TRIHEDRAL CURVES
by
JERALD L/VERE ERIWCSEN
A TÍIF3I3
Subuítted to
OREGON STAT1 COLLEGE
In parti.ai fulfillment of
the requirements for the
degree of
MASTER 0F ARTS
June 191i9
APPROVED:
Professor of Mathematics
In Charge of Major
Chairman of School Graduate Committee
Dean of graduate School
L
AC}OJDG}ENT
The author
to
L)r. Howard
ives,
wishes to
whose
express his deep appreciation
careful guidance and inspiration
have been invaluable in the preparation of
to tharion Ericksen,
who
has made
this
work
this thesis,
possible.
and
TA3L
0? CONTENTS
Page
1..
2
TRoitTCTION
N-TU11DRAL
1, 2
C
JV
2.01
2.02
Lefinition.
Theorei relating arc lengths of a given curve
and ita N-triedra1.
2.03 Tabular data for -trihedrals.
2.014 - 2.07
General t4eoroms on -trih@drals.
2.08 Tabular data for involutea.
2.09 - 2.13 Theororas relating N-trinedrale and
involutes.
2.114
Examples.
3.
3
3
b
S
6
6, 7
B-TRflIEDItAL CURVES
3.01
3.02
3 .03
3 Ob
3 0
3.06
3.13
¿4.
3
Definition.
Theorem relating the arc length of a aven
curve and its 3-trihedrals .
Lennia concerning curves rith parallel principal norLials.
Theoret on plane curves and
3-trihedrals
Tabz1ar data for 3-trihedrals .
- 3.12 General theorems on F3-trihodrals.
Uefinition of 3ertrand curves.
their
3.114
Criterion for 3ertrand curves.
3.1
3.16
Theorem on 3-trihedrals of Bertrand curves
xamplos.
B
8
8
S
9
9, 10
10
10
11
II, 12
B-CURV3
Definition.
Theorem relating the arc length of a given curve
14.01
14.02
arid
its
3-trihedrals
¿i.11
Tabular data for £$-curvea.
- 14.10 General theorems on i3-ourves.
- 14.11i Theorems relating Bertrand curves and
14.15
-
14.03
14.014
3-curws.
14.17
14.18
14.19
14.20
Theorems re1aU.n toral and osculating
planes f a given curve and its -curvee
The enora1 equat;on of all curves of constant
curvature
The oneral equation of aB. tiatad Bertrand
curve3 wiich are not circular helices.
Discussion of the result of 14.18.
Theorem on lertrarid curves.
¿
13
13
13
13
15 - 17
.16
17
13
19
20
20
Pige
1.21
I.22 .
The general equabion of all curves of constant
torsion.
¿4.21k
i3IBLIOGRAPH!
i';aDics.
21, 22
22 - 2h.
2
TRIHEDRAL CURVES
1. INTRODUCTION
In the following we propose to study some space curves related
to a given space curve.
shall enploy vector analysis, using a
e
right-handed rectangular cartesian coordinate system.
Since a vector
U will be considered as an ordered triple of numbers, we can,
ing
convenience, let
ori
shall, then,
IJ
depend-
define either a point or a direction.
often speak of the point
U,
We
or the direction U, thus
identifying the point or the direction with its vector representation.
Capital letters will be consistently used to distinguish vectors from
scalars.
The following notation will be
more than one curve occurs
in
throughout the paper.
Then
a discussion, subscripts will be used to
denote to which curve the symbols refer.
1.
The vector U, having cotaponents u1, u2, u3, will be written
as
tJ
[u1, u2, u311
2.
The scalar product of vectors U and V:
UV
3.
The vector product of vectors U and V:
lu, V)
t.
The determinant of three vectors U, V, W:
,
Arc length:
V,
r4
s
6.
Torsion: t
7
Curvature
8.
Screw curvature: m
9.
Unit tangent: T
:
U,
k
(
always taken positive)
(k2
-,.
t2)2
(always
taken positive)
2
10.
Unit princtpal normal:
11.
Unit binormal:
12.
Darboux
13.
Prines will in:iicate differentiation with re5;ect to arc
N
13
vcthr:
-
D
tT
k3
length.
From
that, if
X
i
-
the study of twisted curves ne ha the familiar result
is the unit tangent
T1
and s
the arc
lnth
of the curve
then
X1
f0T1
xl
da1.
This suggests two related curvos íivcn by
ç.'_
X2 =
j
ti
and
X3 =
j
o
where !Ii and
are
da1,
o
tnit principal rorria1
and
unit bini.inal of th'
orignai curve. The fact that all three curves are nasociated with the
fundamental trihectrl f the íi v'n curvo ugest the nane trhedral
çurves.
Thus we shall have the T-trihedral,
ocii
atad
ol:
ti
or riven, cuì've,
N-trihedral curve, and the ansociatei 3-trihndral
georotry of these
latter two curves,
curve to be discuosed Latcr,
s embodied
ríular
as well
the ass-
curve.
as an'ther
in this paoar.
Th
Sorno
relatd
Rh1l as-
surie
that our iverì curve is
and,
moeove', 5n the regions under consideration, is free of singu-
lar pOj!)tS
(3,
p. 27).
inalytic space curve (3,
. 19)
2.01.
CURVE
N-TItIH:DRAL
2.
DìF1NITION.
Given the curve
3
(s1), then the
=
curve
X2
will be called the
ds1
-trihech'al curve,
or, more simoly,
the Li-trilledral
of the given curve.
2.02.
THFOREM.
nd its T-tìhedra1 have the same arc
A curve
length.
For
3,
1
ds
2
2.03.
a
TÂ3ULAR DATA FOR N-TRIHEDRPL CURVE.
N1
(1) T2
(2)
ds
il.Nl
=
X
2
f
N1,
/
l
(k
T1 -
/
- t1 T1
)
(
/ m1
D1 / m
N2
1,
(14)
132, T21
:r1t/ n1
2
2'
NI
(2,
93 or
.
(-k1 T1 - t1
n. 93)
:3
T1 - t
D1 -
mi
)
/
?
2
(
'
- m1
/ m1 + (i / m
ti Ti + (-kf
ici
k2
(2,
p. 30)
::
()
/m.
t1 + t
)'
(-k1
k1) U1 /
-
t1 B1)(
14
For
N2 (6) t2
Thus k2 N
.-Ix,
X, X'
_(ii
-k1
/
T1-
t1
/ m1
or k2
mr
(xx)
B1,-k
T1 -
.
-(-k'
/
4
i
(t1 / k1)'
Tho N-trhedra1 is a plane curve if and only
T1IrOREM.
2.014.
)2
/
(i
k1 t1) /
t,
t
if the given curve is a helix,
For, if the given curve ia a helix, t1 / k1 is constant (3, p.
Thon, froni 2.03 (6), t2
0,
0.
Conversely, if t2
2).
0. then (t1 / Ic1)'
or t1 / k1 is constant, and the given curve is a helix.
n particular it follows that if the given curvo is a riane
curve, then SO
£3.180
Is its 14-trihedral.
The i-trihedra1 is a circle if the given curve
T1rDrt:M.
2. OS.
a circular helix.
.:1
2.014
the N-trihedraï is a piane curve.
that its curvature is constant.
constant if k
and t
N-trihedral
!
uy 2.33
curve is a helix,
!
D2
T1E0!fli.
only
given curve is a helix.
-(k1 I ¡a)2 (t1 I k1)'
(t1 I k1)'
D2 : D1, then (t1 I k1 )
2.07.
which is
The Liarboux vectors of the given curve and its
equal
(14),
iut k2
are both constant.
ThE0RM.
2.06.
It remains to show
'
0,
and D2
D.
N1
If the given
Conversely, if
0, and the given curvo is a helix.
The N-trihedral of tre U-trihedral is never the
s
given curve.
-(t1
k1
!
For,
if
/ ri.)
B1.
-r.
The
fN0ds2,
X
For
thenTN2/n,:-(k1/)T1
to be equal to
T
±ater is
T1 we must have
impossible since k])
relate
The next few theorens
O,
with the involutes of the given curve
It will be convenient to re-
out subscripts w111 refer to an involute
Tt RULAR
and
of a given curve
cord here the tabular data for involutes (ii, p. 352).
2.08.
O
0.
xn
the N-trihedral
t
oÍ'
Notation wtth-
the given curve.
DATA FOR INVOLUT!S.
(i)
(2)
e
(3)ds(Ii)
-siO
±1 according as e
(c-s1)
Ek,
ds1
T
(s)
(6)
(7)
k:
m,/k1
(8)
t:
(k1 /
2.09.
hedral
THFOR1M.
(c-si)
2
-
The ratio of curvature to
torsion for the N-tri-
.! involute of the given curve are nwne rically equal.
For, by 2.08 and 2.03,
k
2.10.
2
(t1 / k1)' / k1 (c
/t:
THEORFL
I
we
have
)2
k1
The
that
(t1
I
£k2 I
k1' :
osciilatng, rectifying,
N-trihedral and of an
t2.
and normal planes
involute of th given curve are
tarallel,
2.11.
D
:
/ m1
N
N
¡N, dN
vectors of the N-trihedral and of
f the gìven curve are parallel.
an involute
For
The Dprboux
TIIFORFM.
/
2.12.
dsj
/
(
LE14MA.
to be a helix
G
dN
1I2
)
/d -
EN
=
(ds
13
N,
I
/
(ds
1
/ d)
ds). Therefore
D.
6
necessary and sufficient condition
A
is that its
the direcU ori of the
Darboux vector have a
ads
L'or a
curve
fixed direction, name-
of the heUx.
See (2, p. 106).
2.13.
TE0fl. If either
the N-trihedral or an involute of
the given curve is a helix, then so is the other, and their axes re
parallel.
This follows fron 2.11 and 2.12.
We
now
conclude this section of the paper by
11ustrating
some
of the foregoing theory with an example.
2.1L.
ExAL:'LE.
we
cos 2u, cos u,
[
xl
For our given curve
u
-
find
:
si :
T1 :
:
ds1
N1
u du
(1
-
cos u)
[_coS u,
sr
;
-1, sin
[-con u, 1, sin u
[sin u, 0, cos u]
let us choose the
sin 2u]
,
o
<.u<rr.
helix
7
F
D1
cScU, O]
CSC U
OSC U
t1 =
m1
eec u
We note that the helix makes a constant anglo with the direction
[o, i, o.]
The N-trihedra]. 18 given by
X
fN,sinudu,
V
where
U:
O'u
(f
cos
-
<rr,
This turns out to be
X2 :
sin 2u ..TT, O, sin2 uJ
Lu -
whence
(
dI2 / du
jsinu(du/ds1)dscIs.
2
2
sin u cos uJ
u, o,
verifying theorem 2.02.
:
We also have
T2 : Cain U
O
C08
]
B2
[o,
N2 :
1O8
D2
[O, 2 cao u, Q.]
i,
U,
O.]: D1 / m1
O,
k2 :
t2 : O
UJ
..Sifl
N
/
: D1
:
(k1 f
(t1
I
1c1)'
These results verify 2.03, 2.O1, and 2.06.
,
3-TRIkUiJtAL CUI V
3,
FIUiTI)N.
3.01.
Given the curva
then the curvo
X1
X3.iiili be
of the
caUed
the :3-trihedral curve, or, «»re sirip1y, the 3-tr5hedral,
'iven curve.
ffOR'L
3.02e
A cuie and Its B-triheda1 have the
sarip
are
For
83
du
L!Ut.
3,D3,
2
(2)
related that
(131131)
curves, X1
:
s1
()
X
d1
a.
and X
: X
(i) D
:
(3j)
Dj zfld
r
If nrimo
ilj
m
L
:
:
indicate differentiation With resT)ect to
s,
then
4, whence
D
-
!'i,
N
:
(
±Nj, ±I
i
f
Nj,
N'j
I
=
an
mj : ()1.J1)
3.Oh.
?fl}0RPM.
= (n.D)'!
If the given curve i
hodral curve is a straight line pazallel
For, if the Eiven curve is a
vector, and we have
X3 :
a
th
1iìe curve then B
3
j
1ane curve, the D-tri.-
d81 :
:
L
lo a constant
ALAR
3.O.
%-1RIDRAL CURVE.
FOR
DATA
(1) T3
C2 )
i ACCr) r'd?&:
:
(3) 33 =
I9
9
t1
.0
(9.9Y
II
l :
1,
(Li)
fl3
:
()
Ti
:
(,
:
'r3
isna
(ber
(6) k3 :
=
-
(7) t3
(%
X;.
(3)
3.06.
:or,
3.0?.
v.
tora3 ot
3.09.
911
(x;)
/
t1 k1 T1
i
t
/
(by 1orna 3.'3)
inc
taio,
pi
ui1
iirz4, and bínorc1 of
tra1,
para1ìc1, rftn*otive1r, j
!HE01UM.
A
- aur
and ita w_L
U-t,rieura1
-t
-
von
rh
arv
if the
t)XEEtOfl
çntnt
trile(;.r4
T1Oa7M.
--
If
ì! C3tAflt
Driucipal
cwvve.
taigent of tho
TliFOLìt.
-tr2iotri
a
Cij.:'3j)
k1.
:
Tof*k.
_!.tr*edr4
,O3)
N1, t
= - lï%
T-e.
E
torainn
-
-
equal ______
axkoux
-av ____
haa constnnt curvaturc the
oven
as
Ons tarÂt
i
alway
___
cui'Y
rvatire.
the -3-trihedra1
-f -
-
-
L-
atiim.
3.1).
TP'Ot11.
iL tht
gtven curve i
a
hoUx
s.
alao i
the
lo
_trihecr'ai
if the
}3-trihcra1.
ven curve is a
lu either case the
For k3 / t3
- ct1 / k1.
fron theorem 3.07
3.11.
nd
1erxuia
c
to
rcular helix so
a10
is the
helices have parallel axe5.
That the axes are parallel follows
2.12.
TIIEORIT71.
A curve and its
THEOREM.
The J-trihedral of the
3-trihedra1 have eqia1 screw
curvatures.
3.12.
3-trihedral is the given
curve or its reflection in the orzin according a
or
:ven curve is nepative
:3
,)
The final
ositive.
(\sI
r-'
ForY--
ds__EJT
o
bi
-'
therm
ds
1
--eX.
i
of this section Rnd several of those of the
next section refer to i3ertrand curves
For crnvcnionce of reference
we here give a defintion &id a criterion for these
JEFINITION.
3.13.
the torsion of the
A
is
curve
)ertrand
curves.
a curve whose principal
normals are the principal notais of another curve.
CRITFRIOW.
3.lL.
curve
2:
:
r
Q,
i
twisted
nit,
(n :
A
necesary
and
sufficient condition
for a
3errand curve is that there exist constants
,
l, ±2,
.
.) sn.cì
that
rksinw-s.rtcoswsinw,
r being the constant (listnce betreen
tranci curve and its mate,
and w being th
corresponding osculating p1ans of a
(cf., 3,
a:!'t.
23.)
corresponding points of a
T3er..
constnnt angle between
9ertrarìd curve and
mate.
11
3.15.
TH1OR'2L.
If a given twisted curve is
osculating planes are not perpendicular to the corresprnding
whose
osculating pianes of its mate
2,
Bertrand curve
.
.
n = O,
1,
then the B-trihedrai is also a twisted 'ertrand curve.
.),
By 3.1I
there exist constants r
(6) arid
w
-rt3 sin
-t,
k
(7),
t1
cos
k
r E
.p
nd w such that
rO, wniT.
sinw,
rk1sinwirt1 cos
3y 3.u5,
(2n + i) Tr/2,
('y
Thercfore
k3.
- sin
w
w.
Let
T/2...w),q_rtan.
Then
sinfL
qk3
and, aga.n by
As
3.1L, the
in 5ectiorl 2,
used in 2.1L.
X3 :
FXAMPLE.
The
, q
O, .n.
¿
n
Tr
3-trihedral is also a twisted :3ertrand curve.
shall conclude this section
we
soie oí the foregoing theory
3.16.
sinfl.
cosA.
i. qt3
with
by
illustrating
an Exariple.
curve
For our given
choose the
we
saine
curve
3-trihedra1 is found to be
ds1 :
J
J;;4ß1
'
5
u du,
where
u :
r2
4
(
-
i'
Ou
<
iT
This turne out to be
X3 :
1i-(cos
2
u
-1), -(1
4 cos u),
(u -
?:
whence
ax3 I du - [._sin u
cs
u,
sir
it,
sin2
11J
,
sin 2
u -ii )J
12
and
sin u (du /
Thus s3
s1,
verifyng
dz3i)
Uieoreii
d31
3.2.
e also have
T3
1 [-cos
43
4
u, i, Sin
uJ
[cos U, 1, -sin uJ
111
_Ti
u]
: [sin u, u, co
scu,
k3
t
!3
CSC
1
: t1
--: cec'i--.k,
?
(
csc u :
?fl
These results verify theororis 3.06 through 3.11.
IV
LJ
'
13s
u1.fl?I1).
1.oi.
Th
G+
xl 038
ithere
E)
curve.
%rir
if
8:
enet&t
L.O2.
angle,
-trThedrai
g.v3n curve,
th'
f the given
3-curve
nd if
e
'72 the
f the iiven curve.
-
ad
A curve
?1L')t $.
e,
A3 am
L1 be called a
0 the Ji-curve in
i-curve le the
13
-
ita i-eurv
sìe
nave the
arç
;Length.
Aloi
e3
:
e
:
. U1 e n
J"ßi1') co*2e
:
e)
(
eoe
i
e.
sin
¿(I.I) enec3Ie 4
.
sin2e]
=
$1.
UL17.
TA3ULAr SArA FO1
O
(1) Tj3
:
C°
(2)
:
accordilig aa
(::3,
(3)
i:
=
:
:
(1) *13
(t;) ¡1j3
(6) k13
T1
(1
il
'
ain)
Ø
e
I
.
I
(by lemria
(Tj3.Tj3)V
o
;
e t1
:
SInO,
(Ic1
pn
9
° -
:
'
G
siu
eO
03$
O
Xl3X3»
]3I t
l
4 t1
nut
t
a
am
cose.
,
t
dne)
i1ß
e)
e.
.03)
E (k1 oca
e
. t1
airi
9)
e)
de1
11
(7) t13
= t1 co
-
=t1
For
13
sin
CO
k
-
e
13
(3)
The
O
k1
-
m1 (by 1emia 3.)3)
13
foUo1ng
six theorems are irmnediate consequences of the
tábuiar data for a-curves.
14.OLi.
THEORIr.
The
rincipJ. norr.ai o1
i3-cure is parallel
to the principal normal of the given curve.
T}I1OR.
The rectifying plane of
rectifying plane of the
.2
1.o6.
THiOIEM.
v 3-curve is 'arallel
iven curve.
The Larboax vector of
1-curve is ìaraliel
! Darboux vector of the g.ven curve.
4.O7.
also
T1LOREM.
ven curve
!
circular helix, so
evexy 3-curve a circular helix.
14.03.
ti-curve
ThE
helix.
if the given curve is a helix, so also is evexy
The axes of the gLven
urve and its r!-curves are
taral1e1.
rhis follows írom 2.12 and
Li.O9.
THEORI.
1c,
of L.O3.
;-curve we have
6t13 cos e
t1,
cose - e t3 $.dIG
k.
(i) k13 SLfl
(2)
Ö
For
()
q.
1.io.
THELM.
If sorne B-curve is a circular helix, then every
13-curve, includinE the given curve, is a circular helix.
This follows from 1.O7 and b.09.
1.11.
1vey
ThLOFLF&.
tw:i.sted
Bertrand curve is a 13-curve of
curve of constant curvature.
soiiie
Let C be a given twisted 3ertrand curve of curvature k and torBy 3.1I there exist constants r
sion t.
±2,
.
.)
.
O,
w
nw,
(n
O,
j.,
such that
k sin w + t ces w : (i / r) sin w.
Let
E:
l according as sin
w: T1/2.o,
Then
w
O,
and choose
G and
(i)
k1 auch that
(l/r) sinw
la a positive constant and (1) becomes
k.
kcose- etsine:k1.
inau1y, choose t1 such that
k sin
e
+
tS
t ces
e :
t1.
ßy the fundaental theoreri of apace curves (3, p. b6) there exists
a curve 01 having
k.
and t1 for curvature
C is a B-curve of curve C1 .
arid
torsion, and by t.O8
Since C1 has constant curvature, the
theoreii is established.
)..l2.
SOESO
TirEOR121.
Every twisted Bertrand curve Is a 3-curve of
curve of constant torsion.
Let C be a given twisted Bertrand curve of curvat,ure k and torsion t.
2,
.
By 3.11k there
.
.)
such that
cxit
constants r ¿ O, r
nir,
(n
O, ±1,
ksinw.tco5w:(1/r)Sinw.
(1)
Let
E
±1 acco'ding as k cas w - t sin
and choose
ê and
(1/
Then
t.1
is a constant and (i)
k sin
e
r)
w
O,
t1 such that
5mw:
t1.
becons
t cos
e =
t1,
or
k
t cos
sixi
:
t1.
Finally, choose k1 such that
k
cs e
-
t
me
k1.
- t Sin
S -
Observe that
r
ê
k COS
k1
(k co
w
- t sin w)
?
O.
the fundanental theorem of space curves (3, p. 16) there exists a
curve Cj having k1 and t1 for curvature and torsion, and by 14.08 C is
a S3-curve of curre C1.
Since C1 has constant torsion, the theorem is
established.
14
.13
ThThORF
.
ver;
B-curve of a curve of ooxstaxit non-zero
curvature, except those for Which
(2n
..
1)
11/ 2, (n
O,
±,
±2,..)
is a tvisted Bertrand curve.
'or we have,
'
(2) of 1.O9,
k,cos6- E't13s;inê=ek1.
Suppose k1 is constant and choose
e=
Then (1) becomes
(-
17/2),
w and r aucx that
(l/r) uinwic1.
(i)
17
k13 sin w
clearly r
O and w
(i / r) sin w.
t13 cos w
nr,
(n
0,
±1, ±2,
.
Therefore, by
.).
.
3.1I, the B-curve is a twisted 3ertrand curve.
!.iLi.
THQREM.
a-curve of a curve of constant non-zero
Yvery
torsion, except those for which
e
nil,
0,
(n
±1,
2,
.
.
.)
a
twisted Bertrand curve.
For we have, by (1) of 14.09,
o
k13 sin
e
& t13 cos
(i)
t1.
Suppose t1 is constant and choose w and r such that
e:w,
(1/r)sinw:t1.
Then (1) becomes
k13
clearly r
3.114,
O and
w ¿
nil,
(n
(j.
±2,
0, !i,
.
/ r) sin w.
.
Therefore, by
.).
the 3-curve is a tiisted f3ertrand curve.
h.if.
TtikOItEM.
The normal pianes of the given curve and its
B-curve, corresponding to the angle
angle
w:
sin w + t13 cos
e
,
intersect in the constant
e
For
T13.T1
14.16.
THU)RM.
(Tj.
cos
e
+
Bj.
e
e).
Tj.
cos
e
The osculating planes of the given curve and
B-curve, corresponding to the angle
angle
sin
O
,
intersect in the constant
18
Fo r
B133L
TH'ORF!i.
IL.17.
where
V
is
Rfl
'. =
e.
cos
The genera1
eçuation of 11 curves of c'nstant
/ c)fv
r'
is
non-zero curvature c
(1
X
e).
- 'r1
cos
(i
arbitrary unit
dr,
(i)
function of the paremeter
vector
Let us find the curvature of curve
(i).
VIe
r.
have
i
/ C
"i
dr
'
(!.! \2
y
ardr1'
'
wher1cE
cls
(i/c)
(ciX dX
(dV d'fl
.
Therefore
T
V
Thus
dV
N
On
dr
the other hand
non-zero curvature e.
and
kN
dV
ídV.dvr
- dr
drI
T'
dV dV
f _._._j
' ctr dr,'
and
i
k =
For suppose k
e. Let
having the constant
T
!
_
dr da
dr
(
ï
dr tir/
whence
.
dr
o)
= (i I
dr dr/
ì(I
dr =
T
.
dr
V, a
unit vector
Then
!
T' =
C
C.
(i) includes 1i carves
function of a parameter r.
oN
--
(1
/
c)
V
(!.!)1.
;
r
dr
\ dr dr!
d
!
'
19
Li.13.
______
The
Tk
gnr31
equation of
twieted
a].].
ertrand
not circular hlices is
'thich arc
(1/c)caeJ(.!.L)dr
¿ (1 / e) sin
01W,
dV / dr
dr,
where V is an arbitrari unit vector function of r,
fly
¿.0
if
e
(2n
.
1,
c
arc arbitrary cozìst,anta
.
every tvist'xì ;3ertrd curve is a i-curve of
of constant curvature,
(1), Li.17.
1, ±2,
/2, (n = O,
(2n + i)
ni
(1)
3y
With this
TrI2,
1)
o, ±i, 22,
(
curve
n mind let us find the fl-curves of
thco ì-curve will
.13
sie
.
ail be twisted
Jertrand curves
.
e have
=
X,
X"-c!
1- dr
B1
(dVdV)
dr
'
/ ()Ç.Ç)
¿Xi,
= (1 1 c)
V, e
(
I
V,dV/dr
X3 :
(de1 I dr) dr
j
1[(v, dV i dr
(i/c)
f(r,
I(.)4:]tI1
dv/dr
onorul equat on of the
(
n : O, ±1, t2,
.
.
.
),
onht 9-curvs.
(..4)jdr
dr.
(2)
Curvo (2), being the I-tr-ihedral of a curve o
turs C, is a curve of conatant torsion -e.
I a)
Ye
hen
ncyw
constant curva-
obtain (1) as the
(2n
i)
TT/2,
the 3-ourve reduces to a curve of constant
20
non-zero torsion.
It will thus be a
has constant curvature,
ertrand curve only if it also
as is seen by referring to 3.lh..
Jut in this
case it iS a circular helix.
t.l9.
THE SIGÏFICANCE OF c AiW
eL
(1) OF Li.18.
By ¿.O9, the linear relation between the curvature aad torsion
of (1) of
IL.13
is
kcose-tsinec
(1)
or
k -
t tan
9
E e / cos
a
(2)
letting d be the constant distance between a Jertrand curve and its
w
mate, and letting
be the cc,ustant angle between the
planes of the Bertrand curve and its mate, we have
(3,
osculating
p.
(1/c) cose,
d
(3)
cotw:- tane.
otice
we take
tiiat i
e
0,
(14)
then, from
(1),
k is constant,
and the Bertrand curve reduces to one haviri
constant curvature.
From (3) and
i/
14.20.
helix 2
ture C
(14)
rre
then have d - i /
TllEORT.
e,
w
2.
twisted 3ertrand curve is either a circular
J.
!. linearly dependent on two curves, one of constant curva!
2.
constant torsion -e, the second curve being the
3-trihedral of the first.
The nertrand curve is a 3-curve of the
first.
For, from (1) of
lar helix, is linearly
.l8,
a twisted Bertrand curve, not a circu-
dependent
on the two curves
21
(1/c)
I'
1v
dV dV\2
dr,
a curve of constant curvature c, and
/
(i
a curve of cousta.nt
c)
/ dr
V, dV
J
dr,
it was
torsion -e. In the proof of h.i3
that the second curve is the -trihedra1 of the first,
is
r3ertrand curve
h.21.
a .3-curve of the
non-zero torsion e
first.
is
X-(l/c)
is
)T,dV/drI
dr,
(1)
arbitrary unit vector finction of the parameter r,
where
V
is
arbitrary non-zero constant.
If e <O the curve X, being the .3-trihedral of
an
an
staat curvature -e, is
-X
is
a curve of constant
a curve of constant
torsion.
torsion
But the
t, is
/ (x.x!),
t
')n
X
-
by -X
1x
x
results only in
the other hand (i )
iion-zero torsion e.
tion of r.
t
For suppose
t
If
e
3'
=!dr
ds
dr
k
si
-- dr i drdr/
e. Set
dr dr!
_];
and
dr
X
then
are
changes.
curves haying the constant
B
V, a
unit vector func-
\dr dr! ds'
wnence
N
- O,
-X and
Then
cN
e
given by
an odd number of
icludes all
and e
a curve of con-
torsion of
the same except for sign since torsion,
and rep1ac:ug
that the
eneral equation of all curves of constant
The
THhOREM.
and
shown
(i
/ e) urdr!
22
Therefore
T(N,
:(I
j
(L!),
dr dr
.; V, dV / dr
J
X
T (ds / dr) dr
i
IV, dV / drj
(dV dV\][.(l
I-.
dr
/
j
_(1/c)J(VdV/dr,I
)
i
fdV,dV)21dr
dr drj
dr.
The following examples illustrate some of the theory of this
section.
We will use the unit vector function.
y
1.22 .
[cos
-
r, sin r, i).
EXAMPLE OF A CURVE OF CONSTANT CURVATURF.
Let
C:
2
then
X2
V =
-
-
1V
J
[cos
(
[-
-
drdr)
X : 2
(!.dr
drdr
r, sin r,
r,
iJ
cos r, o]
(
r2)(
)J[
r,
sin r,
1]
dr
[hin r, -cos r, r J
To verify that the curvature of X is constant and equa]. to i I o
23
dr
r-
dr
[cos r, sin r,
T
kN
C-sin r, oes r,
T1
i]
o)
thus
i
i
i
-
L.23.
I
-.
EXAMPLE OF A CURVE OF CONSTANT TORSION.
Let
i/o
-2
- -2
dV
r'
-sin r,
r,
L-°°
t''
iT!
f-cos
r, -sin r, i) dr
[sin r, cos r,
verify that the torsion is constant and equal to
X'
r, -sin r, iJ
[sin r, -cos r, O]
X" =
2 [cos r, sin r, oJ
2.
X
To
dr
dr
J
tt,
t
!
. 21. .
Let
[o,
-Lx', x", x"
EXAMPLE Oii'
X1 be
(
x"'l
A
o,
c
i3
/ (x".x")
FAMILY OF BERTRAND CURVES.
the curve of constant curvature o! l.i.22 arid
curve of constant torsion of L,23.
Then
X3
be the
x13:xicoøe.x3sine, o<e''
: [ein
r,
:
r
-cw
i-J
r,
e
ios
ø.
),
co
c-uir* r,
ci
r:]
r,
-sin
r (con
n
),
4 sin ê
-
[cos r (cos
= (cos
-
jn
cot;
j3
S-
e -
81fl
),
[-8m
(cxn;
r (cas
(COØ
(cou
- sin Q
sin r (cas
N13 =
- sin
r
sin
sin G
ê'
e
)2
2
dr
i- (cou
:
e )2,
dr
:
Ic13
sin r (con
),
e
sin
(cos
n
)
)
[
e),
(cou
)
. sin
G)J
- sin E'),
o]
-sin r, cos r,
o]
thus
k13 :
?°
e - sin
ourv.
),
a conutant, so X
is a
ertrand
e
BflLIOUfl.APHY
1. Blaschke, Wilhelm. Vorie3ungen über differenia]. Geometrie.
New York, Dover publications, 191i5. 322 p.
2. Brand, Louis. Vector ant tensor analysis.
Wiley and sons, inc., l9Ii7. Li39 p.
3.
Graustein, fil1iam 0.
Macmillan
Lt..
Scheffer,
co.,
l93.
New
Differential geometry.
230
p.
York, John
New
York,
der differential- und integralBand, Einführung in die
Theorie der Curven. Leipzig, Verlag von Weit und comp.,
1901. 360 p.
Georg.
Rechnung auf
Anwendung
Georietri, erster