction. 3h ‘Ihe Frenet-Serret formulae (2.?) for the vector T1 with respect to C are Tl " k62 ’ I (5.8) 62 . " k'L‘L " 1.03, 3 GB ' 1'82. 'nle vector 1:1 is the generating vector of a ruled surface R along C. DFINITIONS 2! CURVE g! S: A curve on 8 shall be called a geodesic with remt _t3 the vector T1, or a Tl-geodeeic curve, if tug - 0 or a B (Se9) g‘ " FSTGY %- ' On In this case the vector T1 undergoes parallel displacement along C in the sense of levi-Civita,- and 62, the normal to R at the central point, is normal to S at the point P of C. The curve 6 shall be defined to be a ‘rl-Lsmtotic curve if tun . 0 or (5.10) d “Beadufl .. o. g In this case 121 is called an mtotic vector of _t_h_g curve at P. Along a Tl-asynptotic curve 62, the normal to R at the central point, has the same or Opposite direction as the vector 1‘ at P. 2 A curve 0 shall be called a Tl-lixle '9}; curvature when t’tg - O or (5.11) e"“3dwg£3595dn‘r - o. 35 These curves become the ordinary geodesics, asymptotic curves and lines of curvature when the vector T1 is replaced by the unit vec- tor tangent to c; that is, when 9“ is replaced by a“. let W denote the directed angle made by the normal vector N of S at P with the normal vector 62 of R at the central point. Formulas (11.10) hold in this case also, and similar to (lull) and (b.12) we obtain (5.12) sz - tug'r2 + tan, (5.13) tug = 1: sin ti, txn = k cos 1!. Differentiation of the second equation of (h.10) and the use of (5.6), (h.1o) and (5.13) result in (5.11.) 03 ' W' * (”3’32- A comparison of (5.8) and (5.11;) shows that (5.15) t . V. e t'tg. Hence, if the angle V between the normal N of S at P and the normal vec- tor 62 “of R at the central point is constant along C, the torsion t of R equals the Tl-geodesic torsion of C at P. In particular, this is true along a Tl-geodesic (‘V s 0 or n) and along a Tl-asymptotic (1r - 3 u/2). The first equations in (5.6) and (5.8) give the relation 2 2 2 (5.16) k - (txg + tun)” . When the reference vector V is not necessarily a unit vector but is any vector i a i - (5‘17) T‘ t " P 1 ea, ‘ Peaxlsa: 8.199%B e 1, p) o in the tangent plane of S, the T-normal curvature of C (or the normal curvature of T with respect to C) is defined [8 , p. 956] to be 36 . a B 'r 5 fl 1/2. (5.18) an dafip du “875d“ du sfipap ) 'nle T-normsl curvature tun of C is the same for all vectors pa which have different lengths but the sane direction a. Replacing p“ by its equivalent pa“ in (5.18), we obtain - a B Y 6 1/2 (5.19) tun daBG du /(gY&dn du ) , which is independent of p. A T-yg g curvu‘ure for any vector p“, defined by 5 (5.20) eapdwgfiap qu - 0, is then the same with respect to all p“ having the same direction 9. Therefore, in discussing T-normal curvatures and T-lines of curvature, it is no loss in generality to limit the p“ to be unit vectors 9“. Since dio, byanargument sinilartothe one usedbyPan [3 , p. 956], it can be shown that there exist two in}; vectors a“, one file negative of the other (we distinguish between 6“ and ~9a), with re- spect to tdlich the T-normal curvature of C at P has finite extreme val- ues .different from zero. mess vectors are defined to be the principal vectors _a_; 3.1.2 _c_u_r_v_e c at P and the corresponding 1‘-nomal curvatures are called the princgal T-curvatures 93 313 m at P. 'lhese two principal T-curvatures differ only in sign. W 5.1. g 11-_l_:_l_n_e_ 9; curvature has 313 characteristic p33- mssaTlieaamu awareness-assass- £1 curvature 33‘ _g. Tfiii‘fi of curvature _i_s_ g princgal T-curvature . M. the principal vectors of the curve C at P are found from (5.19) by finding the functions 9“ which give extreme values for tun, subject to the condition that g “39%;" - 1. Using the method of Lagrange 3'? multipliers , we obtain dafldufi + 2Aga$63 a o. Elimination of A between these two equations gives (5.11). This com- plates the proof. it this point it miglt be well to point out the difference be- tween a T-line of curvature defined here and the line of curvature of a vector field as defined by Pan [ 8: P. 956]. Pan considers the vector field fixed and seeks the curve with respect to which the normal curva- ture of the vector at P has an extreme value. For our T-line of curva- tmre, the curve C is considered fixed and we seek among all vectors at P the two unit vectors such that the T-normal curvature of C (or the nor- mal curvatures of the vectors with respect to C) assumes its extreme val! use for these two vectors. However, if the vector 9“ or pa is given, there is determined by (5.11) or (5.20) a unique T-line of curvature. Our principal T-curvature is a property of the curve, while Pan' s prin- cipal curvature is a property of the vector field. ‘Ihe motivation for the definition used here follows from the trihedral and curvature matrix of (5.6) and from the analogy with the classical case. the condition 3 (tun)/ 36a - 0 results in tTg =- 0, just as in the classical case (when 6“ is replaced by due) 0151/ Qua - 0 leads to 1:8 8 0. Pan's con- dition 3(txn)/ 3(dna') 2- 0 is equivalent to .21 ‘Bd e?r 5 . (5 ) 5 “T855 (111 '0, which in general is different from (5.11). The curve (5.21) determined by the unit vector GT is» called by Pan 9. line 9_i_'_ curvature g; the field, 38 and the direction du5 of the curve is the WWEE gag _a_t_ P. We now investigate under what conditions on a“ and du“ (5.11) and (5.21) are mivalent in the sense that these two equations would define the same curve for a given unit vector 6“, and the same two unit vectors for a given curve C. a unit vector 9“ and a curve c with direction du“ shall be said to be in 213 reciprocal relation R if 6“ is a principal vector of C and du"L is the principal direction of the vector field 9“ at P. 'Ihe rela- - tion R implies that (5.11) and (5.21) are equivalent as explained above and that C is both a T-lins of curvature and a line of curvature for the vector field. Certain basic relations will first be derived. If 9 is the di- rected angle which the unit vector 11 makes with the curve C, it follows that tun c Gun - dugoutB - dafluacos e + nasin 6)£B- uncos 6 - rgsin e, (5.22) {:8 a 61g udafltbazfls -daB(-£°’sin 6 macos 6)F.‘3- unsin 6 + rgcos 6. These useful formulas give the Tl-normal curvature and the (Pl-geodesic torsion of a curve with respect to any unit vector at P in terms of the angle 0 and the normal curvature and geodesic torsion of the curve. in interesting m at 21:; 112.1155! B.____°°d°31¢ _____torsion as. a sure 3:. sea niesa__a___ne ativeeisammlemwrvmeaiseyimmiaea e an: anal: 3.133 in: as... 111mm 5.2. A unit vector T1 and a curve C with direction du“. —*~ flkh&_2_mi m1 “relation Rheielziianaaiss following three cases holds: 39 (a) _Iisgpsurfacemichigndtherggpngpminim surface, Cipghmgfcurvaturegrligmtgc. 0’) ESiiasnia-s Ciiasubitr _cumi’iSaeéTlii t_a_ngent _tp C. (c) Esggnnimal surface, TlggCgMinclinedE 3 principal direction 9_f_ s; ' that a, _tg 2 line 2; curvature. Proof. minination of do“ between (5.11) and (5.21) results in no «5 6 pa 1' . (5.23) e e daygfibe s dp'tgome 0. By use of (5.2) and the relation (5.211) 51106908“) - 8Y9, equation (5.23) reduces to (5125) have" -= o, where h ar’ the tensor of the third fundamental form for 8, satisfies . 11> Similarly, elimnation of a“ between (5.11) and (5.21) gives where E7 - qu/ds and up is the tangential normal. Conditions (5.25) and (5.27) imply that the orthogonal vectors a“, t“ and a“, n“ must also be orthogonal on the spherical representation of 3. But, for a surface which is neither a minimal surface nor a sphere, the lines of curvature are characterised by the property that they are represented on the unit sphere by an orthogonal net. Hence, case (a) is proved. ho Since a sm'face and its spherical representation are in confor- mal correspondence if and only if the surface is a sphere or a minimal surface, (5.25) and (5.27) hold for any two orthogonal vectors and di- rections on a sphere or a minimal surface. However, once either C or ‘11 is fixed, then the other is also determined by the relation R. For a sphere the components d“B are proportional to g “5 and therefore the geodesic torsion vanishes for every direction due. Hence, from (5 .22) tun - (l/a) cos 9, where a is the radius of the sphere. The extreme values for tun occur when 6 - 0 or it, which proves case (b). The principal T-curvatures for every curve are 1(l/a) - 1001/2, where K is the Gaussian curvature of S. If S is a minimal surface, the mean curvature H vanishes, so that (Se28) K1 . X2 . 0, where XI and x2 are the principal normal curvatures 93 S _a_t P. Euler's equation, giving the normal curvature for any direction d112/ dul, be- 001308 (5.29) an - 110032 '5 - slain2 ‘6 - ulcos 2'6, where '5 is the angle from the positive direction of the line of curva- ture with the principal normal curvature x1 to the direction duz/dul. The geodesic torsion for am direction at P is given by (5.30) 1:8 - (l/2)(x1- - 1(2)sin 2a, which, for a minimal surface, simplifies to (SO31) T8 ' X13111 Z-e-e 33' (5.22), (5.28), (5.29) and (5.31), the Tl-normal curvature of any curve C with respect to a variable T1 becomes (5.32) tun 8 111003 25 cos 9 - Klein 2-9- sin 9 = ulcos(2-9. o 9). The extreme values for tun are attained when 0 = - 25 or 9 =- u - 23. Hence, T1 must be equally inclined with the curve C to either of the principal directions of S. Equation (5.32) shows that the principal T-curvatures at P for any curve on a minimal surface are 3x18 :(-K)1/2. From the results of ‘Jheorem 5.2 there follows COROLLARY 5.2. 93 fl surface, 3 lips 93 curvature 393 3.3g _t_w_9_ mositely directed t_a_pgent _un__i_._p vectors £3 _i_n _thg reciproc_a_l_ relation R. _1_.’n_i-.g preperty characterizes _a. _li_l_l_e_ pf curvature. TEEDREM 5.3. All asmtotic m pf g _c___ur_!g 23'. P 3.2 copjpgate _tp _tpg Ell—"42 _a_n_c_l orthogonal _tp _t_._h_§ principal vectors 9_i_‘_ the, _curIg pp,P. M. The conjugate pr0perty follows from (5.10). A vector_ which is orthOgonal to an asymptotic vector determined by (5.10) is given by the equation “5 YB e dedu-O 768 dB ’ which is the same as (5.11) since 675805 - spang. extremal principal T-curvatures. Proof. Vectors prOportional to the principal vectors are found from (5.11) to be 5. pg: 1/2paaod 7:31/2ap v ( 33) p g e e aygfiadu g g davdu . 112 The square of the length of each of these vectors is equal to ( 5.31.) epoppp" - acitmrciflgez""3chm‘rcm5 -- shvgqudu". Substitution of (5.33) and (5.314) into (5.18) gives (5.35) 5,, - 1(ha3du“dup/aaadu“du‘3 )1/ 2. where in will be used to indicate a principal T-curvature of the curve 1‘. at P. The extreme values of the principal T-curvatures (5.35) at P as the curve C varies are given by those curves for which — a a (.0an awn ) - o. The general method of finding the extreme values of an expression of the form (5.35) [2, pp. 1112-1111;] leads to the condition hduB hdu‘3 lo 29 , . 0,. gmdu‘3 gZBduB which can also be written _ 1/ 2 av B 5 - (5.36) g c haflgyodu du 0. Since [2 , p. 253] . 7 h - d H - K (5 3 ) “B “5 gap 3 where H and K are respectively the mean curvature and the Gaussian cur- vature of S, (5.36) can be written “Y B 5 e d H - x du = . ( 0.5 gap )gYG du 0 By virtue of the relation‘ea‘lgafigT6 - a”, this equation simplifies to GT [3 5 He d dudu -= . c.8375 O 143 Therefore, if H K O, i.e., S is not a minimal surface, the extreme val- ues of the principal T—curvatures (5.35) are attained by the lines of curvature of 3. 'me general method used in this proof fails if the tensor h <13 is pmportional to the tensor g mfi’ which occurs for a sphere and for a mi- nimal surface. On a sphere the lines of curvature are not defined and, as was shown in the proof of ‘Jheorem 5.2, the principal T-curvatures for every curve on S are 3001/2. For a minimal surface the lines of curva- ture are defined but all curves on 8, including the lines of curvature, have the same principal T-curvatures of 3(-K )1/2. However, as shown in 'Iheoran 5.2, the lines of curvature are characterized by the fact that they are the only curves for which the extreme values of the T-normal curvature are attained with respect to vectors which are tangent to the curve . THERE! 5.5. 3313 sguare of £h_e_ principal T-curvatures 3; a curve c 92 S is given 91213 formula - 2 (5038) (tun) ' Hun " Ks Proof. The result follows immediately by substituting (5.37) in (S. 35 ). Some special cases of (5.38) are of interest. If C is an asymp- totic curve, in - o and the principal vectors of c are orthogonal to c. Therefore, fin is obtained from (5.22) for e - 3 11/2, and (ti-n)2 3 T: s 1:2 Hence , for an asymptotic curve, (5.38) becomes 1’2 - -K and thus (5.38) is a generalization of the theorem of Enneper [2 9 Po 2148]. For any curve on a sphere, (5.38) gives )2 = xi 8 x. Similarly, for an arbitrary curve on a minimal surface ( tun 2,-Ke (gin) 'Iheee two results were obtained in Theorem 5.2 by a different method. From (5.38) can be obtained a relation which is similar to the equation of Euler. By Mer's equation and the expressions for H and K in terms of 11 and x2, (5.38) becomes (1:11.13)2 - (x1 0 12)(x1cosze + xzsinze) - x112, or (tin)?! = xicosze + xgsinze, where e is the angle which the positive direction of C at P makes with the positive direction of the line of curvature having the principal normal curvature 11. If we define a principal radius 23: T-normal curva- Iirg to be tfin =- l/t'in, the last equation can be written 2 2 (5.39) 1 cos 6 * sin 9 2 I T (tin) R1 32 m3 analogue of the equation of Euler suggests an indicatrix similar to that of the Dupin indicatrix. By setting 3: = ltfinl cos 9, y - 'tfinl sin 9, and following the method used for the Dupin indicatrix I 2. p- 21:11: we thair: the equation hS 2 (5.1.0) 37 0 R1 which is defined to be the indicatrix I £93 _thg principal radii pf T-pp}; ma]. curvature 33 P. As shown in Fig. l, the magnitude of the two prin- cipal radii of T-normal curvature for the curve c is given by the length IRzl’ \fv‘i Q P W Pig. 1 of the line segment P9. Unlike the Dupin indicatrix, the indicatrix for the principal radii of T-normal curvature at P is an ellipse both for an elliptic point P and for a hyperbolic point. 'Ihis verifies an earlier Statement that the principal T-curvatures cannot be zero when d K 0. Equation (5.39) shows that the principal T-normal curvatures are equal in magnitude for two curves which are equally inclined to a prin- cipal direction of s. THEOREM 5.6. _'I_!_3‘_e_ m 23 _tth gguares pg _t_h_e_ flcipal T-p_u_r_v_;_a- _t_1_1_r__e_§_ pf _t_w9_ orthogonal m 31:: P 3.3 constant 93 i3 indgpen- dent pf php choice 93: the t__wp orthogonal curves. This constant gguals 2 H -2K-x§¢xz 2. Proof. Let Cl and 02 be any two curves orthogonal at P. Let lt‘n and 25m be principal T-curvatures for the respective curves, and 146 lxn and 2xn their normal curvatures. ‘men by (5.38) and Eller's theo- rem, it follows that - 2 - 2 (it"n) " (2txn) ° H(lxn ’ 2*n) " 2K - H(x1 + x2) - 2K ’H2-2K3‘K§*X§e mm 5. 7. 1133 Tl—normal curvature £13 33; Tl-geodesic torsion 93 _a_ curve with regpect _tp _a_p arbitpgy unit vector T1 at P satis- fy 213 relations tun - tun cos a, tTg - txn sin a, where a 3.3 213 directed _apglg from _th_§_ principal vector M 313 principal T-curvature tin _tp in; vector T1. Proof. Let B be the directed angle measured from c to a princi- Pal vector of C. ‘lhen, with 6 - a 4 B in (5.22), we obtain tun - xnms(c 4 B) - 18811101 * 3) . (uncos B - tgsin B) cos a - (unsin B + vgcos B) sin a. From (5.22) and the preperties of a principal vector, the last equation Simplifies to (Soul) tun . tin COS a. By a similar argument, tTg .- xnain(a + p) + 'rgcos(a + B) = (uncos B - 'tgsin B) sin a + (unsin B + tgcos B) cos a, so 1i'hett 1L7 (5,1,2) 1: a 5: sin a. This completes the proof. It follows from (5.141) that the Tl-normal curvatures of C at P are equal with respect to two vectors which are equally inclined to a principal vector of C at P. Similarly, (5.112) implies that, for two vectors which are equally inclined to a principal vector of C at P, the two Tl-geodesic torsions of C are the negatives of each other. Equations (5.111) and (5.h2) indicate that the sign of a principal T-curvatmre of a curve determines the signs of the Tl-normal curvature and the Tl-geodesic torsion of the curve with respect to a variable vec- tor T1 at P. Either principal T-curvature may be employed, and their signs are determined once the positive direction on c is specified. A change in the positive direction of C interchanges the signs of the two principal T—curvatures and, consequently, changes the signs of all T1- nornal curvatures and Tl-geodesic torsions with this one exception - the classical normal curvature and geodesic torsion are unaffected because they involve expressions which are quadratic in the dud. However, it is true that with each principal direction of S there are associated two principal T-curvatures, the classical principal normal curvature, say 11, for T1 in the positive direction of C and 41 when T1 is in the Opposite direction. The principal T-curvature could be made single valued for each curve, but only at the cost of limiting the vector '11 to a half- plane and thereby losing the generality of some of the formulas and re- lationships. As equations (5.22) indicate, the normal curvature and geodesic torsion of a curve determine the value of the Tl-normal curva- 118 ture and the Tl-geodesic torsion of the curve with respect to am vector T1, with results consistent with the classical case when 1'1 is in the positive direction of the curve. macaw 5.8. 3.9.2 Tl-geodesic______ torsion p_i_‘ 2 curve C attains its. extreme values _a_t_ P when T1 _i_s_ _a_n mtotic vector pf the curve; 333., when 0 g 2 Il-agzptotic curve with m _tp 1:1. ‘Ihese extreme values _a_r_g equal 29 313 principal T-curvatures pf _t_h_g curve _a_t_ P. Proof. From (5.111) and (5.112) the condition for the extreme val- ues of fitg is a(tvg)/aa - txn cos a -= txn = 0. Hence Tl must be an asymptotic vector for C. 'Jhe extreme values for t’tg occur when a is 3 11/2, and hence are tt‘n’ This proves the theorem. 'Iheorans 5.1 and 5.8 indicate that tun attains its extreme values when tTg - 0, while tTg has its utreme values when tun = 0. This is also brought out by the relation (5.13) (Mum) 45,12), which follows from (5.111) and (5.1.12). Squaring and adding equations (5.22) results in (5.1114) (any + (9312 '- x: + 17:. - Combining (5.113) and (5.11.11), we obtain 2 .. (SOhS) (tun) § (t'cg)2 I X: '9 T: = (tun)2e ms. its .2212 .2: .212 Lustre a: 21.2 T1-____..normal _mature 222 in: 131-522- desic torsion pf C with respect _t2 5 variable vector Tl at P is a con- 149 aLaniaiP- wflcommtiaauamiheflssem es 21:21.2 99151:; curvature _apd geodesic torsion 2; 1h: m, p}; _i_g _a_qugl pg _t_h_e_ M pf _tpg principal r-curvature 2; c g P. DEFINITIONS 9-}: _TE EXTREMAIS £95 23—3 G-EQLMAI: CURVA'NRE LN}; fl e-GEODESIC TORSION: 'Jhe 9-31.31 curvature exn pf g _c_u_r_v__e c _a_t P is defined to be the T 1 tor T1 which makes the directed angle 9 with the positive direc- tion of C. For a fixed angle 9 there is then associated a e-nor- -norma1 curvature of C at P with respect to a unit vec- mal curvature for each direction of S at P. be directions with respect to which the G-normal curvature at P has its extreme val- ues are called the e-principal directions pf S §_t_ P, and the cor- responding e-normal curvatures, ein’ are called the 6-principal curvatures pf S a}. P. A ‘curve on 3 whose direction at each and every point is a 9-principal direction of S is known as a 9-ng pf curvature g}; S or as g extremal 3‘35 _t_h_e_ B-m curvature pp 3. Similarly, the e-geodesic torsion 91g 93 _a_ m c §_t_ P is the 'fl-geodesic torsion of C at P with respect to a unit vector T1 which makes the directed angle 9 with the positive direction of- C. For a fixed 9, the extreme values for the O-geodesic tor- sion shall be designated by fig, to which will correspond the extremals £93; _t_t_1_e_ e-geodesic torsion pp 3. mm 5.9. 31313 6-_1_i_._n_e_s_ pf curvature 93 3 £921: 2 orthoggnal 222212942. ammnrcfiimrace seisiihs earn- cipal directions makes _a_n angle 93 -O/ 2 with _a_ corresponding So principal direction pi S. 11333 extremals $23 the B'E‘E‘Eiifi Lop- 2.1.92 9.1.92. £25! an _a_.ortho ml in. ibis _____bisects as Bites 2:. curvature. M. «let a be the directed angle at P measured from the prin- cipal direction of 3 with principal normal curvature 111 to the positive direction of a curve c at P. Then, by Euler's equation and (5.30), (5.116) Gun -= ancos e - 18811.11 .9 - ulcos a cos(c+6) «I» xzsin a sin(a+0). Hence, ( 5,1,7) 3(exn)/Ja =(12 - x1) sin(2a o G) and, with the proper choice of positive direction on c, the extras val- ues Gin are attained when (5.118) a - - 6/2. a - "/2 - 6/2. For the two opposite directions (5.119) ' a - n - 9/2, a . 317/2 - 9/2. Since the principal directions of S are perpendicular to each other, (5.118) proves the first statement of the theorem. At an umbilic point, ail-112 andthe edprineipaldireetionsOISareindetemnate, Justas are the principal directions of S. In a similar manner, (5.50) erg - xnsin e o tgcoa e ?' xlcos a sin(a+o) - )1 sin a cos(c+0), 2 and (5.51) 3(evE)/&a - (ll - 2) ees(2¢ + 6). lbs extreme values 818 are attained when (5.52) c - "/11 - 9/2, c - 3n/11 - 0/2, or, for the Opposite directions, 51 (5.53) a - 511/11 - 9/2, c . 711/11 - 9/2. Equations (5.h8) and (5.52) prove the second statement of the theorem. Again, at an.umbilic point, the extremals for éTg are indeterminate. Substitution of (5.h8) and (5.h9) in (5.b6) gives the following results: a - -o/2, aim - alcosz 9/2 - 12:11:12 6/2 - (112)05 - 1,) 4 (ll/2)»: 6! a - u/2 - 6/2, éifiz - ~[x1cos2(u/2 - 9/2) - xzsin?(n/2 - 9/2)] (5.51;) ' “(l/21(Xl - X2) + (H/2)cos 9; a a u - 9/2, Gin]- I X10082(fl - 9/2) " 1:281:12“ ' 6/2); d = 3u/2 - 9/2, 935112 - -[x1c032(3w/2 ~6/2) - xzsin2(3n/2 -9/2)]. Thus, for each fixed 9 there are two distinct esprincipal curvatures of S at P, which become the ordinary'principal curvatures of 3 when 6 = 0. From.(5.5h) it follows that aim . 932112 e H cos 9, (5.55) ._ _ 9"nl " 9"n2 " "l " ‘2' Similarly, from (5.52). (5.53) and (5.50) we find the following results: - 2 a = “/11 - 9/2, 91: gl - alcos ("/11 - 9/2) - xzsinzhr/h - 0/2) = (you, - x2) + (m) sin e; a a 31/11 - 9/2, 9th - -[xlcosz(31r/11 -9/2) - xzsin2(3n/11 4/2)] (5.56) = «(l/22X»1 - x2) + (fl/2) sin 93 52 a. = 5.1/1. - 9/2, 9'21 = )tlcosz(5fl/11 - 9/2) - x28m2(5n/h ~e/2): a = 711/11 - 9/2, 9:82 c —[lv1cosz(7u/11 -6/2) - xzsin2(7n/11 -e/2)]. ‘Ihe sun and difference of the two distinct extreme values for the e-geo- desic torsion become 6;g1*6;g2 =Hsin 9, (5.57) 9:31 " 6¥g2 " "l " "2' Rom (5.511) and (5.56). it is m that — 1 + cos2 a. sin2 e‘n efin I11 R2 ' (5.58) z . l..t(0082 a - 81112 a) 6 g - Bag R1 R2 when the appropriate value is substituted for a. From (5.58) we obtain an indicatrix 19 _f_o_1_'_ 39.9. 0-principal curvatures pn_d_ _f_9_r_ _thp extreme E- ggs. pf _th_e_ e-geodesic torsion 9.1: P. Again by the method used for the Dupin indicatrix [ 2, p. 2111], at an elliptic point Ie becomes the two conjugate hyperbolas 2 2 (5059) "x" " "z" ‘ z 1 31 R2 ' while for a hyperbolic point I0 is given by the ellipse 2 2 (Sew) 1%1T " fir - 1. The indicatrix 19 at a hyperbolic point P is represented in Pig. 2. Consider the rigid frame consisting of the four half-lines 1.1, L2, L3, L1‘ frou r in the directions for e - o, 11/11, "/2, 311/11, respectively. 53 ' L Q4» 'Rz.‘ Qf‘l’ Q2 1 Pig. 2 To find the absolute values of efin and of Gig for an 9, rotate the frane through an angle -6/ 2. 'lhen the values of 'P—Qi and .1791; represent 5 , while the values 55% and a give the the two absolute values of o 1: two absolute values of efig‘ A new definition of the lines of curvature can now be given by me 5-10- 9:; 2 ”will“ £12 .12; ”with” .9. 929.259. 2.1: e 9&- 2}. “surface. .919. .1132 2!. ___.___chmtnre 22 s 2.1-: 2.2 6-322 2.: 22:- mnnwmmmae ewwcmatmgm Riggs. gee-_linegggcurvatureggindetellinategggg £29213! £13933 mini-a1 surface all 9-_l_i_ne;s__o_f_ curvature have the £33 e-principal curvature, w 3 11. 3933. Erom (5.51.) it follows that Maid/3° - -(H/2) sin 0. Hence, Gin attains its extreme values when 9 vanishes or is a multiple of I. It follows fro. (5.51;) that a vanishes or is a liultiple of 13/2, and therefore the required G-lines of curvature are the lines of curva- " 1 z ture of S. The extreme values for B‘n are 1‘1 and 12. Sh For a minimal surface H - 0, and the preceding argument no longer holds. In fact, equations (5.5M show that for a mm surface a 1 Gun ‘ '1’ which is independent of 6. Hence, all O-lines of curvature have the sale e-principal curvatures. In the case of a sphere, 11 - x2 and equation (54:6) becomes Oxn : x1 cos 9, whichisindependentof a. Therefore, onasxileretheO-nor-alcurva- tureatPforafixedeis the saneforallcurvesons, andthe e-lines of curvature are indeterminate. “m 5-11- Let 01 29.92 2229:3322 swam- gn_a_l_ g: P. m g lfixn E 19% _th_e_ $92.22. curvature 93 3.1.2 e-ggodesic torsion £2; 31; m (:1, g; by 291.1! §_n__d 20¢g _t_h_e_ similar quantities _fgg 02. it; the; follows 3952 19%. ’ zexn ‘ a °°° 9' (5.61) T 4 198 t-Hsine. 26 g Proof. In (5.156) let cl make the angle a with the principal di- rection of S, and 02 the angle a:1 11/2. Then 19"n an]. cos a cos(a + 6) + x sin 9 sin(a + e), 2 zexn - x1 sin a sin(a o 6) + 12 cos 0 cos(a + 6), whence 19*n*29"n'(r1*12)cos6-Bcose. Similarly, tran.(5¢fl)) we Obtain 55 1618 . 11 cos a sin(a 4 e) - x2 sin a cos(a + 6), 29Tg - «lain a cos(a 0 6) 0 x2 cos a. sin(a + 6), 1913+ 2918- (11 + x2) sin 0 - H sin 6. ‘Ihis last guation 2:9. 3 generalisation _o_f _t_h_e_ theorem that 2‘22. M torsions (0 -= 0) at; 5 point _i_n _t_w_9_ perpendicular directions £3 nega- tives of each other. 6. THE RELATIVE T-GMDBIC CURVAJ'URE (how [6, p. 50] and Pan [9 , p. 211] defined a first curvature of a curve on a surface relative to a congruence of vectors associated with the surface. This relative first curvature becomes the geodesic curvature when the congruence is the normal congruence. The definition of the relative first curvature involves the derived vector of the unit tangent vector of the curve. We shall now generalize this notion by using the derived vector of am unit vector in the tangent plane to the surface. ‘ Let the unit tangent vector to the curve C on 8 be given by i du i i (6.1) To: to - F X 'a - €61 ,a , and the unit vector normal to 3 by N - n1. Consider the congruence of unit vectors F associated with 3 (not tangent to S) and given by (6.2) P: f1 - phi.“ * qni. (empa‘p’B + <12 = l, q 1‘ 0). where the functions p‘! and q are of class not less than 2. Let the orthogonal projection of vector P on the tangent plane at P be a vector in the direction of the unit vector T1 which makes a directed angle 9 with the tangent To to the curve 0. If T2 is the unit vector in the tangent plane which makes with To the directed angle 0 - e 2. u/z, them we have T1: ti " 90:19“ a (Sweaep ' 1: Pa " Peas P>0)s (6.3) i a 1 T2: t2 ’ ¢ 3‘ ’a ’ (gestalt? ‘ 1). We note that p = sin a), q - cos do, where a) is the angle between the 56 S7 ‘ 4 4 vector F and the nomal N to the surface such that O -= a: - 11, but ex- cluding a) - 11/2. Denote by T a unit vector in the tangent plane which makes a directed angle wk with To, and hence (60h) T: t1 ' taxi,“ 3 (Baffin)? ' 1)e Let M be a unit vector at a point P of C, which is linearly de- pendent on F and T and which is also orthogonal to T. 'lhen (6.5) mi - ati + bfi - (awa‘ + hp“) x1,a + bqni. From M ° ‘1' = 0, it follows that a (6.6) a - - bcapp #3. Substituting (6.6) in (6.5) and using the fact that M - M a l, we obtain b g g _ a p B a 1/2 . *1/11, A (1 gagspopp‘vw) #0 Eluation (6.5) becomes xi.a(- gfirppt‘rv“ + p“) 1* an:L (6.7) mi - e A 01‘ (6o8) M ' : -p[T c030? - 9) - T1] + qN [-p sinw - (9)11}, it all s 4 [1 - pzcoszfl - 9)]1/2 - [1 - pzcoszw-Onl/z ’ where Tn is the unit vector in the tangent plane which makes a directed angle of 11/2 radians with T. The vector M will always make an acute angle with vector N if in (6.7) or (6.8) we select the arbitrary sign to be the same as the sign for q. If F is linearly dependent on N and T, then it = 6, T1 - T and (6.8) reduces to M - N. The derived vector of T with respect to C is i 1 s i i (6.9) (t ) = z vipx .a + any? . 58 Elimination of n1 from (6.7) and (6.9) gives (’61). - 23.46511“. B - 01va Eapa/q + dpavptasmmrapp/q) (6.10) . midpavpz‘H/lql . which can also be written 1 ' i B a. a B Y a i (6011) (t ) . X ,C(£ * DB - *xnp * *xngfiyp * * ) * ‘ *‘nA/|Q.9 where p“ . (p/q)e"L and *xn (or tun) is the normal curvature of c with respect to the vector T: ti - taxi, 6. Thus, when the derived vector of T is deconposed along H and a direction in the tangent plane of S, the tangential component becomes (6.12) V33”? - sz“.5 - *xnp“ . ,xnghpBW“. where Ill.“ is a tmit vector in the tangent plane. We call 1'18 (or egg) the T-geodesic curvature _o_f_ c _a_t P relative _tg _t_h_g F-comence or mere- ly the relative T-geodesic curvature of C at P. Since the derived vector of T and the vector H are orthogonal to T, ft“ must be orthogonal to t“. We select its direction so that [La - chirp and therefore ' ,‘0 " V " 'lzs where ,u. is the angle which the vector/La makes with To. From (6.12) and (5.7) it follows that a (6.13) ,lg ya.“ gig/e - V‘s - gluten a) sin(9 - v), where *x8 is the T-geodesic curvature of C with respect to the vector T: t1 - taxi”. then the vector T is tangent to c, V - O and the 59 relation (6.13) becomes (6.111) 11 In -xtanaosin9. "8 8 n In this special case for V - 0, x is Grove's curvature Kn [6, p.50] and .8 Pan's relative first curvature is [ 9, p. 211]. sequently, from (5.16), both THERE! 6.1. g 213 _gu_1_e_c_l_ surface generated by the ge_c_t_9_r_ T _a_; _i_t m _a_l_ql_lg G g 3 cylinder, M 313 relative T-ggodesic _cur: m 2.1: c is. m. M. The curvature k of the ruled surface must vanish and con- ng and Vxn must be zero. Therefore, (6.13) shows that x is zero. 0388. H THERE! 6.2. The relative T-ge_o_desic curvature 25 C at. P _i_s- SEEET‘bmicwcmamfipgflfligfifififi mfollowingmm: 1. mcggmtoticmflremctgh 2- awmacmmaeum mem- MEQPEUJIW Maymwnrgna flimflmfisfifi m. In(6.13)a*§8-*xsifvxn-O, which provesthefirst The second case covers the situation when either tan do - O or aide-1)-ObecauseeitherF-3Nwhenac-Oorw, orFisalinear combinationofllandTwhen e-v-Oors. mm 6.3. 35; relative III-geodesic curvature 95 c at P, where Tn .13 the unit Eggntial normal vector at P whose directed 231.5 _with 2.2232223: "/2 _nradiw insignia ate-tx-ttan cose. 11-2 s s m 60 233 relative Tn-geodesic curvature. gt 0 _at P ta; gual t9 the ge_o_- desic—_theiCEEPEflSeillEanaziyifiemu - inseam sale: 1. mcggagtcurvaturegts. 2. nannies; Gimmes: We ence assen- ttgdflPtglheutzdgendentggtlfiangtngthg £12m N genial. is s at. P. M. The unit tangential normal vector Tn which makes a di- rected angle of 11/2 radians with To is given by (6.15) Tn: t: - maxi,“ , nu - swap - eflagfiyi‘r. Substituting n“ for t“ and -g“ for y.“ in (6.12) produces _ a g B a o. _ p v a (6.16) E: 113-8 6 n .9 . 1; p/q ngfive 11 n p/q. where the geodesic torsion 'cg c -11xn’ Consequently, a (6.17) 7158 (-€Q)(-£ )nlg we - tgtan m cos 6. when 6 is a line of curvature, 'cg - O and nag - 118. then tan as c O, the vector P = 3 R. Finally, I“ is a linear combination of Tn and N when a - 11/2 or 311/2 and cos 9 - 0. From (6.17) we observe that th_e_ relative Tn-geodesic curvature gt 3 geodesic ling 9}; curvature 1.2 3953. A curve on S is called a T-pseudogeodssic 93 S relative t2 .932 _ P-co ence, or simply a T-pseudogeodesic .92 S, if the relative T-geo- desic curvature at each point of the curve is zero. From (6.11) a curve is a T-pseudogeodesic on S if and only if the normal to the ruled surface R, generated by T, along the line of striction is parallel to 61 the vector M at the corresponding point P of S. It is seen from (6.12) that T-pseudogeodesics on S are defined by a 'duB (6.18) éfi" 0 (Far " dBTpa * dBYp6W6*a) *1. an;- 8 00 Thus, for a given F—congruence, through each point P of S there passes i a i a unique T-pseudogeodesic for a given vector field T: t - 11 x ’a' men the vector T is tangent to the curve, the T-pseudogeodesics become Pan's pseudogeodesics [9 , p. 213] 6 a a a $1“)qu .0 (J9) y‘Wa'deP ”yearn- Far 7 main. tmgShimmmmgr-Mggw- 8‘00 mtotic curve relative t9 3 unit vecto E E; geodesic 2.9. S. M. hon (6.13) #8 must vanish when *xg - #1111 -= 0. Foragivencurve ConSandagivenvectorfieldTinS, is it always possible to choose an P-congruence so that c will be a T-pseudo- geodesic on 3? The unit vector P is determined by the angles 9 and a1. Equation (6.13) and Theorem 6.2 show that, if c is not a T—asyaptotic curve and if 0 - t 1‘ 0 or 11, for each value of 6 selected arbitrarily the relative T-geodesic curvature will be zero if (6.20) tan a) - “1!an sin(9 - 11)]. Hence, there exists a one-parameter family of r-congruences for which the given curve C will be a T-pseudogeodesic with respect to the given vectors T along c. However, if C is a T-asymtotic or if F is linearly dependent on T and N, C will be a T-pseudogeodesic with respect to T 62 if and only if it is a T-geodesic with respect to T; that is, if and only if T undergoes parallel displacement along 0 in the sense of Levi- Civita. dud IBtheanarbi‘h‘arycurveonSwith unit tangentvector-a-g, and it“ the contravariant conponents of a family of unit vectors T along 0 in S. We define the vectors T to be parallel along 0 relative to the F—congruence if and only if equation (6.18) is satisfied; that is, if and only if the relative T-geodesic curvature is zero. Such parallel- ism of T along 0 is called a relative malleliem. By meoran 6.2, a relative parallelism of '1‘ along 0 reduces to the parallelism of T along C in the sense of Levi-Civita when G is a T-asyaptotic curve or when the vector P is linearly dependent on the vectors T and N at P. A com- parison of equations (6.18) and (6.19) shows that the unit tangent vec- tor to a pseudogeodesic undergoes a relative parallel displacement along the pseudogeodesic. Hence, pseudogeodesics 933 S 333 relative _agtg—parallel m 23.1 3. Pan [9 , pp. 213-2111] defined a relative parallelism and a rela- tive associate curvature in a different manner - by replacing in (6.18) *5 and t“ with dub/ds and dua/ds, respectively. we have selected our definition of relative T-geodesic curvature because it occurs geomet— rically as the tangential component of the derived vector of T relative to the F-congruence, just as Graustein's associate curvature (or angu- lar spread) is the tangential component of the derived vector of T rela- tive to the normal congruence. Besides, our relative T-geodesic curva- ture plays an important role in further generalizations of curvatures discussed in Chapter 7. 63 THEREH 6.5 2132 relative T-geodesic curvature 93 _a_ 12939.? _a_l__9_ng sauna. 632229.111? Pie mricallezlkleari-EEEME th_e SEES? between th_e_ m vectors diglaced 100g _a_n_d; parsi- lel-lz arse as 2.99.9.1; as P ales 0- 23933. The proof follows the method of Pan for his similar theo- ran [9 , pp. 2111-215]. Replacing his dufi/da and dua/ds by +5 and t“ to give us formula (6.18) does not affect the result that d6 2 1'2 as ' H ’ where e is the ange between the two vectors displaced locally and rela- tive parallelly. In this proof it is necessary to seems that all the functions are analytic along C in a certain cannon interval of s. It is known [ 5, p. 1113] that the T-geodesic curvature of C (or Graustein's angular spread) is given by d (6.21) was = x8 + I! , where 118 is the geodesic curvature of C and v is the directed angle from the tangent vector to T. A generalization of this result is given in mm 6.6. 1113 relation between th__e_ relative T-ggodesic curva- turegtctttPtggEregtivefirstcurvamregtcttPtg gtveng (6.22) #8 a 3‘-g + £ + {intan a) sin V cos(a + V - 6), wheretin _igths £26128]. T-normal curvature 33 CE “3922 directed 9319;? from tht correspongg princigal vector 9}: 0 _t3 3.1.2 pgsitive direction 9.2 C. 611 Proof. Substitution of (6.21) in (6.13) yields (6.23) 1358 - x8 4 3% - tun tan a fin“) - t)- Subtracting (6.1h) from (6.23) and substituting for tun the expression given in Theorem 5.7, we obtain (6.211) V58 - lg - 3% 4 tin tan ab [cos a sin 9 - cos(a 4 v) sin(9 - (1)]. By trigonometric simplification, (6.211) reduces to (6.22). For the case when the F-congruence becomes the normal congruence, (6.22) simplifies to (6.21). 6 7. THEGMEBALCASE: WRVANIUNITVETOR To each point P of a surface S associate an arbitrary unit vec- tor 61 given by 2 2 (7.1) a: - p651.“ . uni. (smafiafi'3 - l. p + q = 1). wherep,qandeaarefunctionsofulanduzofclassnotlessthanz The functions 0“ are the contravariant comonents of the unit vector T1 which has the same direction as the orthogonal projection of 61 onto the tangentplanetoSatP, whilep-sinm andq-cosao meromis the directed angle from the surface normal N to the unit vector 61 such that Ofimsw. As in Chapter 11, we consider a' curve 0 on 3 imersed in a one-parameter family of curves satisfying (11.2) and (1.1.3). 'men toeachpointPofc thereis associatedaunitvectorGl, andthep, qandeaarefunctionsofthearclengthsalongthecurveC. The angle which a“ (:1) makes with the tangent g“ to the curve 0 shall be denoted by 0. Let H denote the unit vector in the plane of N and T which 1 makes the directed angle of - 11/2 radians with 01. It follows that H " Pu " q a (7.2) :1 61 ' qu " PTIS and N - pH 4 q (7.3) 01’ a - «w + .61. The unit vector T2 orthogonal to T1 is defined by equation (5.2). Since the unit vectors T1, T2 and N, in that order, form a left-handed 65 66 coordinate system, by (7.2) the same is true of the unit vectors 61, T2 ' and H. With the use of (11.1), the derived vector of 61 along 0 at P is ' ‘ i a. i {3 i (3:) - 12.39“sz .a + p6 .383: ,a + pause“; n (7.1:) i i + (1.363 n - quYgYaEBx ’a. By virtue of the relations 1 1 i x ,a- eatl +¢at2 , (7.5) a a eaea - 1, 6 13a . o, o ,Bea - 0, equation (7.1;) simplifies to (7.6) " (pdaaeazfl ‘ quEB )ni. Differentiating (7-7) m - arc tan (p/q). we obtain (7'8) "03. ' (9P2? " 111:5)455. Hume ' “$ng ' (1(qp35 " m’fi)€fl ' (103': (7.9) Qafigfi " -p(qp.B "' 13%ng " ~poo'. huations (7.9) and (7.2) simplify (7.6) to (7.10) (gif - (p 9x8 . a. erg”: + (Gun - w'mi. where exg, Gun and 91: are defined by (5.7). 8 67 From (5.6) the derived vector (ti). along c at P is given by 2 (7.1.1) (t;L ) - - exgti- etgni. By means of (7.3) equation (7.11) becomes (7.12) 63' - -

'15 and 0‘ c1 f 11. If 61 and El lie in opposite half-planes h and h through the normal line to S at P along 0, .5“ = -GG and addi- 1 2 tion of the equations in (7.28) gives I .—I 60 *Q .00 Therefore Since along C both to and E are continuous functions of the arc length 5, it follows that .. 4 < .- m_m=cl, clip-11, (Glandclinhl): _. 4 ¢ .- m+m.=c1’ ogmncl, (Glinhlandfilinhz), m-macl, 0-m-11-cl,(61andGlinh2), _. 4 .- m+m=211-cl, n-cl-msu, (Glinhlandclinhz). In each of the four cases in (7.29) the angle between 61 and '51 is the constant cl. 75 ”comma: 7.6- snwmssmmm 32 19.2122. 11:: in. 322 secs mired. :22 s 3.2.121 eases: m names-inseam, fizflflwimeflectflm mmfiswcommtwgfimmm Mammal: 2125 1'19. sma- P_1_'_03_f_. Since the tangent to an asymptotic curve suffers normal parallel displacement along the curve, the corollary is a direct con- sequence of heoren 7.6. We note that in normal parallelism the asymp- totic curves on 3 play a role analagous to that of the geodesic curves on S in Levi-Civita parallelism. . THERE! 7.7. _I_f p 339393; 01 suffix-spit}; _a_ TZ-parallelism £915- mesamwmwenemmflmelmamam c, i; 13 parallel _i_n__ _t_1_1_e_ Euclidean 321.133. 2.122;. The V-geodesic curvature '11. 8 = 0 by Corollary 7.11, and vxn - O by definition of normal parallelism. Hence, 0; - 0 from (7.111) and 01 is a constant vector. 8. 0011631131703 01" LINES ASSOCIATE WITH 1 SURFACE S We now apply some of the results of the previous chapters to the study of a congruence of lines associated with a surface 3. Since the conditions derived in Chapters 2 and 3 will be needed to study the ruled surfaces of the congruence, we first obtain expressions for the cosines 91.1 52 and 53- As the point P on 3 moves along the curve c defined by (11.2) and (11.3), 61 is the generating vector of a ruled surface of the congruence of lines g:l in the direction of G1. The second equation of (7.2) is (8.1) 61 :- pTl 0 qN - T1 sin to + N cos co. From (5.6) and the first equation in (7.16) it follows that 1:62 = G; - -(cos 0))(exn - m')T1 + (9118 sin a + Otg cos my!2 I (8.2) ‘1 (sin “)(e‘n - an )N - -T1vxncosm + 1.2ng o vinsinob. Hence (8.3) kGB-Glkaz--T x cosm-T x '1vi sinao. 1 v g 2 v n g The vector To tangent to the curve c on S is given by (8.11) To - T1 cos 6 - ‘1!2 sin 9. From (8.1), (8.2), (8.3) and (8.11) it follows that 131 8T6 ° Glasinmcos 9, (8,5) 1‘52 -= To 1 1:62:- - cos 0) cos 9 vxn - sin 9 'xg, kB3=ToakGB--cosoocose 11. tsinev . vg n 76 77 By use of (7.15) and the fact that p . sin a), q - cos a), cos a . gafieagfi, sin e = capeazfl, the tensor expressions equivalent to (8.5) can be shown to be B1 = p scaGGEB. (8.6) m, . (pm, - q daB)E“EB. 2 1‘33 '["' N¢Y9Yga9fi ‘7 q dTG¢YeB " dWeT¢B " ¢B(Pq,a. qp’a) ]EQ£BO We shall henceforth assmne that k f 03 that is, that the ruled surface R of the congruence (or vector field) is not a cylinder. THEDREM 8.1. g geodesic curve C _c_1_n_ S will also b3 3 geodesic crating vector 61 _i_s_ i__n_ _t_h_§ tangent plane pf S _a_t P 31; C 33 2 straigt line. I Proof. By virtue of (8.5) the condition 81 - kBZ = O that C be a geodesic curve on R becomes (8.7) (exg - 6.)sin 9 sin a.) 0 (Gun cos 9 + 91:8 sin 9)cos co = O The use of (5.22) and (6.21) sinplifies (8.7) to (8.8) wgsinesinm-txncosm-O. Since G is a geodesic curve on S when 118 c 0, (8.8) will be satisfied if cos co = O and hence if G1 is in the tangent plane to S at P, or if 1th = 0. But 11g - O and xn -= O inply that c is a straight line. (8.9) 78 THEOREM 8.2. £1. mtotic _c_1n'__1_rg_ C 213 S E Be 3 geodesic magwsurfaceRgfficonggencegflglzgc some assessments mu, n22.12212.22- tify_i_ng pl_§_n_§_ _o_f_ C. m. In this case “n - 0 and (8.8) reduces to 118 sin 6 sin a1 = 0. If x8 -= 0, C is a straight line since ”n also vanishes. The condition sin e - 0 implies that G is in the plane determined by the tangent to 1 the curve and the normal to S, which is also the binormal of the asymp- totic curve. Finally, sin to - 0 is the condition that Cal be normal toSatP. DEFINITION 92 _TLII; LINE _QE STRICTION 1E1: 911: 3: he net of curves on S which are lines of striction for the ruled surfaces of the congruence is called the line _o_f. striction _n_e_‘_t_ pp 3. THEDREM 8.3.. _hglinegf strictiongggpsg Evengz 8.10 - d d a p . O. ( ) (Pa’B q CB) ‘1 du 29; gpnormal W this pat becomes _thg _a_sEtotic 3133. hen 9L9. ggneratipg vector 01 _i_s_ _a_ unit vector ‘11 i_n_ _thg taggnt flame _t_o_ S _a_t P, 3113 line 2; striction p_e_t_ consists 9_f_' 213 £191 2f Tl-geodesic curves 29.2 _t_1_1_e_ family of curves of the vector field. Proof. Since 82 - 0 is the condition that the curve 0 be a line of striction for some ruled surface of the congruence, (8.10) follows from the second equation of (8.6). For a normal congruence p - 0, q - t 1, and (8.10) becomes the asymtotic net d duathifl - 0. When 61 as lies in the tangent plane, p - l, q - 0, and (8.10) results in 79 (8011) . 6Q,B duaduB C 00 By use of the relations a. (8.12) ea - ea cos a '71; sin 9, one ’B .. 0, equation (8.11) can be written as 5 (8.13) (0a6a,pdufl)(e Y61mm ) . 0, which proves the last statement of the theorem. mm 8.11. 213 necessgz _a_n__d sufficient condition that 3313 line 22 striction 933 pg orthoggnal i; (8.111) p3,? - qH - 0. £1.13 3 minimal surface, the line _a_f striction 931. 1:111 93 orthog- Engmw dim encesiEsaa-msi‘filssa _Ehg 39.18293 plane 95 353 P vanishes. Proof. he net (8.10) will be orthogonal if and only if “3 .. . 9 - . (8.15) a (pad, qdap) p .3 qH 0. For a minimal surface H c 0 and the condition (8.15) becomes paw s 0. THIDREM 8.5. _‘lfpg 923 9_x_1_ S correspondiitg _t_g the developables pf 31.2 comm” 95 lines g1 i_s_ Evenpz . 6 v 2 Y r a 5, . (8 1 ) [‘P‘I¢Ye saga“! dya¢ Op-QYGB ¢B‘¢B(P‘ba‘qp’ a) 1d“ d“ 0 £93 3 comme 93 lines _in_ Eh: tangent plane 93 S, this _n_e_t_ consists 2; lh_e_ family _a_.g Tl-mtotic curves 9; 213 E 9.1. 2111121 algae _which :92 .._....._..V°°’°°rs T1 2.29. 29.8921 is 22 _CWB- Proof. muation (8.16) follows immediately from (8.6) and the condition for a developable surface, namely {33 - 0. hen 01 is a unit 80 vector T1. in the tangent plane, p = l, q - 0 and (8.16) simplifies to (dYaeYduaXe 95m?) - o. 513 This proves the second statement of the theorem, which Pan also verified [8, p. 962] by a different method. Incidentally, for a normal congru- ence (p-O, q-11)wegetthewell-knownresultthatthelines of curvature correspond to the deve10pablee of the congrueme. TIEDREH 8.6. 29; g conmeme 9;: 1353 tangent _tp S, 393 psi correspondig _t_.9_ £133 develggables fl _bg orthogonal _i_f 3513 93111 _i_f 313 12503 33 _ip _a_ principal direction 2;. S. M. he inner product of the expression in brackets in (8.16) and of the tensor g“5 results in the orthogonality condition a 2 c1 c1 (1 - m9 (1797.“ 1 q (174179 - cine“) - I (mac - c119,“) - 0- hen p - l, q a 0, this condition simplifies to Y a. _ dTae 4) 0, which states that the orthogonal vectors a“ and ¢a must be conjugate. Hence the vector T1 must be in a principal direction of S. BIBIJOC‘RAPI'H 1. Eisenhart, L. P. , _A_ treatise 93 the. differential geometg _o_f_‘ curves and. surfaces, Gin and Company, 1909. 2. , _a_g introduction to differential omet vi _us_e_ 93 the tensor calEIus, PrEcefin‘fimversity Press, 1 h . 3. Graustein, w. 0., " Parallelism and equidistance in classical dif- ferential geometry, " Transactions 93; the American _Igg'xematical 30018121, V01. 3’4, 1932, pp. 551‘5930 h. , Differential geomegxz, Macmillan 00., 1935. 5. Grove, V. 0., Lecture notes 93 differential geomegz, Michigan State University, 1930. 6. , " Sobre curvaturas gensralizadas,‘ Revista Serie _a_, Universidad Nacional del Tucuman, Vol. 9, 1952, pp. 57-33. 7. Olmsted, J. H. H., 30mm £99935!!! D. Appleton-Cantu? 00., 19h? . 8. Pan, T. L, ' Normal curvature of a vector field," American Journal of Mathematics, Vol. 7h, 1952, pp. 955-966. 9. , " On a generalization of the first curvature of a curve in a hypersurface of a memannian space," Canadian Journal 9_f_ Mathematics, Vol. 6, 1951:: PP. 210-216. 10. , " Torsion of a vector field, " Procg%fi of _thg Amer- ican Mathematical $00192, v01. 7’ 19 ’ pp. " e 11. Struik, D. J . , lectures on classical differential gm, Addi- son-Wesley Press, 5950. 81 MICHIGAN STATE UNIVERSITY LIBRARIES 0 3502 8693 3 1293