JOSEPH W. JEROME. ABSTRACT. .... similar such sequences; and, every minimizing sequence is bounded in W. To see that JU is .... S. D. Fisher and J. W. ...
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number I, August 1975
SMOOTHINTERPOLATING CURVES OF PRESCRIBED LENGTH AND MINIMUMCURVATURE1 JOSEPH W. JEROME ABSTRACT. exceeding
points, Thus,
It is shown
a prescribed
there
is at least
we show
curvature
among
bound
viz.,
for at least
all
which
one which
to be sufficient
a condition,
necessary
that,
upper
minimizes
for the
curves
set
which
has
extends
not of planar
in the
of the problem
length,
The proof
of length
a finite
the curvature
solution
prescribed
a decade.
smooth interpolate
L
sense.
of minimum
been
known
immediately
to be
to curves
in
R", n > 2. Introduction. of minimum planar
Garrett
integral
points,
provided
of the interpolating jecture.
shown
earlier
necessary curves
That
to Professor
this
condition
by Birkhoff
is prescribed
note
Birkhoff
1. Interpolating
curves
a set of points
of minimum
in R
with
for the length
on the occasion length
this
con-
of his sixty-
is necessary
It is now seen
the point
set of
is to confirm
for the existence
provided
of a curve
a finite
bound
of prescribed
condition
curvature
the existence
interpolating
of this
and de Boor [l].
and sufficient
hypothesis
upper
The purpose
of minimum
describe
has conjectured curvature,
a feasible
curve.
It is dedicated
fourth birthday.
basic
Birkhoff
mean square
was
to be a natural
of smooth
interpolating
set is not collinear.
curvature.
Let
N> 1 and let
9 - \(x ¿, y .)}■„
LQ > 0 be given.
Our
is the following.
(H) There
is a function
(i) L < L0, (ii) (x,y)£W2'2(0,
r(s) = (x(s),
L)xW2'2(0,
y(s)),
0 0
such that
.(t)\ < e/2 fot each n = 1, 2, • • .. Then, if nQ satisfies
and l/i(s*)-/„0,(s+)|
l/,(sJ-*J 11 I * Thus
of \s
by the equicontinuity
\s - t\ < 8 =*\f j(s) -/
kn0-sjinf !||Tx||y.:
exists
Banach
space,
T be a mapping
to Tx in Y if x a bounded
x £ U\ = a, there
Y a normed
space
with the property
is weakly
sequence
linear
convergent
to
\x n \ C U such that xQ £ U satisfying
is an element
\\Tx0\\Y- a,. Remark.
interval,
In the case
it was shown
continuous
curvature
d K./ds
+k'/2
of variations quiring
ture as well
locally
= 0. Forsythe
(open) curves
as the (loe al) defining
For the deduction
of these
solution
F,
Theorem
1 is valid
guaranteed
the usual
expression
they
the curve
Remark
x = x(s)
added
of the existence
of a global
tion to [l])
in the technical
report,
pseudosplines
Thomas
and elástica,
as General
Motors
authored
Research
the points
equation
re-
consec-
of the curva+ k
) = 0.
methods finally
for a that
in this
case,
is
" •, dx /ds)\ by arc length.
minimum
Nonlinear
to the author's is mentioned
interpolation
by G. Birkhoff,
Publication
attention also
H. Burchard
that
(in addi-
by splines,
468, Warren,
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
February 3, 1965.
of J
l3j
calculus
here,
id k/ds
calculus
We mention
It has been brought
the failure
out a formal
considered
n > 2. The curvature,
is parametrized
in proof.
of minimum
equation
the continuity
1, cf. [4].
in R",
k(s) = \(d/ds)(dx/ds, where
through
by function
by Theorem
a solution
carried
curves
deduce
differential
conditions
for curves
that
on a compact
the differential
[2] have spline
pass
conditions,
functions
derivative
satisfying
and Lee
the spline
As necessary
of interpolating
via the Gâteaux exists
for the nonlinear
only that
utively.
of graphs
and D.
Michigan,
66
J. W. JEROME REFERENCES 1. G. Birkhoff
approximation,
and C. R. de Boor,
Approximation
Piecewise
of Functions
(Proc.
polynomial Sympos.
interpolation General
and
Motors
Res.
Lab., 1964), Elsevier, Amsterdam, 1965, pp. 164-190. MR 32 H6646. 2. G. E. Forsythe
and E. H. Lee,
Variational
study
of nonlinear
spline
curves,
SIAM Rev. 15 (1973), 120-133. 3. Joseph
functions.
W. Jerome,
I: Existence,
4. S. D. Fisher
the problem
Minimization
and J. W. Jerome,
of minimum
problems
and linear
and nonlinear
spline
SIAM J. Numer. Anal. 10 (1973), 808-819. curvature,
Stable
J. Math.
and unstable
Anal.
Appl.
elástica
equilibrium
and
(to appear).
DEPARTMENT OF MATHEMATICS, NORTHWESTERN UNIVERSITY, EVANSTON, ILLINOIS 60201 Current
address:
Oxford
University,
Laboratory
of Computing,
England
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