WWI r 1 k t ‘ WI 1 1 ll 124 488 HTHS SURFACES ASSOClATED WITH A SPACE CURVE Thai: for the Degree of M. S. MICHIGAN STATE COLLEGE Lawrence Edward Schaefer 1941 ' "linen“ .‘ Michigan State” University ~*" ‘FM'rrrr'v. ' ' z”.- . firmer ' l ’ '~ .l " ' " . I ‘ ”‘u * \A - ~ ‘ . 'I' ,‘ 1 I . I ' II I ' RETURNING MATERIALS: )V1531.J P1ace in book drop to LIBRARJES remove this checkout from M your FECOY‘d. Eflgé will be charged if book is returned after the date ‘ stamped be1ow. —’ T "v £j.§;.~f}?§”“ '7‘“ ' E; "' -, . 3 q t , , i i ' I i } 4 a 1 l . ! i g I If“ 3 ‘ 1 . -V 9 h 1 0| .“-’.'""?,V. «4 VI" ”a, 1 il‘ , r, . .‘ ' V . ‘Ily . v ~~ _ V . . ' ' |i. ' U‘ 1! , . \ . \ ,' ‘ ' l v I I | x ACKNOWLEDGMENT To Professor Vernon Guy Grove whose forbearance, encouragement, and numerous suggestions have made this thesis possible. SURFACES ASSOCIATED WITH A SPACE CURVE by Lawrence Edward Schaefer A THESIS Submitted to the Graduate School of Michigan State College of Agriculture and Applied Science in partial fulfilment of the requirements for the degree of MASTER OF SCIENCE Department of Mathematics 1941 Contents Introduction . . . . . . The Torus Defined . . . . Preliminary Computations Some PrOperties of the Torus Maps on the Plane . . . . The Normal Congruence . . Loxodromes on the Torus . U1 .p F‘ 19. 26. 31. SURF CES ASSOCIATED WITH A SPACE CURVE Introduction It is the purpose of this paper to discuss some of the metric differential prOperties of a torus. Let us consider a curve C» , a point ‘P on C. , a local trihedron of C at ‘P , and any point Q . The local trihedron is defined by the tangent line, principal normal, and binormal to C at ‘P , wherein these lines are referred to as S -, VK": and ‘Q -axes in a manner similar to the x-, y-, and z-axes of a fixed cobrdinate system. Let 1’ have cobrdinates (x,y,z) in a fixed system of cobrdinates and codrdinates (6,0,0) in the local cobrdinate system of C at 1’ ; similarly let G have cobrdinates (X,Y,Z) in a fixed system and cobrdinates (§,Vr,§) in the local system of C: at 1? . The projection of the line segment Joining —P and Q on the x-axis can be represented analytically in two ways, prOJ° PQX=~_X-—X ‘ DPOJoPQX= Set + v12 + fk (X,\(Z) 3r 9‘ Y F'\<3.\ Hence ._ X’s-x: 3d+n9+ ‘gA Similarly and Z-z—Sx +‘1V‘ “' 4942. wherein (1,6,3 , Lyn,“ -, Ava)» are the cosines of the angles made reapectively by the tangent line, principal normal, and binormal to (I at 1? with the lines through 'P parallel reapectively to the x-, y-, and z-axes; that is they are the direction cosines with respect to the fixed cobrdinate system. Hence the equations of transformation between the fixed cobrdinates (X,Y,Z) of (Q and the local cobrdinates (f,\1,-g) of Q referred to the local codrdinate system at 13 with fixed coordinates (x,y,z) of C are* X: x+d§+£wl+>.%, (1) Y: «aw-(BS-xv‘mni-flg » z: ry+3§ +Mw +11?- For example consider a circle C” of radius <1 and lying in the normal plane with center at the point "P on C . The local cobrdinates ( 30mg) of a point Q on C’ 3 are 3:0 ) (2) V13 amu) §,-= (x AL“ u. . wherein \L is the angle from the (in! principal normal to the line through 13 and the origin. The fixed cobrdinates (X,Y,Z) f=i§.z of W? are 1 Xe.- x+dQcmut+d>xMu , (3) Y=A1+a~n¢°W+aeri 'Z = v} -* o.y\ szu.'* dmv Imhnia *V. G. Grove, Metric Differential Geometry of Curves and Surfaces, Notes prepared for use in lziichigan State Colleg e, p. 35, eq. (74). Hereinafter referred to as Grove, geo metry. 4 The Torus Defined If in equations (2) and (3) above (3 is a circle of radius b>0., with d and b both positive and finite, the locus of the circle (7 is a torus. Let ‘V be the arc length of the circle (I measured from a fixed point on <2 . (O, 0,0) (o,a,o) C F’i5.3 n 7(O)b)°) For the circle (L, the radius of curvature (o is b . Also being a plane curve the torsion #7 is zero. Grouping these last restrictions and observations together for convenience we have O/(;. 40" an) and 754%, .Q—‘rm,>\—+V J{-~H'X"’*Z. Preliminary Computations Several computations can be made which will prove useful in discussing some of the properties of the torus. By means of the Frenet-Serret formulas* a_ 2L (1" _~%_ ) (5’: 3%:— ) 3‘ {a ‘ I ’_._ 5-1. Z , (5) 1 =‘<%+$)’ W‘ ‘ (%+T>’ “' (NT) .A' =: é% > /*’ g z? ’ ‘v’: 2% wherein primes denote differentiation with reSpect to the arc length, and the relationships expressed in equations (3) and (4) we readily find the following partial derivatives of the cobrdinate X for the torus: * L. P. Eisenhart, A Treatise RE the Differential Geometry g: Curves and Surfaces, New York, Ginn and Company, 1909, p. 17, eq._(505. Hereinafter referred to as Eisenhart, geometry. ‘()k = 1. *‘(LQ.QD°\L -+ <1)\ néw.-, )§.= —.QJl;aL~ u. i- d)\ QDQLL , x,= iii-"Timed . (6) {Xm‘= “fiQQmm-O~>\A3w~u , X..= 3,—in , (X..= 39(“t‘w0 Also certain fundamental coefficients defined as follows will be used: * 2 ..Xu + XS 4 zfi , {‘1 F = xuxv + Yqu + zuzv , 2 G = Xv-+ Y3 + 23 , H2 = as ~ F2 , ** qu Yuu Zuu l XV Yv ZV * E. P. Lane, Metric Differential Geometry pf Curves and Surfaces, Chicago, The University of Chicago Press, 1939, p.75 eq.(3.3). Hereinafter referred to as Lane, Geometry. ** Lane, Geometry, p. 75 sq. (3.6) *** Grove, geometry, p.75 eq.(13®), with Lane, p. 123. XUV YUV Zuv 1 M = H Xu Y1). -'u a X. Y. 2.. Using the notation suggested at the end of the last section these relationships become E=ZX: a F=quxv ) G'ZXV‘ 3 (7) _|_ quk _ _\_ xu" ‘ va \, = H xW > M‘ H Xi . N " K Xu XV Xv xv It will further be noticed by comparing equations (4) with (7) that HL, HM, and HN are all of the form Q where cm + bfl. +C,>\ (1‘6” by. + W“ 6.3+ bm + 6.0 d,oL + b,9+c,>~ eqp + hang/u. CLJ + Vow-icy m (8) Q (.134 +539+C3>x c.3(3+\,3\m+C3/u “33"".3“+c39 But Q can be written in the form of a product, namely C». b, c:I CL (5 Y Q 2: 1 b1 Q3. R N“ M . Q3 ‘33 C5 >\ la V 8. However the second determinant of the product“ Hence 0., b, c, (9) Q = 0., b, c, . a be C's We may therefore for ease in computation state the following introductory THEOREM. Any determinant of the form Q expressed in 18) can be evaluated as a determinant of the coefficients of dw.l,)\. The evaluation of these Q determinants agreesfavorably with the X value notation already adOpted and expressed in equations (6) and (7). Making use of the introductory theorem Just stated and the relationships expressed in equations (6) and (7), we may derive the following fundamental coefficients for the torus: E: 0.1 , =-Q. \ F: o , M=O ‘ lO ( ) 6" (\..£-_. mu)2 ) '-' as)“ (\- flew-am) , Lane, Geometry, p. 18. 9. Some PrOperties of the Torus Theorem 1. The parametric net on the torus is an orthogonal net. The parametric net is formed by the families of lines u.= constant and v = constant. The curves u.= constant are the circles whose planes are parallel to the osculating plane of C.. The curves v = constant are the generating circles C]. Proof: A necessary and sufficient condition that the parametric net on a surface be orthogonal is that F = 0.“ By equations (10), this condition is satisfied for the torus. Fig.4. V= Constant ; Lane, Geometry, p. 116. 10. Theorem 2. The parametric net on the torus con- stitutes a conjugate net. Proof: A necessary and sufficient condition that a parametric net be a conjugate net is that M = O.* By equations (10), this condition is met for the torus. By definition a net of curves on a surface is a con- Jugate net in case the tangents of the curves of one family of the net at the points of each fixed curve of the other family form a developable surface.** On the torus, for a fixed v, the tangents of the curves u = constant form a circular cylinder. For a fixed u, the tangents of the curves v = constant form a circular cone which degenerates to a plane for u =‘q-a and a circular cylinder for u = 0. Theorem 3. The asymptotic curves on a torus can be found by quadratures. Proof: The curvilinear differential equation of the asymptotic curves on a surface is given by:*** LOLAAI +2Md-d~4r+Nd~Ar"=O . Making use of the relations in equations (10), the differ- entail equation for the torus is (ll) -O.ob.~2' -\- 00;“. <\——g’b—Qmu)o(”‘=o ) * Lane, Geormetry, p. 138 Th. 3. ** Ibid., p. 135 Def. 1. “*4“. Ibido, p. 123 qu (406). 11. from which we find d». (12) V -'-"- i hr; 0 Jaau‘b_o.mu)‘ Equation (12) can be shown to be an elliptic integra1* by use of the following substitutions: Let u.= cosZ‘x , then x V" ,_ b I ‘dnc / - 1‘ b1 -— b I‘ . ‘ 7‘ /Ha‘ -(‘L R) ) let )c- £i. :: f%fsin 9 , then M-|<1&I_‘ v " b T ) d9 = + 1. ‘ . M"(‘-b&--\) ‘/‘—2?;1§l+lu.m9)z ) and let 9 a %} — ¢ I then __s;“‘(lo.x_ ') &¢ (13) v = i b 4 ET V/|_.§L1 cc: . q z 11!-5m"(1_:-‘) a. For convenience we shall call the curves u = constant parallels of latitude, and v =constant meridians. * Harris Hancock, Lectures on the Theory of Elliptic Functions, New York, John W'iley & sons, 1910, p. 187. 12. Theorem 4. The parametric curves are the lines of curvature. Proof: The lines of curvature on a surface not a plane or a sphere are parametric if and only if F = O, and M'= Of That is, the lines of curvature are parametric from equations (10). But the lines of curvature form the 9351 orthogonal conjugate net on a surfacejb*It follows then from Theorems 1 and 2 that the parametric curves are the lines of curvature. This theorem follows readily from the differential equation of the lines of curvature. For any surface this differential equation is*** (EM — FL)du2 + (EN — GL)dudv + (FN .. GivI)dv2 = c. which becomes for the torus by equations (10) (14) d(\-%—mu) MM=O The solutions of equation (14) are seen to be u = constant and v = constant. Theorem 5. The points \u\ >51: are elliptic points; the points \u\a.from relationships (4), it is readily seen that for \u(>3§ , K >0 since the factor (b- (3%“- )>0 ; also 1'01" |u|<1§ , K‘12|:. , and real for \u\< 1;: . Theorem 6. The parametric net on the torus is isothermally orthogonal. Proof: Necessary and sufficient conditions that a parametric net be isothermally orthogonal are** — __ £5.) = o (18) '- " O : Md‘ (“Q03 6- u.v From equations (10), F = 0, also it will be observed that E E ) — = C 4 r '— = ° G a fun tion of u only , hence {in} 6 W 0 Theorem 7. The torus is an isothermic surface. Proof: By definition an isothermic surface is a surface on which the lines of curvature form an iso- thermally orthogonal net.*** Theorem 7 follows immedi- ately from the statements of theorems 4 and 6. * Lane, Geometry, p.162. ** Ibid., p. 120 Eq. (3.21). *** Ibid., p. 145 Ex. 6. 15. Theorem 8. The parametric net onfa torus is iso— thermally conjugate. Proof: Necessary and sufficient conditions that the parametric net be isothermally conjugate are“ 1 H— M «Lo-3. L“ =0 (9) -O ) va From equations (10) we see readily that M 2'0, and that L _ h - since N —a function of u only ’ (9”? W )u-v - o . Theorem 9. The minimal Curves on the torus can be found by quadratures. Proof: The curvilinear differential equation for the minimal curves on a surface 18*“ EMI +2FMOLU+GM1=O. By equations (10), the differential equation for the torus becomes 1 l (L 1 2. (20) com +(\—-5-u.u.) at .0, from which we obtain u. b I = i a. o O (21) V k a b-acmu. Performing the indicated integration we find that (22) V: i 0.sz i ta..." ”1"": M“ V 5 -CL bC~which would make mu>\ for this factor. The third factor equated to zero yields .._b__. Thus we have the following Theorem 10. In the neighborhood for those points for which Cmu=i€3 the torus resembles a minimal sur- face. The torus could never be a minimal surface, since the only surface of revolution not a plane that is a minimal surface is the catenoid.“ If we take the partial derivatives of the functions E, F, and G defined in equations (10), we find that E == F' = E = F; = G; = o , \A- U. V 2r (J) Cy“: 2(\—%muX%-MUL) - Hence the Christoffel three-index symbols of the second kind for the first fundamental form** r“ = ,—_‘-——.(c-Ej + Fla-zen) . P: = 1‘H.(-FEu—EE,+2€FL) \i H I \ I r e fi.(GEv-F6u) . r“ =l—H=.(ee,—FEV), ‘1 \1 P‘ = {b.K-FGY*GG.,+ZGFV), ['11 11 11 I .— If” (E e, + FG-ulerV, * Lane, Geometry, p. 165 Ex. 8. ** Ibld., P.132 Eq. (5.4). 18. become ‘ _ i -1‘ _ _ G,“ P“ _ O ‘ Pu = O ’ ‘33 - 2E ) <26) . G Pii : C) ’ Pi: ='— if' ) r‘:a=: 0 ° Assuming the parameter along a geodesic curve on the torus to be u, the curvilinear differential equation of the geodesics is“ ,3 o v" =“P; “(211,: 'P.‘.)v' +1211; "’ P11, )V'l-i- F‘z'lv which becomes by equations (26) (27) QCL -_- 3.55:. 9&1: _. ithly. am 5 out 2E am In order to find solutions for equation (27$ we shall use the following substitutions: Let 95.! P 3 du- ’ then equation (27) becomes -3 _g_ ~2~° awev =% let now ~1. P ='} ) and then we obtain wt 3 E G- This last equation has the solution _ GS+SEC - ’3’ SEG“ ’ where C is an arbitrary constant. Lane, Geometry, p. 147 Eq. (8.4). 19. Retracing our substitutions, equation (27) may be written (28) A. - i C’ w; SE 1/6-5 +sec Theorem 11. The geodesic curves on the torus can be found by quadratures. Proof: In view of the above discussion, the geodesic curves v, which are solutions of equation (28) are u. (rtobu. (29) V: t’VSEJI/GS‘1'SEE‘ . It has been remarked already (under Theorem 9) that the minimal curves on a surface are geodesics. Hence the curves V = .:___£fllh__. -\ 51TCL Ag”.u. ) (22) “ha-Ql' bmu-a’ are solutions of equation (29). Maps on the Plans The element of arc length is represented by* oLs‘= EobUL} + lFoutow + Go‘du’l) which becomes by equations (10) for the torus (30) as‘ -—-. eta...“ +(I- %~e~m~)‘ael. * Grove, Gegmetry, p. 69 Eq. (119). 20. If we make a transformation 3 a 3(u) and V = v, wherein __ ab (31) M = b-qmu, d“ ) the square of the differential of arc length equation (30) becomes (32) 45‘ =<\— -°‘-’b- mu)1<fl‘+otfi‘) . From the transformation equation (31) we obtain ab _ 55-0." mu. (33) U = tam' bt—oj' bonus-Q. Finally if we let 3 - y and V = T we may state the following 1 ‘L ~ Theorem 13. The transformation y : -—EJL—-to " b"“-A~*“- b“_o'1 bWW'Q- and x = v represents a conformal mapping of the torus on the xy-plane, wherein the parallels of latitude map into lines parallel to the x-axis and the meridians map into lines parallel to the y-axis. Proof: A transformation between two surfaces is con- formal if and only if the minimal curves on the two surfaces correspond.“ For a plane, the minimal curves are given by the differential equation (34) Mt + OLA-(t '-"- O . The minimal curves for the xy-plane are thus 21. (35) x-tvq- But from the transformation eXpressed in theorem 13, the minimal curves on the xy-plane are «Hf-a. ml... u- . ab _, (36> 7" = * 04—M— W W... Finally comparing expression (36) with equation (22), we see that by the given transformation the minimal curves on the torus and the plane are identical. Also by observing the expression for the transforma- tion it is readily seen that for u =constant, y = constant; and for v = constant, x - constant. 'E o ’1 6 V8 C.<)V\3"v““.t 1 1‘.) 1% 3 p i “i 3 1 I e Y (La- Consul-“t , F‘33.511 * Lane, Geometry, p. 193 Th. 2. 22. 41 “:11 Z :31 “ 3 uzf-E ‘b “ “biz-:01. b‘G- 1(_}_22_¢) 0 fl 1 E’ X. c, 3 Fig.55b For some other possible mappings of the torus on the xy-plane, we recall that the transformation x = x(u,v), y = y(u,v), between a surface referred to parameters u, v and the xy—plane is an equiareal map of the surface upon the xy-plane in case the functions x, y are solutions of one or the other of the two partial differential equations* (37) Xqu - xvyu = :t H . Using this relationship and those of equations (10) we have for the torus (38) xw‘iv‘l‘v’ifii “(“tmrl Let x = v, and y = f(u), and choosfihg the negative sign before H; then equation (38) becomes (39) .o.(\- fimw) - 5M)- Hence <40) {my = aku- ewe). * Lane, Geometry, p. 203 Th. 2. 23. £5 b represents an equiareal mapping of the torus on Theorem 15. The transformation x = v, y (1(1L-' Ibgau) the xy-plane. Proof: The transformation (40) satisfies the partial differential equation (38); therefore the transformation expressed in the theorem represents an equiareal map of the torus on the xy-plane. This transformation also maps the meridians into lines parallel to the y-axis and the parallels of latitude into lines parallel to the x-axis. fl 9 2 'fi uzT‘l’ ‘ :n’ 3 l I l o 1 l S 4 ’%b I 2‘ / “I. // // //“ g5 / / 1 O // x 1 ’ 0 3gb ¥b The map on the xy-plane extends from x.= O to x 24Tb and y = 0 to y = 21rd. is represented on the plane by the area of the rectangle Thus the surface area of the torus 24. which has for its dimensions 21.. and awe , or Mf‘ab. Comparing this method of finding the surface area of the torus with the method of the calculas, the advantage of the equiareal map just described is seen. Let the torus be generated by revolving the circle 2 6 2 8. U (X—b)2 + y about the y-axis. Then the total surface S is ex- pressed ( (41) 8:: 23V 9 wherein x = b + a cos 9, zr S = 2’WJ(b + 3. cos G)a d9 0 therefore =0 S: 4’lT‘ob. A1 a. 9 (bxfi Fig} 1 9=2Ar X. and ds a d9 . 21 arexb9.+ a sin e] , Another equiareal map may be found as follows: let y g u and x a v§(u), (42) O.( |_ then equation (38) becomes W“) = X'v It should be noticed from equations (10) that Xv is a function of u only; x = vf(u), or by equations (10) (43) x: Qv(\_ xv = f(u) G. b has as a solution em). 25. This map is not as convenient a representation as the other equiareal map already discussed as seen pictorially. 8 fl 0 “2‘ a fig 3 I 3’ w '1 o X- o v-1§ ‘4 Fig.8a. F‘s-8b For the entire surface, the map appears as follows: ’rWfi W l I i i l l , i \ !\ J' ,‘\ l ‘I ‘ j V . 1 K I I l O 1 l R I q-(b 1TQbU‘3fimq) I 26. For this equiareal map the x ranges in value from zero to Z’ITatb (i- %’-Cm‘1) , and y from zero to 211’. As a check, the area under the curve (44) 1': Z’iTCLbKi-G‘? CAI-34x1) i represents the area of the torus. 2‘1" 117 Q o ._ __9.__ = - “M A - Jo I’WQ.(\ b £40641)d.~\ 2WQb\_“1 b 140) therefore A: 41716.1) The Normal Congruence Let us now g choose a line nor- mal to the circle in the normal -’->- I plane C. with cofi 6rdinates of a point on the line (_)_{_,_Y_,§) and having ’ (Zita) f direction cosines A, B, and C. Let D be the distance C. from the point Ff 9. 9 (X,Y,Z) to the point (22.1.2) . 27. Hence we have (45) _)$=X-\-DA and similar expressions for i, and 7 The direction cosines of the normal to a curve are given by the relationships* J\ -3: = ) (46) A:-T-\_ . 13- H . C I; I wherein J‘=A1u_%y—A1v3u ’ and J2 and J3 are similar eXpressions using the notation at the top of page 5. Combining the relationships eXpressed in equations (3) with those in equations (45) and (46), with some algebraic simplification we have as a result (47) A:((3M‘UW)MKL+(F/u-(5V)mu) and similar expressions for B and C by permitting and (Sea. PM. wwwflrem Haw-‘33. (340k, X—rp >/"’—,>‘ ’ V—*/0~, M41, M9M3A—bc. But in the determinant 0L (33' 1 MM =\, >\ #12 each element is equal to its own cofactor.** Hence, (48) A - .. )(Itbm\1 +-.K QDGKL - * Lane, Geometry, p. 79 Eq. (4.8), and p. 64 Eq. (1.4). ** Eisenhart, Geometry, p. 13 Eq. (40). 28. Therefore we have combining equations (48) with equations (45) and (3), But for D any constant value, the surfaces for which the cobrdinates of a movable point are (y,y,;) and (X,Y,Z) reSpectively are related by x = x - At where t is a constant. We may recall that the parametric equations of a surface S parallel to a surface S can be written in the form §I=’x — At, y = y - Bt, E : z - Ct in which t is the constant algebraic distance from a point P(x,y,z) of S to the correSponding point §(§,y,§) of S and A, B, C are the direction cosines of the normal of S at P.* From equations (49), the surface for which a point P(§,X,;) is represented is seen also to be a torus since it is of the form of equations (6). We may now state the following Theorem 16. A surface parallel to a torus is also a torus. An interesting geometrical interpretation for the principal radii of normal curvature may be develOped. Let us construct a line in the normal plane perpen- dicular to the osculating plane at a distance b from * Lane, Geometry, p. 208 Th. 1. 29. (x,y,z), that is the polar line. Also for any angle u, construct the normal to the surface of the torus and ex— tend it to meet the polar line. The normal line also passes through the center of the local cobrdinate system, that is through the point (x,y,z). To show this we make use of equations (49); comparing the definition of the torus, equations (6) we see that X = x + aA , or X-XrlLl—L E... A B ' — +8“ p ill ( which is the equation of a line with direction cosines A, B, and 0 through the point (x,y,z). Call the line seg- ment on the polar line intercepted by the principal normal and the normal line T; and the line segment on the normal line intercepted by the torus and the polar line Q. S Q 0. Wm.) rr .. 3 3 b Fig.10 ’l/ 30. From the figure, T is given by T = b tan u . Squaring, and making use of trigonometric relationships we have T2.: betan2u = b2(seceu — l) = ‘02 —b2 coseu (50) 2313-3005311 5,2 ::h2 . (cos... + Also by the Pythagorean TY meor m we see the folloadng re- lationship: )2_b2 . (51) T =(C-ii—a Equating equations (50) and (51) -— Wfi.<.b G‘cu‘L” -+ 0-)2 = (CQi- o31- we see that _ E>--1)which is certainly true for all a and b Qno”‘ by relations (4). Eisenhart, Geometry, p. 76 Eq. (24). Bibliography Eisenhart, L. P., A Treatise on the Differential Geometry 93 Curves and Surfaces; New York, Ginn and Company, 1909. Grove, V. G., Metric Differential Geometry of Curves and Surfaces, Notes prepared for use in Michigan State College of Agri- culture and Applied Science. 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