smooth interpolating curves of prescribed length and minimum ...

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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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