II IIIIIIIII III III I I II III CONFIGURATIONS ASSOCIATED WITH THE NORMAL CONGRUENCE OF AN ARBITRARY SURFACE Thesis for the Degree of M. A. julia C. King 1928 IDIIIO ll'II c I IIEIIIIII'VILV'IIIV .UJIIII...NI . . . a , O I . I I t I I . . I I. \ I I I I I I I I \ . . I . I n . I . s. I I . . .. . _ .I I _ I I t I I . b I r I I I .. u I ' I r u l - I I I . I I R r I I I I. I I u I / . I . . I . . I I II I _ u . I . n I I - ~ I I . I I I n .. . n I Ina. . .II/ o . I. I . I I I I I I o . n I I l D I I . I . . . C . I . . I . I . I . I . . I I l . I I V I C l I . b I I I ‘ 'Q , . \ l I \ I I l ',\ I O .I . I4 I ' I I i I . I I I I A ' I f. :1 .— I Q I \ I I I I .7 I I I I I O I I .I I ,I . . I O I I I It . I I I I . . Q I. I . I I I I I I I I . I I. I b I . . O I I I V I . . . I . I I I I I I I I O I .I . I I I . I I I A I . I I I I4 I x I II... . I . . I I I I I I v a O I t}. cat’l‘l’ll‘lro KIINI. . . . \ I .._ . I I. I I. 1 . lI . I . ‘ I I a I I , I I a ‘ . I I . . I . l I u . ( . 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I r .I I . II I. I r. J . I. . . I I I I. I. n I i l I O I ‘ ~ ' IA . 4 . I h . I C I l I tI. . I. I I . I A. I I I . n \n .‘ I I . I I II .. \ I I ..I I N. , \I 11.9: _ a I I. h r \I I I I v I . a \ I 4.. .I . . ‘rfl of I I . \ O I I . u I I I v I.‘ I . . 47.. r I I . l I a c. . . .OII I O I I I II- I . I . I o . I r ( I! I \ I I . I. I I . I . O a. I I o I ._l_ I .I v . I I . 0 c.— I O D . .I In .I . r... I I A . q I O \F MSU LIBRARIES .—:—. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped beIow. IT IS A VERY GREAT PLEASURE TO ACKNOWLEDGE MY INDEBTEDNESS TO DR. V.G. GROVE FOR HIS PART, NOT ONLY IN THE EXECUTION OF THIS THESIS, BUT ALSO IN HIS SUPERVISION OF MY GRADUATE WORK; FOR WITHOUT HIS ENCOURAGEMENT, CONSTANT AND SYMPATHETIC AID THIS PAPER WOULD HAVE BEEN IMPOSSIBLE. CONFIGURATIONS ASSOCIATED WITH THE NORLTAL CONGEUENCE OF AN ARBITHAPY SURFACE A Thesis Submitted to the Faculty .ytgff"ifl MICHIGAE STATE COLLEGE of AGRICULTURE AND APPLIED SCIFNCE In Partial Fulfillment of the Requirements for the Degree of Master of Art? by . " Julia GI King CONFIGURATIONS ASSOCIATED WITH THE NORMAL CONGRUENCE OF AN ARBITRAHY SURFACE Introduction The purpose of this paper is to study certain con— figurations associated with an arb trary surface with Special reference to the normal congruence. We make the lines of curvature parametric, and find the Laplace transforms of the given surface with respect to this net. We investigate the effect of imposing the con- dition that the parametric nets on the Laplace trans- forms be lines of curvature on these surfaces. We find that the given surface is plane. We discuss also the similar problem with respect to the surfaces of center of the given surface, and find that a normal congruence cannot be a congruence of Guichard. We also discuss parallel surfaces, and introduce the concept of ranges of nets. We find as- sociated with a given range a certain parabola tangent to the parametric tangents. This parabola degenerates into a system of parallel lines in case the parametric curves of the range are radial transforms. Consider the system of differential equations __ (5‘) (It) (I) Oak—a, cal—é oy-ufl’Q, 9“ v = awe“ + £63!" + O/(MCP , (1) 9,, = 05.2.09“ + 5m)4,+6/m@, (9.: Wu + ’7 4"» 1018fo 42. = WW W», 27h er sin 0:"): '13.; #(GE +,,g ~2FF) (2a)1’6‘):{lfii‘z =Ig‘i(-FE‘;EEV +2E F“), J"): b s XXL... J (unzil’lgzlfl ( (SHAW): {5}}: ;%(EG: _ng”)) CI“): bi: 2‘ X. kc. H 34% wore? GGu-rzCF) <2.) L“’=§":&{ =1 :‘L:(EC~..+ FGu-QFFQ) 01"": b" = ELM, H . _ [TU-6D n . )‘4" Ha, ) 3111.: FD‘HG D ) (3d) .. _F___~___I) E I) . 7:51. £1)” 71— J )7 = H; . If we impose the condition that the lines of I curvature be parametric, that is, F: D :0, we shall find that the coefficients of system (1) reduce to (1:) Eu (3‘) g“ a : —— , 0» = " . 1E 2E (2a') “A E (ob!) (33) £1.— 5 "IE? ' " 5 ' 3.? ’ 3. a023_ 5., 3—D=~i—, - JLéE ) '7” E: Efl (BOWEN): fig ’ (3‘1”)? :0 . )yfl:: a . ‘ Cf)»: D'ZO )1‘... _-D-——=~J“1 ‘ ’ G P. wherein PL and PL are the principal radii of normal curvature. The integrability conditions for system (1) with coefficients (3') are m by— '."2D+f‘if b"=°) 3.:4- 1‘2? D._S’:}D‘=o H ihe Laglacejgransforms of the Lines of Curyature We shall now find the coordinates of the focal point on each line of the congruence of tangents to the curves Cu . (4) These coordinates will be of the form K=X+RX, 7.23.511Y’ io=E+R2. The direction cosines of the tangent to Cu at Kg: 2) are (5) k0. , ‘19:: ) ZR fie“ E ‘I?’ Hence the coordinates of the desired focal point will be of the form 7v 0 I -k+n.’-‘—“&-= K+ILKu r’ 2 LJ-HZ fEE— ?+n 7k 2, : +J‘L Ease-+1133». o 2 f A 0) ‘1 (>4- 0 ll Similarly for the tangents to the curves <3ythe co- ordinates of the focal point will be of the form _ Xv _ u Xo—K+f\.——'-X+ +11. kv r— , (7) go 74A%= H-HL"? ‘2'», 2° {ml-fl. ‘v :-2-+IL"'ZV (LT [I H The locus of each of these points is a surface called a Laplace transform of the given surface. The parametric equations of these surfaces may be written 3; =K+n'ku ’ g“: Ic+m"¥v , (a) ”kg-+117 , 37‘374137» . :: =2+IL Eu , 5+! é‘thgv , H I We now proceed to determine the functions F- and F3, . Equating the expressions for the direction cosines of the tangent to Cu at No.73), we obtain L oh- .8, i=2??? 5: 4; <9) 3%:éfif- afi‘dx.’ a 4L gzj:__§?da, Expanding the expressions (9), we have sf: figfiusm.)Jag/w») V] In? , (Ea) fit‘fiflrflfln‘.)ngfi[o“WM3'; ; . a « . +A‘me' v ' EVE=fiog£LQiMJédfil h } Multiplying by k, , . , and 2-, respectively, we obtain LEI-2f§[n'(czt°‘in‘.)vlm,+[(a+n'Whit}? ' i ‘ (l) ' t; 3 (10) “Ev-fi; R‘Gofifi‘v1v13u2'*&Hnt )']7 l (M . ‘E . 2,, an 05%(a‘flm ,.]%.3.+RHRL 3V1“ (15‘4" Adding these equations, we obtain 0 :(I+/1'LM) v' G. Since neither v‘ nor G is zero, it follows that '4 n" L(Ul) : o, . .1.— . (11) n 2 — £03) Similarly we find that u_ __|____. (183),.L '- — (IA) CL The parametric equations of the Laplace transforms are therefore tv {,2 )0. £633 9 5-1-4: of“) ’ V (13) 72., -.: ‘7‘- ‘22.) ) 7-H: 7 ICED?) 2 2 v i,“ -5, §+¢=é- aged) Let us denote the coefficients of the first and second fundamental forms for the minus first and first Laplace transforms of S byE.. ,E, , etc. andEHJ-j,” etc. For the minus first Laplace transform F; :Z[xu.-J_ gm.) awku+ é() V+DX)+LLQ ”jg" BLEéY +190? *ftii 0:5] ’ )béofl)‘ (14) F:— rib—$03”: amt”) £i11J>k k“ _T¥_ gikpgfi- (a) All») )zu:t] ’ F:— __(Lv+4 40»£(.j( 02):a(n)é01_)ké:’d>E~= 0. Am.) 4 In order for F; to be zero, either the irst or second factor must be zero, since E10 . If (IA) 0;.) (u) A 4... A .— V it follows that §v=o , 1 O ’ firzo , whence integrating we find that §=§<->, 22': 72 (a) ) £;-= a; (0%). The locus of(s )7, 6’) is therefore a curve. Therefore the condition that E. be zero, that is, that the minus first Laplace transform of the lines of curvature on its sustaining surface be a net of lines of curvature, is (.2.)_ QWU” Am) a- (15) 6,,— In a similar manner the condition that E. be zero, that is, that the second Laplace transform be a net of lines of curvature, is w.)_ (anywa. 02) e v . (16) From (15) we find 6:2) ("J ((2.) (17) It: ‘- “I '5 - Using (3), equation (17) may be written either in the form (18> 4:53: L7A°”‘=t%%F-t§t%6’ or in the form (u)L (19) figs? é§"6=°' Hence integrating we obtain (20) (LL—t: =£rg l/ where T7 is an arbitrary function of v only. 8. If use he made of (3b), equation (20) may be written G ._ gee-v Similarly, if condition (16) holds, it follows that ENE-1U where U” is an arbitrary function of u only. Hence if the parametric nets on the Laplace transforms of 3 are both lines of curvature it follows that 91:7 , H (21) Ev AH " U ' 13.12:: .L[,§_{_§_ 3.. 5., EG‘ H ‘* 1 3v 57)] (23) bbn'zo Suppose now that fill—:0. Then from the integrability conditions (3), we find 11 bp‘g l E D301 _¥Al% Dec, I (12.) szfzz ::-j§¢L-:- gLéF-:£.cv But since )9 it follows that 3:0. We may state our results in the theorem: Li; the w _f_i_r__s._t_ and gigs}; Laplace transforms 9_i_’_ 9 surface S with respect 32 _a_n_ orthogonal conjugate parametrig pet have M lines. 5;; gurvature parametric, the. surface S _i__s_ a plane. 3. The Focal Surfaces of the Egrmal Concr, ruence Consider the surface S, with equations g:k+—P, K ’ <23) )Z:(z+P,Y, §=2~+ ?.Z n where w. = is} The condition that the parametric net on S, be the lines of curvature on 3, is that I DG-E D' ._ (34) F=zgu §V=ZPNK V kt+ PIVXI= Pu. Ply ‘ 0- Then either P,,,_=o, or I ”=0. But if Pas-0, it follows. from (24) that S“: )Zugfuzo. The first focal surface would be a curve. It follows therefore that PW =0- Henoe the parametric net onSCP,)is the lines of curve.— ture if P, is a function of u only. Similarly for the surface $09,) , PL is a function of v only. 10. Suppose now that the parametric curves on both of the surfaces of center are the lines of curvature on these surfaces. Then 310’ EU l 319) A elm ~_/’ u C) Lg L at» 1)" But .3. 5.»: E~ E v 3v D D 13"“ ’ -Ga_§1> 3%(13'3‘ T ID": Hence . EV DV g— IL. (25) g” 75' ‘ s 1)" Substituting these values into the second and third of equations (3), we find that bi-JL_ E_\ID__ Evb =6 mm E 1E 26 ’ I: V b GK~ GHD_ GKD __ G 1G 2—5 From (26) we obtain the followdng equations - $-12: - Ev 3-5 :LG)'O’ (.21; ._ .2. .. G! L; 16. XE .. O ' H i... __ L = o ’ XE 2G this implies that P, and PL are equal. Since this is O . If not true, we have (27) EV=G£~:O' 11. Substituting these values into the first of equations (3) we find that 11>"20. We may state our results in the theorem: The congruence g; normgls canngt pg g congruence 9.; Guichard. D , (1' a Q 4. The ioint Coniu :23 surface m Consider the surface 5; with parametric equations if==kQ+-A.EC , 7Z1: -+ 1.]? , (as) E:“ -+ A.75 . where X is an arbitrary function of u and v. The point (§.>),€) will be the harmonic conjugate of MM}, 1‘) with respect to the focal points on the normal if, and only if "PP; _| (89> Rn " a,- x 1 Q .+ (T1 (3 were ’5? JL Hence for a minimal surface.X=fl°, that is, the point conjugate surface does not exist (i.e. at infinity). The point P6933) bisects E F, . 13. 5. Parallel Surfag 3 « -m -~ ‘ - _L_ - J— ~i~-+ - kérs. A ’ 10?. t)’ ,1 If k:- constant, J: = constant, and Hm?— constant. Hence if the point conjugate surface is a parallel surface, S has constant (#0) mean curvature. We may now find the expressions for E“: PW and G“? t )— EN: 2' (Led- X KL.) , EW=Z A‘XP.=1PL=W- 3 Hence 3“ is a radial transform of 3 and located symmetrically with respect to the common focal surface. 7. The General Surface [)0 Consider the surface 3 defined by S'zkv-AZ, 7:71-1:17 1;:f-I-XZ. where A is an arbitrary function of u and v. We now find the functions EW) Fm, 21:. We find readily that Eva: Z (kgi-AKH'F A“ E)z, EN:E+LL;E+XL1P,\ E) [N = E\Ey-+c\ 9K), 3: Xu k V Q%)=Z(¥v+t\ty +kv KY: (34) Gm 6.0“?)‘4. A: D“) = :- rename +A market“ L3, D?“ = 130.3%}, Akk' 16. b‘(n.t {(L» +(\ytu+(\ Kuv+t\uvK+AugV )1 D‘Q‘): b‘(l‘%)+k%bl D‘Q)‘=‘X\AV' D"“"= 2' when X. annex/i + A. ‘3 3’ 3““): D"(“ 13;) + va ‘ Consider the local trihedral composed of tangents to the lines of curvature and normal. The tangents to 300‘) the surface , and S intersect in the lines % Joining the points R:(k+.c\_@ AKA the?) (“A LOP-«Ar a_.J3 E‘ and S:(k _L:(P7~ QKA) hr.) ‘(k'f‘i)’ (RC kA) Tris—LP» VG.— Put " V? e a \ P and X I? X. I? 8' Then earrings we fit] and S 5 [Su- Bfleckfi %1. The coordinates of P and S with respect to the local trih edral are 17. 5.... MP.- u) {2:0 8 {2128" W In the tangent plane (9») and in local coordinates (Ema) the equation of f is 2/, '2‘: Va .... 4—. 309.401) Ba’dol) Then I; - B(P-kA)§+H(P.~kl)>)~HB(E~BA)(Rckrl) = 0, 95—: =BX§+RM+MEMQ a) we k,\)]=o Consider the case in which RzPL. Then § = 13mm): +90%sz n 3012490“: 0, €= BE—I—Rh—fi B (P.~l< A) =- o, gig—2 R B A: 0 52 Since neither R nor 3 is zero, we have i=0. When E; [a , we may eliminate K and obtain :: H