nl‘lllllllll'W ‘

‘I
ll

l

f
i

ll

l
I

fl

122

 

METRIC DIFFERENTIAL GEOMETRY
ON A CONDID

THESIS FOR THE DEGREE 9F M. A.
David Francis Randolph
1933

 

    

Wuv -5

”film 50..

5- 71-:

Q \
a

MSU

LIBRARIES
.—;_

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES wil]
be charged if book is
returned after the date
stamped,below.

 

ACKNOILEDGMINT

To Doctor Vernon Guy Grove without
whose suggestions, aid and encouragement

this thesis would have been impossible.

 

(II-II.

METRIC DIFFERENTIAL GEOMETRY ON A GONOID

RcThesis
Submitted to the Faculty
-of
HIGHIGAN STATE control
of
AGRICULTURE AND APPLIED scxzror

In Partial Fulfillment of the
Requirements for the Degree
of

Master of Arts

. by
David Francis Randolph

1933

 

 

1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.

13.

CONTENTS

Introduction .............................
Defining Differential Equations ..........
Fundamental Coefficients and Forms .......
Asymptotic Curves onta gonbid ............
The Quadris of Lie .......................
normal Polar Reciprocal Quadric ..........
The lormal Quadris .......................
Ohasls's Correlation .....................
The Parametric Osculating Ruled Surface R7
The Focal Conoid .........................
The Osculating Conoid ....... .............
Quadric Oonoids ..........................

3.

8.
13.
16.
19.
34.
36.
38.
32.
39.
41.

NITRIC DIFFERENTIAL GIOHITRY ON A OONOID

1. INTRODUCTION

It is the purpose of this paper to discuss some
of the euclidian metrical properties of ruled surfaces
belonging to a linear congruence. As is well known,
such surfaces have as their flecnode curves two straight
lines‘. For the purpose of our study we shall choose
one of these lines as the ideal line in the xy-plane,
and the second lying in the yz-plane and passing through
the origin. Such surfaces, from the euclidian point of
View, are ccnoids.

To repeat, a ccnoid is a one parameter family of
lines, the generators, which always intersect a given
line, the directrix, and are parallel to a fixed plane,
the directing plane.

* I. J. lilczynski, Pro ective Difrsrential Geometr of
Curves and Ruled Surfaces, EeIszg, E. 6. TeuEner, T955,
p. 155. fiereafter referred to as Wilczynski, Geometry.

we may write the parametric equations of a ccnoid

as follows:

 

 

2
tafifiA
X :: VI,
3 Nye)
(1) y = VhHAu. :u :
- A“ : fig
2 7-1 Ll, “s“.
~4

wherein L1 is the distance of the generator from the

directing plane; v' is the distance of the point,
(x,y,z) from the directrix, measured along the generator;
A is the tangent of the angle which the directrix makes

with the z-axis; and I and Tn are the direction cosines

of the generators. Both./e and ”1 are functions of

only.

The curves u : Cth‘l. are the generators. The
curves v : Con at. are the curves cutting the generators
at a constant distance from the directrix. The equation
of the direotrix is, of course, V : O . If the
directrix is perpendicular to the directing plane, the
ccnoid is called a right ccnoid. It is evident that

awmmWMamm
amhtminu Azo-

3. DEFINING DIFFERENTIAL EQUATIONS

The parametric equations of the ccnoid in homo-

gensoue coordinates may be written in the form

(3) X =v1

1 I

x2: vm+Au, stu)x.,=l.

From (8) we may determine the coefficients of the follow.
ing defining differential equations

 

 

qu : dlqu+b,xu+C,Xv+d,X,
(3)

va :. d,XW+b,JCu+C2Xv+d2X.
wherein

a, : HIM—mi”) , C. :— vQ'm”-—)n'1") ,

1m' eml’ '-m1’

(4) 1m

13,: d. 2 a2: hazel: C1130.

Since fl2+7772 1" , and if we let A = 1771"”11/ we

we may show that

‘5’ A = ’l=——’"', A’ = jmem, A3zf’mtm'1’:
V m 1

we may now rewrite (3) in the simple form

“Q

X = LA: X“ -- VA” X, ,
(6) A
O .

}( ::

VV

Another set of defining differential equations
may be found by choosing two curves on the surface, say

V - v. and y _-.-. vza; v, . The parametric equations of

these curves in homogeneous coordinates are respectively

X=v,1

1 X,:v,m+Au, X3: LL, X,=I;

(7)
y: vz’e) yr=v1m+Aul ya: u’ y‘zl.

It is well hnown‘ that the pairs of functions (x,y) of
(7) satisfy a system of differential equations of the

 

' lilcsynski, Geometr , p. 136.

5.

form

X”+ P..X’+ P..EI’+ ck} + $.51 =0,
(8)

y" + Puxl +10“l g' + $196+ cfiuy = 0,

wherein

X, : 4X " (1*

Z— I X =1 , e‘I'C.
u’ aLu‘

 

we find the coefficients of (S) to be

 

 

 

 

 

 

 

I 1 . _ a
—,_ ’ , _ V3 _..As\’
I)" - A v. ’ 12 - A v. I %n ‘- A ) $71.- . )
A (v.-v.) A (V.-V.) .v,- v, v. -' v.
B. 7. ’ A’ V1 ) 12 Z A, v1 ) = 42 V!- : ‘ A, v1 .
Amt- V, P A(V,-V,) $3, V1 " V. ) 17“ V1 — V,

The defining differential equations known as the

'Gauss lquations' are.

 

‘ L. P. lisenhart, Differential Geonetrz g; Curves and
Surfaces,New York, Einn and Company, 1 09, p. I53.
HereaIIsr referred to as Eisenhart, Geometry.

 

 

x“ -_- {271“ + {’2’} X, + DX,
(9) x“ = {’31, + {'m. + D'I,
I... = m x. + PM. + D”X,
wherein
{' '} ; y’éA’+AV7n’+AVTnA“ 02‘ =- VA’m z,
' I + (VA + Al)‘ I “MM“

Am'(v‘A‘+2Avm'+A’+D " ATMV‘AA’d-Avm")
I + (VA-M1”
-- (g‘g‘izAvm'+A'-H) (V‘A’+AM’) ,
l 4- (VA + A “1
111’ + Am’ , {'1’} 3 .. Am(vn’+Am’) T,
l+(VA+A1.)‘ I+0lA+A1)2-

{ '2'}

m
-
'3
W
0

{2'2} : O, {.122}:,'+A&2:n;‘>2‘ D"X : O.
DIX = ~.._A—m. e

 

/ + (VA+A1)T

Let us make the transformation of curvilinear

coordinates

u : @(U,V) 1'3 U , V : ”20430

Then the differential equation (3) becomes

 

 

)(m1 = E, xW+'5,xq + ex, +d,x,
(10)
XW = d. x.“ + b1 x—d + c, x,- + d,x,
wherein
25: : ‘3 LVV :2 LVH. I ‘51 :. " Luz:
v
(11> C, ' c w. +aww new-r WWW, wigs).
v
b, = d. =2 a,=b,= (11:0,
wherein the values of a. and C are given by (6) as
a ‘-"- V4, ’ C :2 - VA“,
£5
Therefore (10) may be rewritten in the form
x“ 201—? w. — z w.) x—-
(12> + PTA w... -vA=w.— wmww $3: - algfnxv
XVV :2 "" WVV XV 0

W

8.

3. FUNDAMENTAL COEFFICIENTS AND FORMS

It is well known that the functions defined as

follows':

I I

A=Iyw2yl=~m lezq,xv1=,(
(13)
C =11“, yvl = ~(VA+A1) -

are proportional to the direction cosines of the normal

to the surface at the point (x,y,z). If we let
A2+B‘-{-C’: '2, then the direction cosines X; Y) Z

of the normal to the ccnoid may be written in the form

 

X =.__’5.: -__72L__ ’ Y-‘JA: l i)
(M) H ”(mmm H when“)
Z ':.._C_.: —- VA‘I’AL

H 1+(vA+A!Z)’~

Consider, on a ccnoid, any curve C: whose equation

is of the form (paw): O . Then the element of arc

 

* Bisenhart, geometry, p. 114.

 

.IV .I pll..e..1w J: H" .rbfi'... l,, 1.! .lel .l ‘ .5 I . e ... . n'! W.» , If. I, ‘33 I‘nloi.

9.
of C is given by
-(15) HS)‘=f-= E du‘ + 2 F dudv +6 dv".
The differential form (15) is known as 19; £1331 _f_\_I_n_-

damental form, and E I F, and G as the first fundamental

coeffisients.’ The latter are

E = X3, + y: +2.3 = v*A‘+2Avm’+A‘+I,

A
H
0'3

v

‘1

H

XHXV + 9"- 5v+zuzv 3 Am I
.. l 1 _.
G " XV + yv‘ + iv - I 0
Hence 1:32 first fundamental form £93.; 3.9.2. ccnoid 1.;

(17) f: (mu-2mm +A’+ I) du‘ + zAm dudv + alv‘ .

It can be shown that

9.

EG-F = H‘= A‘+B‘+C‘.

u

The net defined by f 0 is called the minimal net.

* lissnhart, Geometry, p. 70.

10.

The minimal net is imaginary on all real surfaces.

Therefore the minimal net lg imaginary on‘g ccnoid.
The quadratic differential form
(18) q); Ddu“ +2D’oLu aLv +D" dv’

I II
is called the second fundamental form, and D ) D , and D
are the second fundamental cosfficients.’ The second

fundamental coefficients are defined by the formulas

(19) HD 2 11cm yu, Zvl, H0531“, ywzvl I HD"==IX,,, yMZJ.

Therefore the second fundamental form for the ccnoid‘;g

(20) (D: M’ ain’t-A 2A dqdv.
\fHLvn-IAI)‘ \fH-(vA-MU‘

 

 

The differential equation of the asymptotic curves
in curvilinear coordinates is (P: 0 . The asymptotic
curves may be found by quadratures. A discussion of the

asymptotic curves will be given in the next article.

 

* Eisenhart, Geometry, p. 115.

11.

We shall now find the condition that the curves
V: Const. , or the curves that out the generators at a
constant distance from the directrix, be plane curves.

The equations of these curves are

X v1) yzvm+Au) Zzu)

wherein V is constant.

A necessary and sufficient condition' that a curve
be a plane curve is that its torsion be zero. The con-
ditioa that the torsion of a curve C‘u on any surface be

zero is '*
‘ Jrq.s jfiuu_1 53in.» ' = (9"

For the conch! this condition becomes

jllm’l’ -__ m" I!” = 0 .

 

' lisenhart, Geometry, p. 16.
*' Iisenhart, Geometry, p. 17.

18.
We may integrate this differential equation and obtain
in :2 c,l+ Clo. +C3.

Since 12+M‘=l and/Z :: Sin c( we may write the last

equation in the form

0( = 6” drccos[(c,u+c,)sinxj’
:: :gfl —— (x..

wherein X) C 1 and C3 are independent constants. Hence

5 necessary and sufficient condition that the curves

hold.

A necessary and sufficient condition* that a

surface be a develcpable surface is
n ’2
DD --D = 0.

For the ccnoid this condition becomes A = 0 .

 

* Eisenhart, Geometry, p. 156.

13.

By use of (5).and integrating, we find that the

necessary and gufficient conditions that'a ccnoid‘pg
5 develcpable surface are

Under these conditions a ccnoid i_s_ a plane surface.
Since plans surfaces are of little interest to us, in

this paper, 33 shall hereafter assume A x O .

4. THE ASYNPTOTIC CURVES ON A CONOID

The differential equation defining the asymptotic

curves on a ccnoid is

(31) VA’ du’ + 2A dudv : 0.

Evidently, u = con st. is one of the sets of solutions.
Therefore the‘ggnerators form one famihy‘gg_asygptotio
curves 93 a ccnoid. Removing the factor Au and separ-

ating the variables we find

A’du+£dv: 0.
'25’ V

14.

Integrating we obtain the family of curved asymptotios

on the ccnoid
(28) v‘ A = c

Necessary and sufficient conditions‘ that the

asymptotic curves on a surface be parametric are
II
0:0: 0.
For the ccnoid these conditions are

VA’= O

The curve V = O is, as we have seen, the directrix.

The condition A’= 0 implies that

szizlzc.
1

m

Integrating we find

(as) 1: Sin(c.u+6.), m=Cos(c.q4CQ.

 

* Eisenhart, Ggometry, p. 139.

15.

This is the condition that a ccnoid be a helicoid, since
a helicoid' is a surface generated by a line which is
rotated about a fixed line as axis, and at the same time
translated in the direction of the axis with a velocity
which is in constant ratio with the velocity of rotation.
If the values (23) be substituted in (1), it is readily
seen that the above definition is satisfied. Therefore
9. necessary 2339 sufficient condition Egg; _t_l_1_e_ curves M
igtersect 3 g generators gym; constant distance fyggwyhg
directrix pg asymptotic curves i; _t_1_1_aj_t_ Eh; ccnoid 25 g
helieoid.

A necessary and sufficient condition" that the
tangents to the asymptotic curves separate the tangents
to the minimal curves harmonically, is that the harmonic
invariant I of the first and second fundamental forms

vanish. For a ccnoid
I = ED“-.2FD'+GD = vn-zAmA.

The condition I: 0, AI 4 0 implies V = RAMA/0' .

Therefore 9}; g ccnoid,whioh i; not 2 helieoid,there _i_._s_

 

* lisenhart, Geometry, p.146.
** lisenhart, Geometr , P. 139.

16.

gg,unigue curve along the points g§_whioh the asymptotic

tangents sepgrate the minimal tangents harmonically. ‘Gg
‘2 right ccnoid the curve is the directrix. The condition

I "E 0 13131168 A’ = 0, A = 0 , therefore it}; only:

 

minimal concidIGG 3 right ccnoid and g_helicoid.

5. THE QUADRIC OF LII

The quadric of Lie or the osculating quadric along
a fixed generator, is the surface generated by the tangents
to the curved asymptotios at the points where they cross '
the fixed generator.

By (32) the equation of the curved asymptotios on

a ccnoid are

(23 bis) va = C .

A point on the tangent line to a curve at a point (x,y,z)

is of the form

(34);: x +t(x“+xvf‘fi)’ )2: 3 +{(Hn"5v§{)) I: 2+t(zu+2vfi).

The coordinates of the point (r,y,z) are given by (1).

l
loreover from (31) we find the relation 51.. -: -- 1.4.. .
Au 2 A

17.

Therefore a point on the tangent to the curved asymptotic

through the point (x,y,z) has the coordinates
(25) 3 : v1 +vt(,£’ 229%); 7:: vm +t(m"1;’%')+A(u+t)) I: cut.
ls may rewrite (35) in the form
(26) 3 = VU-H‘H) , N 2' v(m+Pi)+At , f: I,
wherein we have made the following translation

ET:: .3 2 fil = :§.—-/414 ‘, if = L3"l4 )

and have adopted the notation

_ ’viA’ _, ml,)77A’
(27) R- 1 72—2?’ P 7.5..

The coordinates C5213} in (36) are functions of V and
t' only. Therefore we may find the locus of this point
by eliminating the parameters V and '2' . We obtain

(as) ART+P3i—R>zf+)n§—,et+uj= o,"

This is a second degree equation, hence the locus 21 the

tangents 33 the curved asmptotics g_t_ points 91 .9. fixed

18.

generator‘yg‘g,guadric surface. This quadric is called
the gggdric‘gg Lie. For the right ccnoid the equation of
the quadric of Lie becomes

(29) Fifi-REE +m3—172‘20.

To determine the nature of the quadric of Lie, we
shall find its intersection with the ideal plans. If we

put the equation of the quadric in homogeneous coordinates

6,353,53) caution (38) becomes
(30) A Rf*+P§i —— R??? + midi-15(7) +A1Ia3zo.

We find the intersection of the quadric of Lie with the
ideal plane 53 = O to be the two straight lines

(31) 3:0, 2'0 :0; MIME—Mm, "66:0.

Therefore the quadric gg_Lie for a conoidIGGWg hypgrbolis

pgrabglgid.s

 

‘ L. I. Dowling Pro ective Geometr New York HcGraw-
Hill Book Company—Tar , p. 1167—1 ' '

19.

6. THE POLAR RECIPROCALS OF THE NORMALS TO THE
QUADRIC OF LIE OF A RIGHT CONOID

The polar reciprocal of any line with respect to
the quadric of Lie, is the line of intersection of the
two tangent planes at the points where the given line
intersects the quadric of Lie.

In particular, the equation of the normal to the

ccnoid at points of a fixed generator is

(32) 3-.— v1+)nt,>z= vm—lt, 3 = nvi.

wherein the trihedral of reference is the same as the one
used in developing equation (38). To determine the values
of 1' for the points in which the normal intersects the
quadric of Lie, we shall solve equations (39) and (32)
simultaneously. We find

(33) PvA(v,(-+ mat - Rvnt(vm—1t) + mod +mt) -,€(vm -1t)=0.
If we use the following relation

(34) Pl—Rm A ) Pm+R£ a—A—C

215 )

Fl“

20.

'9 may reduce (33) to the form
(35) vn’t’ —£(I+V’A‘)t .-= 0.

We must notice the two cases, A’r- 0 and A’ It 0 . If
A, = 0 we have only one finite solution, i' = 0 . If
A, t O we have the two solutions

- - v‘A‘
t. ' O ’ t " 2 + 2 ’ e
V A
If we use these values of ‘t' in (33) and let (5'7'3')be
the point when "l" = 0 and (3:7:3'fie the second point,

the two points are

3’=v1, 7'=vm, 1:0;
(as)

24:
3" = f)?! ___1*3y“t) ": V 41 ___"”vabt) ": 2__A+1V .
v1 (vA' ,7 m (vn' ,I vb.

The tangent plane to a quadric surface f(§,7’3’ so) = O
at a point @:’7’:3: 91s given bye

 

*V. Snyder and C. H. Sisam, Analytic Geometry 21 Gain
New York, Henry Holt,1914, p. . '

21.

(38) 59.3.3" + ggw+ 53%!” + Big :0.

Therefore the equation of the tangent plane at the point
(217;!) to the quadric of Lie of the right ccnoid is

(as) (PI“+)n)5 - (mum + dos" — RM; + hwy-,0," = 0.

listing use of (34) and the value of (3'; 7",3')from (36),

we may reduce equation (39) to the form

2+1V‘A” pg- .3; I, l +m§-1’1+VAI=O.
(4o) vA’ [VM R’I) 2A +1

Since (3'7'5') is the point found by using {- = 0 , it

is therefore on the ccnoid; the equation of the tangent
plane to the quadric at (5:91!) is
(41) m(s ~s’) ~ 1(7-7’)+ MOS-3') : o.

By the use of (36) we may reduce (41) to the form

(42) M5‘X’7—fvn320.

38.

By examining equations (40) and (43) it is readily
seen that they are the equations of two planes of the

pencil which has as its axis the line
(43) ynpr —VAR7—393'5+l =0) m§—,e7+vnz=o,

Therefore the intersection of the two tangent planes is
the line (43). we will call this line the normal polar
reciprocal line. Numbers proportional to the direction

cosines of this line are found to be

(44) K/\ = 15+ zv'A’R , K4 = mn’+2v=A3P, Kr: 211’.

Let us find the point in which the normal polar
reciprocal line intersects the fixed generator. This
point is the point of intersection of the two tangent
planes and the plane determined by the normal to the
ccnoid and the fixed generator of the ccnoid. The

equation of the latter plane is

(45) mvns—xvnh—sr-O.

Hence the coordinates of the point of intersection of
the polar reciprocal line and the generator may be

found by solving equations (43) and (45) simultaneously.

23.

we find the coordinates to be

- _ .;£_ - _. In 25 z:
(46) 3.. m, )7‘723'2’ 0.

Therefore the equation of the normal polar rgciproca;

line is
47 = —- 1 I . J = — m o’+2v'4‘ . "11’1-
( )3 m”““ +2VAR)’)7 V53“)?! 0,!

As the point (x,y,s) moves along the generator the
normal polar reciprocal line generates a surface whose
parametric equations are (47), with parameters V’ and
d , since u is fixed. Eliminating these parameters
we obtain the equation of the locus to be

.2 1 m A (9-7") - A’MN‘HW' +3)
(42)
+[2imn' + 2A(>77’-,€')]§’1 —2A3 = o.

This surface is, of course, a quadric. We shall call it

the normal polar reciprocal gpgdric.

The condition that the normal polar reciprocal line
be perpendicular to the generator is, that the sum of the

products of corresponding direction cosines be zero.

34.
This condition is
(49) 1(1A’u vWR) + mpmv + 2V'A3P) = 0.
Solving we find

V2.1.) V=-__(_,,
£1 £1

 

Therefore pp each generator there exists two point; pp
opposite side; pf the direotrix and egpidistant from pp,

pplar reciprocals perpendicular‘pp he generator.

7. ran NORMAL QMADRIC

The equation of the normal to the right ccnoid at
the point (x,y,z) on a fixed generator by (33) is

(32 bis) 3 :v1+mt ) 7)=Vm-',€i' , 3 == t(vn +A1).

wherein the trihedral of reference is the same as that
used in deriving equation (38). The coordinates (3,5,3)
are functions of v and 1’ only. Therefore we may find

the locus of this point by eliminating these parameters.

35.
Ne find the equation of the locus to be
(so) Am1(3‘-7*) + ADM-195’) + A1(fl1£'1’2)-3 = o.

This locus is, of course, a quadric surface. We shall
call this quadric the ppppp; guadric. To determine the
nature of the normal quadric we will reduce the quadric
to standard form*. we find that for proper choice of
trihedral of reference the equation of the normal quadric

may be written in the form
(51) A (3FW-I)” (rt—’71) :42.

Therefore the normal quadric surfaoe'pp p hypgrbolic
pprabolcid.

The equation of the normal polar reciprocal quadric

by (48) is

umAar-w) - A’(m*s‘+,£‘7‘ + I)
(4Sbis)

+ [2 IMA’ + 2A1(m*—,L*)]£); - 243 = 0.

 

‘ Snyder and Sisam Analytic Geometry 21 S as New York
Henry Holt, 1914, p. . ' ’

36.

The equation of the normal quadric for a right ccnoid

may be determined from (50) to be of the form
(52) 77111161429 +A(m‘-,€‘)3’z — 3 =0.

It is easily seen that the condition that the two equations
(48) and (53) be identical is

Therefore the condition tppt the normal polar reciprocal
quadric and the normal guadric pp”; right ccnoid pg ident-
ical Gp that the ccnoid pp“; helicoid.

8. CRABLI'S CORRELATION

The tangent plane to the ccnoid at the point (x,y,z)
1.

(53) D13 - 1’2 + VA(.I—u) 2 0.

wherein the trihedral of reference is the one used in deriva

ing equation (1). we notice that as V' varies along the

87.

fixed generator equation (53) represents a pencil of
planes with the line

(54) ”73-1)? =0, I-u:0,.

as axis. For every V’ there is uniquely determined a
plane. Therefore the points of contact and the tangent
planes are in one to one correspondence.

Consider a pencil of planes (0.1.) + A (bx):0. The
cross ratio of the four planes determined by the param-
eters NAME» A, is the cross ratio of the parameters.‘
Therefore the cross ratio of the four tangent planes
determined by the values V. ,V,,v,, V, used in equation (53)
is

(55) (V, V, / V, v,)

The cross ratio of four points on a line is the cross
ratio of their distances from a fixed point on the line.

Since gives the distance of the point of contact from

 

R. I. linger |Gp Introduction to Proaective Geometry
New York, D.’C. Heaths n3 Company, 3 p. . '

28.

the direotrix, it is evident that the cross ratio of the

points of contact corresponding to the four planes is

(56) (V, V, /v, V,)

Therefore the tangent planes pp”; ccnoid along pointp‘pg

5 generator, are projectively related.yp the corresponding
points 2; contact. This is known as Chasle's Correlation.‘

9. rs: PARAnTRIC OSCULATIIG noun scarier Rv
The equations of the tangents to the curve

are of the form
(57> 3 = v1+vrtnz =Au+vm +t(A+vm‘), x... n...

This surface generated by the tangents as we allow the
point to move along a fixed generator is, of oourse,a

ruled surface,f1 . We shall show that the surface Fiv

v
cannot be a developable surface. A necessary and

 

elilcsynski, Geometr , p. 136.

29.

sufficient condition that a surface be developable is
DD"-D‘2- o

The values of D, D' and Du may be determined by the use
of (19). We find them to be

(53) 0:0, Hbz—A, D"=O.
Thereforoiheeamsirinmnsnm manna
developgble surface.

Any point on the tangent to the curve V: Const. at
the point (x,y,s) on a right ccnoid is found from (57) to .
be

(59) 5 = v(e +152), 7: v(m+)n't), 3 =u+t.

The locus of this point is also a surface (?v° To deter-
mine the quadric of Lie for this surface, we must find
the equations of the asymptotic curves. The coefficients
of the defining differential equation are the values of
(58). Therefore the asymptotic curves are the parametric

curves defined by the differential equation

A dvaH’:0.

30.

The curves v= Cutters straight lines, and generators of
the ruled surface RV . Therefore the quadric of Lie of the
surface is determined by the tangents to the curved asymptotes

1’ = Cons-t. . The equation of the tangent line to the curve
t = a...“ at the point (5,7,3) is

(60) f: (l+l't)(v+al), 7 =(57Hrn'tM/vtd) , if: out.

The coordinates (5,5,?) are functions of f and A only
since we are keeping V fixed. Therefore the locus of
the tangents as the point G,7,$) moves along the curve
v :. Const. may be found by eliminating the parameters. We
find this locus to be a quadric

(61) W31. -1'7,I +7275 —17 = 0.

wherein we have made the translation

The quadric (61) pp the quadric f is f t e ruled

 

surface RV .

The equation of the quadric of Lie for a right

31.

ccnoid by (39) is

(39bis) P53 — R7); +7713 "17 =0.

we notiee that the condition that the two quadrics (39)
and (61) be identical is

3:2." P=m'

Since

R : ,e’-IA', P: Inl—m-A'
24 2A

I

we readily see that the condition that the two quadrics
of Lie be identical is

A’ZO

Therefore the condition that the guadric p£,Lie of t e

right ccnoid,pp_identical pp_ he quadric f is f t s

ruled surface R, i; that the ccnoid p_e_ p helicoid.

 

 

33.

10. TH! FOCAL CONOID

The equation of the tangent to the curve v : Cami.

at the point (x,y,z) by (57) is
(57 b1.) 3 -.- Nib“) , )) =v(m+m't)+A(u+t) , .1; = u+t.

Ne wish to determine a curve v =V(u) such that the tangents
to the curves V=V(u) are the same as the tangents to the
curves v 2 Court. . ‘The curves V = Wu) so determined are
the curves which correspond to the developables of the
congruence formed by the tangents to the curvesv: Coast. .
As the point (x,y,s) moves along the curve V: V0.0
on the ccnoid the point (3,7,3) will describe a curve
V=V(U.) on the surface generated by (3,7,3) . We will
find the direction cosines of the tangents to the curves
described by the point (3,7,3) .in the manner just stated,
‘ at the point (3, 7,3) . , then determine the condition that
these direction cosines be proportional to the direction
cosines of the tangents to the curves V = Const. on the
ccnoid. When the two sets of direction cosines are pro-
portional, the two lines are identical since they both
pass through the point 6, 7,3) . Numbers proportional

to the direction cosines to the curves V =V(u) on the

33.

surface generated by (3,)? ,3) at that point are

I u I ' ' ‘
€§é€:= VOL 'f V/L t"f V.l‘t '+ (l-+.‘(3£I& I

(63) fl = A+v1n'+vm”t+(A+Vm‘)t +(m+7n’i‘)a47‘;.
i5 = I + It 0
dmt

Numbers proportional to the direction cosines of the

tangents to the curves v = Conston the ccnoid are

(63) v1' A+vm’, /.

The conditions that the direction cosines of (63) and (63)

be proportional are

Vi"? + «MM-2%. ‘-' 0.
(64)
VM"? + (m+m‘t)fi= 0.

We notice that the equations of (64) are homogeneous in

g and l , also in T and I . The conditions that they

have a common solution are
(65) t(A’+A’t) : 0 , jg (VA’+A§&) : 0.

Therefore the conditions that the direction cosines of the

34.

tangents to V=V(u) be proportional to the direction

cosines of the tangents to v: const, are
I
(66) tzo’ t=-.§T, V=CIV= e

Let us now examine the conditions (66). If
1‘ = O the focal point- on the tangent to V:- Cohst. at
the point (x,y,z) is the point itself. Therefore the
locus of the focal points, when i’ = O , is the original
ccnoid. If I” = - 4' the equation of the locus of the

‘3’
point is of the form

(6?)§:v(1-%‘)17:V()n-1"Z.-%-')fA(ur.£-;), I : u—fi; .

This equation may be simplified by the translation of

coordinates

1 _.
(68) I: i, V=7-A(Q-.§3),I=J‘u-
Equation (67) may now be written in the form

<69) 3‘: v(1-%),’i=v(m--’Z’,§§,f=-%;.

The points whose coordinates are given by (69) are, of

35.

course, the focal pcints‘ on the tangents to the curves
V 7- cons t. . The locus of (69) is a ruled surface, and
its equations are similar to equations (1) defining the
original ccnoid. Let us see if (69) is also the equation
of a ccnoid.

The conditions that a surface be a ccnoid are

1. The surface must be a ruled surface;
3. A11 generators must pass through a fixed straight line;
3. All generators most be parallel to a given plane.

Ie have previously stated that the locus of (69) is a
ruled surface. If we eliminate the parameter \I , we get

the straight line

u
(D

(70) (m —"2§')§ -(1- -’=—-"},)’)= o I +

 

”ale

This line always intersects the s—axis, since it is satis-
fied by 5:0 and 7:0 . As we allow .‘4 to varywe
get a family of lines passing through the s-axis. These
lines are the generators of the new surface. The gener-

ators 11s in the plane I+§= 0. Therefore they are

 

Iisenhart, Geometry, p. 398.

36.

parallel to the xy-plane. Hence the locus of the point
(3,7,3) is a conoid, which has the z-axis as direotrix,
and the xy-plane as directing plane. It is evident that
this oonoid is a right oonoid. Therefore the surface ‘22:
m by the focal points 29 the. tangents 33 the curves
v:const_i_g 2 313131 oonoid. We shall call this ccnoid the

focal oonoid.

A generator of the focal conoid is a curve u: tenet.

Its equations by (69) are
:0 ,I I
(sens) 3 = v(1-£—4),7=V(m-EA$), Iz-A,

wherein u, is fixed. The normal polar reciprocal quadric
for a right conoid by (48) is
41mA(§‘—*z’) - #0721311» 1‘7“!)
(48 bis)
+[zimA' +2A=(m‘-1*)157 —2A3 = o.

If we substitute (69) into' (48). we find that the generator
of the focal eonoid will lie on the normal polar reciprocal
quadric if and only if

A' = 0'.

37.

But it is evident on examination of (67) that the focal
ccnoid is identical to the original conoid when A’: o .
If

I 3 __
(71) A — A —- O

I

the generators of the focal ccnoid are tangent tg_the
normal polar reciprocal uadris,.;f the given ccnoid

;g_not‘g helicoid. The equation (71) may be integrated

and written in the form

 

 

(72) I = SMfCleHU) >77:co.s["<“¢‘““].

q+€g LL-t Ct

A necessary and sufficient condition that a sonoid
be a plane surface is that it be a developable surface.
We have previously seen that the condition that a surface

be developabls is
a
DD“—~ D’ :0.

The condition for the focal ccnoid is

1

AA”-.3A’ - I = o .
A?

as.
From (73) we find
A' ='- A3(u+c) .
If we substitute this value into (69) we find
(74) S =v[1-1’(u+c)], 7: vfm-m’m-ocfl , I = c.

Therefore if the focal ccnoid is a plane, it is a plane

parallel to the xyeplane. From (73) we find by repeated
integration that the right ccnoid, whose focal ccnoid is
a plane surface, is the ccnoid whose parametric equations

are given by (1), wherein

A=O,
(75)

 

7/2 =. €¥u§lfl9 V/T::?:'- ‘i‘ C, V’) —.(}L%f%)z ’

 

 

m z 1 W (/,..(u_Hm-c,(u+ct).

Ct

39.

11. THE OSCULATING CONOID

The equation of the osculating plans* of the

curves at the point (x,y,z) is

at?

4E;

(75) a u.
auf’ du" dm‘

 

 

The equation of the osculating plane at (x,y,z) of

V '5 c°“5t- on a right ccnoid is therefore

ll 3 ._. I ~—
(77) hrs-x 7—VA(3: «+2133)-o.
we notice that for any particular value of V' equation

(77) represents one member of the pencil 0‘ 1319319.B having

as its axis the line

(78) mac—17:0, I-u‘+24_3:o.

 

* Iisenhart, Geometr , p. 11.

40.

This line intersects the z-axis, since (78) is satisfied
by 5 2 O, 7':- O . As we let u. vary we get a family

of lines passing through the z—axis. These lines lie in
the plane 3 = u'fi; . Therefore the lines‘ are parallel
to the xy-plane. Hence we see that this locus satisfies
the conditions that a ruled surface be a oonoid. The

equations of the ccnoid are

(79) 3 = 1%, 7: W7, 3 = LL-nfiL; .

He shall call this ccnoid the osculating ccnoid, cor-
responding to the curves V = C0715 t .

The equations of the focal ccnoid of a right ccnoid

from (67) are

_ -_‘£Qy _ . a _ a
(67b1s)3-V(x ‘ZT" , ’1-v(m-TZ;4 , 3- 14—213., .
The equations of the osculating ccnoid are

a
q u _ u = __
(79) swat, 7..»72‘, :S u g. .
For any fixed value of q_ the generators of both of these
I
quadrics lie in the same plane, ,3: Q -%3 , parallel to

the xy-plane. Also both generators pass through the z-axis.

41.

Therefore the condition that the generators of the two
quadrics be identical is that their direction cosines be
proportional. The generators are therefore identieal if

and only if

8 .4: As! '- IIA’
( 0) m“ Arm __ M'Ar

 

He may readily show that (80) is an identity in L; .
Therefore the osculating ccnoid corresponding tg_the
curves v : const, and the focal ccnoid _q_f_ a right ccnoid

are identical.

13. QUADRIO COHOIDS

One system of differential equations defining the

ccnoid is the system (6), namely

(6bis) X“ = .VZéle—VAXV, X”: 0.

The solutions of (6) are of the form
XI = x0(ulv) I x; = xz(u; V); X; =XJ(U.,V), x9. 7- X,‘(H,V).

Let us choose any point I , with coordinates a” XIII” X30

43.

corresponding to some fixed value of u and V . On the
curve V = const choose another point X, . The coordinates
of the point 1, may be represented by a Taylor's series

of the form

X, = X + XuAu + Luna 1 471...-
2!

 

The point defined by

Li
A M
is any point on the secant line through I and X, , since
this expression is a linear combination of x and I, .
The limiting position of this secant line as X, approaches
1 along the curve V: Constis the tangent line at X .
The limit is

{:32 erg") = "u

Therefore the tangent to the curve v : Cons t at the point
x is determined by the points I and In.
Choose another point I on the curve v s can: in.

The coordinates of X,may be represented by the series

fist-0'

uuu 3!

 

)cl :. x +x,Au + xuud‘fl‘+x
.2!

43.

The point defined by

«2 ( )C1 4‘Jr -‘x:uJAl!) D
‘51::

lies in the plane determined by the tangent line of the
curve V =conct. at X,and the point I, . The limit of
of the above expression as I, approaches 1 along the
curve v: conct. is In“. Therefore the osculating plane
to the curve V: const at X is determined by the points
1) Xe, and X'uu. °.

Suppose that the parametric curves u. = const and
v :: can: t. are the asymptotic curves on the ccnoid. Then
the osculating planes to the curves V : const. are identical
to the tangent planes at the same point. Any point in the
tangent plans may be expressed as a linear combination of
the points I, X“, Xv. Therefore when the asymptotic curves
are parametric the coordinates of the point X satisfy
differential equations of the form

}: : °( 1:“ ‘f 63.x‘l 7' f).r )

MK

X” = YXu-fSXy-rgx,

 

* E. P. Lane Projective Differential Geometry g£_curves
and Surfaces: cago, hIcago Univers y ress, 1535, p.11.

44.

Therefore when the parametric curves are the asymptotic
curves, the coefficient of X,” in (6) must be zero. Hence

the condition for the ccnoid is

Therefore only when the conoid is a helicoid are the gen-

conetant distance from the directri; asymptotic curves.

Now let us make the following transformation of

the curvilinear coordinates u and V :
(81) u = cp(u,v) I V = Lp(u,v) .

The differential equations (3) under this transformation

assume the form

Earn? +B0Xu + EXV'I' Jx.

X
|

521117 +Eixu + szv“ 3:1.

In order that the new parametric curves E = const. and

V = echct'. be asymptotic curves , we must have

45.

(83) i=0, 51:0.

Suppose that conditions (83) are satisfied. Then the
conditions that the asymptotic curves be straight lines
are

(84) C,1: C) , I), = (3 .

When conditions (83) and (84) are satisfied the surface
is such that the parametric curves are the asymptotic
curves and are straight lines. The surface under these

conditions Lg‘g quadric, since quadric surfaces are the
only doubly ruled surfaces.

In particular let us make the transformation
Ezu ’VT—‘V‘A.

We obtain the system

 

‘- y u )2‘_ ll _
(as) x“ = “14.114 man“ x.W .0.
24A
We observe that in (85):,9- O and 3,: O for all values
of the parameters, Therefore £133 curves U. 2“ “gt,2_1:£

straight lines and, 9; course, asymptotic curves.

46.

If we equate to zero the coefficient of XV in the
first of (85), we obtain the condition that the ccnoid

be a quadric, namely
(as) 715‘ +34"—2AA"=0.

Equation (86) may be considered as a third order differen-
tial equation in.,i . If we integrate this differential

equation, we obtain

e -‘
cues“ 1' 99-. c, + if

Let a! and G be the angles which the generator makes
with the x-axis and y-axis respectively. It follows at
once that Sin"! = 6. We may now write (87) in the form

 

- du+b
(88) tonne. c +I

wherein

a -.- c.) b -.- c,c,+7c, ; c : c,c,) dsc.c,c,+7 .

Of course, d,b,c , and d are not independent. The relation

existing between them is

d = dbc -4(d‘—c‘) .

4?.

le may rewrite (88) in the form

-la
(89) = I (C-1)Ll +d : [La u+r .
5 4 a (c+a)u-+H+b 4 qu-rS

wherein

 

Psc-a, q: c+a,r=d-b,$=d+b.
The relation existing between the constants of (89) is

s = 2r(P‘~7'i-'32F9 .
P"7‘+"

We may write (89) in the form

(so) e“: am, a = 12:- 5,
q u.-+ S 2

Therefore the ccnoid will 22”; guadric if and only if
the equations (90) hold.

We have seen that if the asymptotic parameters
are parametric, the homogeneous coordinates of a point

DC on a surface are solutions of differential equations

48.

of the form

If :: CKL.X'u 7' (3 va 7' f>.X',

MIL
(91)

X

VV

rxu+sxv+qx.

And furthermore if we make the transformation of curvi-

linear coordinates
U=u, V=v‘A,

that the defining differential equations (3) assumes

the form

(85 bis) X“ = v‘('IA" + 3A”-2AA“)X., , kw = 0.
2 ti
Therefore '11 = const and V = CONS t. are the asymptotic
curves.
Now let us find the condition that the curved
asymptotios belong to linear complexes. (223,22312g

asymptotios V: Const. belong Lo linear complexes if
and only if "

 

‘O. T. Sullivan, Pro erties of Ruled Surfaces, Whose
As m totio Curves Belon to inear Com Taxes Trans—
ac¥ions ofégmsricafi‘flitfieiit ca ooIegy, V l. 15,
191 , p. l .

49.

 

(92) 3%;5 I»; V2(5A4+34’:— 2AA”) = 0.

24

we may readily verify that equation (93) is satisfied
identically for all values of Ll and V . Therefore

pp the ccnoid the curved asymptotios V44=const b81095

1:3 linear complexe .
Since, from a projective point of view, a ccnoid

is a ruled surface belonging to a linear congruence, we
may state the more general theorem: the curved asymptotios

pp any ruled surface which belong 32 3 linear copgruence

belong 12 linear complexes.

BIBLIOGRAPHY

Bell, R. J. T., oordinate Geometr Of Three Dimensions,
London, Macm an and ompany,.l9

Dowling L. W. Projective Geometry New York IcGraw-
Hill Booi ompany, 9 . ’ ’

Eisenhart, L. P., Differential Geometry pf Curves and
Surfaces, New YorE, Ginn and Company, 1909.

Pcrsyth, A. R., Differential Geometr p; Curves and
Surfaces, Cambridge,_5am5ridge University Press, 1912.

Grove, 7. 0., Metric Differential Geometr ‘2; Curves and
Surfaces, Notes prepared for use n Michigan State
GOIISEO.

Grove, V. G.,,Qp Canonical Forms of Differential B nations,
Bulletin of the American Mathematical Society, Iug.,1930.

Lane, I. P., Pro ective Differential Geometr p£_Curves and
Surfaces, Chicago, Chicago finiversity Eressj'l935.
Snyder V. and Sisam O. H. Analytic Geometry pf Space
Dew York, Henry Holt an Company, . ’
Sullivan, 0. T., Properties of Ruled Surfaces Whose
Asymptotic Curves Belong to Linear Complexes, Trans-
actions of the American nathematical Society, 701.16.

Weatherburn, C. 1., Differential Geometr g£_Three Dimensions,
Cambridge, Cambridge finiversity Press, 19877'

Wilczynski, I. J., Fojective Differential Geometry pf
Curves and Ru e Sur aces, Liepzig, B. . eubner, 1906.

Winger, R. I., An Introduction pp_Pro ective Geometr ,
New York,-D. 5. Heath and Company, 3.

. - ‘ .vv‘;v’. It'
\ '1", h" (#-'{.,“}.‘~‘..‘) I: '
("1' _.;, '.« ‘ ‘.
t" o

. w
I.... ‘x‘o

v

. "I. I .
w . _ .'l“~
o3}; .

i',
gent“?

-"‘.w.

‘ '1‘ w"
.K‘,‘ -2 'v
I I

A .

4-.

. ‘ s
b',.s It
I (“"lJ

‘ .

' _A
‘1’)“
R'

‘.

‘ikfl'ii s I.
I}.

n

. ' ‘ fie ' .

. . - - I o" . . ’ ‘ ' . .‘I‘ ..
'1‘ -‘-. v.1. ..~,. .1" 1’; 4,, “Y (43}
. . \' ‘w ‘ 4 ~. ' " ,
:(1‘. - *v 5’. 31.7w? x‘ «35$! («51: .- (is
., h I - ' 0; A . ' w > ‘
_ 51w ”VJ-'15s. 343.35. lei-("if d

e"
i L.
4‘.

 

 

ut((me((menuluuyufllmm