Pacific Journal of
Mathematics
THE BENDING OF SPACE CURVES INTO PIECEWISE
HELICAL CURVES
JAMES M C L EAN S LOSS
Vol. 32, No. 1
January 1970
PACIFIC JOURNAL OF MATHEMATICS
Vol. 32, No. 1, 1970
THE BENDING OF SPACE CURVES INTO
PIECEWISE HELICAL CURVES
JAMES M. SLOSS
It is the purpose of this paper to show that a regular C3
space curve Γ in a Euclidean 3-space, whose curvature /c Φ 0,
can be bent into a piecewise helix (i.e., a curve that is a helix
but for a finite number of corners) in such a way that the
piecewise helix remains within a tubular region about C of
arbitrarily small preassigned radius. Moreover, we shall show
that the bending can be carried out in such a way that either
(a) the piecewise helix is circular or (b) the piecewise helix
has the same curvature as Γ at corresponding points except
possibly at corners, of (c) if the torsion of Γ is nowhere zero,
then the piecewise helix has the same torsion as Γ at corresponding points except possibly at corners.
Also we shall show that if, in addition, Γ has a bounded fourth
derivative, then an explicit formula can be given for a sufficient number
n of helices that make up the piecewise helix, where n depends on Γ
and the radius of the tubular region about Γ. In this case, we shall
also show how the determination of the piecewise helix can be reduced
to a problem in simple integration.
1*
Bendability*
DEFINITION 1. A curve is called a piecewise helix if it consists
of a finite number of segments, each of which is a helix (i.e., a curve
whose tangent makes a constant angle with a fixed direction). A point
at which two consecutive helices meet will be called a corner of the
piecewise helix.
If, in particular, between corners the helix is a circular
helix, then the piecewise helix will be called a piecewise circular helix.
REMARK.
THEOREM 1. Let Γ: r(s), s — arc length, 0 ^ s ^ I, he a regular
C3[0, I]1 curve whose curvature κ(s) is nowhere zero. Then for any
given e > 0
(a) there exists a piecewise circular helix Γ*: fef(s), s — arc length,
0 ^ s ^ ϊ, such that:
\r(s) - h*(s)\ < ε ,
1
3
0^ s ^ I
(I.e., r(s) can be extended to lie in C on some open set containing 0 ^ s 5Ξ I.)
231
232
JAMES M. SLOSS
(b) there exists
0 ^ s <Ξ; I, such that:
a
piecewise
helix
Γf: h*(s), s = arc
length,
0^ s ^ ί
\r(s) - h*(s)\ < ε ,
ami JΓ2* Λαs £/&e same curvature as Γ at corresponding points, except
possibly at the corners of hf(s)\
(c) provided the torsion τ(s) is nowhere zero, there exists a
piecewise helix Γ*: hf(s), s = arc length, 0 <ί s ^ I, such that:
\r(s) - /&3*(s)| < e ,
and Γ} has the same torsion as Γ at corresponding
possibly at corners of ht(s).
REMARK.
0
^s^l
points,
except
In each case the curve Γ is "bent" into a piecewise
helix.
Proof. We shall prove (b) and indicate what minor modifications
are necessary to prove (a) and (c). Let κ{s) and τ(s) be the curvature
and torsion respectively of Γ.
Then φ) e C^O, I] and τ(s) e C°[0, I]
since r(s) e C3[0, £]. By hypothesis, rc(s) Φ 0; therefore,
is continuous and thus uniformly continuous on [0, I].
(1.1)
I κ(έ) I ^ κmκ
Let
on 0 ^ s
and
(1.2)
and choose d(ε) > 0 such that
(1.3)
|/(s 2 )
provided | s2 — s t | ^ d, where
a = </cmax?V6 exp {i* m «i/2(l + /Lx)})" 1 .
(1.4)
Let
n — n(ε) = smallest integer ^ —
δ
and
Jo = {s: 0 ^ s ^ d}
(1.4.1)
/,- = {s: iδ < s ^ (i + 1)3} ,
J,_! = {s: (TO - l)δ < s ^ 1} .
i = 1, 2,
,» - 2
PIECEWISE HELICAL CURVES
233
Then 7,(0 ^ j <Ξ n — 1) form a disjoint covering of [0,1], each of length
Let
(1.5)
τ,(β) = fjφ),
β e J, ,
i = 0,1,
,n - 1
where
f
=
J/IO" + 1)«] for i - 0,1, .. , n - 2 .
for j = n — 1
By the fundamental theorem for space curves there exists a unique
curve hj(s), s e /,-, for which:
( i ) its curvature and torsion are respectively κ(s) and TS(S) as
denned by (1.5), and
(ii) its position h3(s), tangent ί3 (s), principal normal %(s) and
binomial bj(s) satisfy the initial conditions:
(1.6)
= φ'δ)
htfδ) = r(jd), tj(jδ) = eAJ8), ns(j8) = e»(jδ),
where ejβ), ez(s) and e3(s) are the tangent, principal normal and binormal
of r(s) respectively, and s is the arc length parameter of hs.
Moreover, if
0
(1.7)
Φs(s) =
1
Λ(«)J
1
0
0
/,
-/,
oj
then Φj(s) satisfies the differential equation:
(1.8)
Φ'M) = KWAJΦM
.
Also, because TJ(S)/K(S) = f} = constant on 7 i( hj(s) is a helix on JΓ, .
By the Frenet formulae for Γ, we have:
ψ'(s) = /c(e)A(e)y(e),
(1.9)
0
where
(1.10)
Ψ(8) =
Φ)
, A(8) =
0
-1
.0
1
0
-/(«)
0
f(s)
0.
Considering both (1.8) and (1.9) as differential equations on Is, we
obtain:
(1.11)
Φj(s) =
Φj(j
58
i = 0,1,
,n - 1
234
JAMES M. SLOSS
and
(1.12)
*® =
¥
H
^
j = 0,1,
••-,%-!
Since by (1.6) Φj(jδ) = Ψ(jδ), we see that if
II(C«)II = / Σ ^ - ,
then
| | f ( s ) - Φ,(s)|| ^
(1 18)
'
|/c(ί)| | | A ( ί ) - A , | |
+V
J jδ
\\¥(t)\\dt
£ί
But by (1.7), (1.10), and (1.3)
11 A(ί) - A, 11 = l/2[/(ί) - / 3 ]2 < i/2αe
for ί e /,-
Also we have
l|y(ί)|| = VΎ
and by (1.7)
11 A, 11 = 1/2(1 + /)) ^ 1/2(1 + /Lx)
Thus
(1.14)
|| ¥(s) - 0,(8) || < ilίδeα: + ivΓ || Ψ(t) - Φf(t) || d ί ,
Ji'δ
where
N = κm
Let
C = sup||?Γ(ί)-Φi(ί)||,
telj
then by (1.14)
(1.15)
|| ¥(s) - Φd(s)\\ < Mδεa + NC(s - jδ)
from which we see upon combining (1.14) and (1.15) that
β e /,
PIECEWISE HELICAL CURVES
235
l|y(β)-0,(8)11
< AfδeαΓl + N(s - jδ)
(1.16)
, (s -
+N
2!
jδ)k+1
k ( s
~
m
Mdsae
Mlεaem
by the definition (1.4) of a.
If we let
Φ*(β) =
t(s) n(s)
, s e I*,
i = o , l , ••-,% = l ,
then Φ*(s) is piecewise continuous on [0, I] with discontinuities possibly
at
j = 0,1, 2
s = jδ ,
, n — 1,
and by (1.16) since /,• is a cover of [0, I],
(1.17)
|| y(β) - Φ*(β) || < e/ί
for 0 ^ s ^ {.
Let
(1.18)
h*(s) = r(0) + [t(σ)dσ ,
for 0 ^ s ^ I .
Jo
Then A*(s) is a piecewise helix Γξ for which
for s e Iit j = 0,1, . , n - 1 .
h*'(s) = h'ά{s) ,
Thus for 0 g s ^ Z:
I »•(«) — Λ*(β) I ^ Γ I ^(β) —
Jo
^
Jo
l|y(8)-0*(8)||d8
<ε
by (1.17).
Next we note that s is the arc length of h*(s) since
for s e / , .
Moreover on the interior of I5:
I h*"(s) I = I h'/(s) I = curvature = φ)
JAMES M. SLOSS
236
by construction of hj(s).
This completes the proof of part (b). For the proof of part (a)
and part (c), only obvious slight modifications are necessary. In part
(a), we need only the additional fact that a helix is circular if the
curvature and torsion are both constant.
2* Explicit results*
derivative we have:
If we allow r(s) to have one more bounded
THEOREM 2. // in addition to the assumptions of Theorem 1, we
also assume that r(s) has a bounded fourth derivative on [0, I], then
we can choose n(ε) in part 2 to be
a*l
n(ε) — smallest integer > ——
(2.1)
as
where
(2.2)
a = <<£maxZV 6 exp {k max l/2(l + fL*
r' jr" x r"") _ 3[r"' r"][r' (r" x r'")]
<g*,
O ^ s ^ l
,
(2.3)
REMARK.
A similar result holds for parts (a) and (c).
Proof. Since
\*V — i
\ /
V /J
and
_ r '.(r» x r "')
the expression in the first inequality in (2.3) is simply the derivative of
Thus
g*[s2
If we choose
(2.4)
δ =
aε
-
PIECEWISE HELICAL CURVES
237
where a is given by (2.2) = (1.4), then
\f(s2) - / ( * ) !
whenever |s 2 - sλ\ < δ.
gives the result since
<aε
This, by the proof of part (b) of Theorem 1,
a*l
I
n(e) = smallest integer > — — = — .
aε
o
THEOREM 3. Let Γ: r(s), s = arc length, 0 <: s ^ I, be a regular
space curve with bounded fourth derivative and nowhere-zero curvature. Denote the curvature, torsion, tangent, principal normal and
binormal of Γ by /c(s), τ(s), e^s), e2(s) and β8(s). For any given ε > 0,
let n(ε), δ, and I5 (j = 0, 1,
, n) be given by (2.1), (2.4) and (1.4.1),
respectively.
Put
tj(s) = ^ - { [ / | + cos (gd(s)™>)]ei(jδ) + [m sin (gj(s)m)]e2(jδ)
m2
/ y [ - cos
where
)<?]}, m= + Vl + f'U gό{s) = \ fc(σ)dσ
and let
t(s) = tj(s), s e Iά, j = 0, 1,
,n .
Then the curve
Γ*: h*(s) = r(0) + I t(σ)dσ, s = arc length, 0 <; s ^ I ,
Jo
is α piecewise helix such that
\r(8)-h*(8)\<
and Γ* has the same curvature
possibly at the corners.
Proof.
e,0^s^l,
as Γ at corresponding
From (1.7)
nd(s)
satisfies the system of differential equations
points
except
JAMES M. SLOSS
238
Φ's{8) =
(1.8)
where A3- is given by (1.7).
Ψ(jδ) is given by
on I, ,
φ)AjΦj(s)
The solution of (1.8) for which Φ3-(Jδ) =
Φ3-(s) = egi{s)Aiψ{jδ)
.
The eigenvalues of A3 are 0, ίm and — ίm and the corresponding
eigenvectors are:
Γ 1
1
0
ίm
1
_ -ΛJ
— im
Also the matrix T = {TX1 T2y T3) has the inverse
"2/y
0
1
—im
—/y
1
ίm
—/,-_
2m 2
2
Thus
where
0
0
0 "
0
im
0
0
0
-im
and
| + c o s ^ (s)m, m sin ^
m
From this it follows that
'tj(s)'
nά{s)
"[/? + cos gj&mfaijδ)
+ [m sin g3 (s)m]e2{jδ)
x
which gives (2.5) and the theorem is proved.
- cos g3 (s)m]ez(jδ)'
PIECEWISE HELICAL CURVES
239
By using the definition of torsion as given by Hartman
and Wintner [1], p. 771, [3] p. 202, the continuity requirement of
Theorem 1 can be relaxed from C3 to C2. A question of further interest would be to consider the bending of normal curves, see for example, Nomizu [2] and Wong and Lai [4].
REMARKS.
BIBLIOGRAPHY
1. P. Hartman and A. Wintner, On the fundamental equations of differential geometry
Amer. J. Math. 72 (1950), 757-774.
2. K. Nomizu, On Frenet equations for curves of class C°°, Tohoku J. Math. 11 (1959),
106-112.
3. A. Wintner, On the infinitesimal geometry of curves, Amer. J. Math. 75 (1953).
4. Y. C. Wong and H. F. Lai, A critical examination of the theory of curves in three
dimensional differential geometry, Tohoku J. Math. 19 (1967), 1-31.
Received March 17, 1969, and revised form June 13, 1969. The author wishes to
acknowledge support for this research by the National Aeronautics Space Administration, NASA Grant NGR 05-010-008.
UNIVERSITY OF CALIFORNIA
SANTA BARBARA, CALIFORNIA
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
H. SAMELSON
J. DUGUNDJI
Department of Mathematics
University of Southern California
Los Angeles, California 90007
Stanford University
Stanford, California 94305
RICHARD P I E R C E
BASIL GORDON*
University of Washington
Seattle, Washington 98105
University of California
Los Angeles, California 90024
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K. YOSHIDA
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*
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Printed in Japan by International Academic Printing Co., Ltd., Tokyo, Japan
Pacific Journal of Mathematics
Vol. 32, No. 1
January, 1970
Robert Alexander Adams, Compact Sobolev imbeddings for unbounded
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Tom M. (Mike) Apostol, Möbius functions of order k . . . . . . . . . . . . . . . . . . . . .
Stefan Bergman, On an initial value problem in the theory of
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D. G. Larman and P. Mani, On visual hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R. Robert Laxton, On groups of linear recurrences. II. Elements of finite
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James B. Lucke, Commutativity in locally compact rings . . . . . . . . . . . . . . . . .
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James McLean Sloss, The bending of space curves into piecewise helical
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James D. Stafney, Analytic interpolation of certain multiplier spaces . . . . . . .
Patrick Noble Stewart, Semi-simple radical classes . . . . . . . . . . . . . . . . . . . . . . .
Hiroyuki Tachikawa, On left QF − 3 rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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nonlinear evolution equations in Banach spaces . . . . . . . . . . . . . . . . . . . . .
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