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
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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