Is Shur's arm lemma true for nonconvex arcs?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Exercise 8
![1.16 Shur's Arm Lemma
The following result describes how the distance between the end points of a
planar curve is effected by its curvature:
Theorem 7 (Shur's Arm Lemma). Let a1, az: [0, L] → R³ be unit speed
C' curves such that the union of each a, with the line segment from a,(0) to
a,(L) is a conver curve. Suppose that for almost all t e [0, L], K, (t) is well
defined, e.g., a, is piecewise C², and
Ki(t) 2 K2(t)
for almost all t e (0, L]. Then
(n1(0), an(1) < dist (02(0), a{(1).
dist
az(L)
a(0)
a,(L)
az0)
az(6)
a(to)
Proof. After a rigid motion we may assume that the segment a1 (0)a (L) is
parallel to the r-axis and a is rotating counterclockwise, see the picture
below. Then there exists to € [0, L) such that a (to) is horizontal. After a
rigid motion, we may assume that a, (to) is horizontal as well. Now let 6, be
the angle that a makes with the positive direction of the r-axis measured
counterclockwise. Then 6, € [-x, 7] (for 0 this follows from convexity of
a1, and for 02, this follows from the assumption that k2 < K1). Further note
that
10,(t)| = |0,(t) – 0,(to)| =
Thus |61(t)| > |62(t)|, and, since |0,(t)| € [0, #], it follows that
cos (61(t)| < cos |02(t)|-
Finally note that, if we set e := (1,0), then
||a1(L) – a1(0)|| = (a1(L) – a1(0), e1)
cos |0,(t)| dt
cos |02(t)| dt
(az(L) – az(0), e,)< ||a2(L) – az(0)|l-
Exercise 8. Is Shur's arm lemma true for nonconvex arcs?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a4dcd82-baf4-45bc-b40b-693a3e683492%2Fa41f1efc-552e-4b3c-8b7f-8268e8450da6%2Fegp5qmg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.16 Shur's Arm Lemma
The following result describes how the distance between the end points of a
planar curve is effected by its curvature:
Theorem 7 (Shur's Arm Lemma). Let a1, az: [0, L] → R³ be unit speed
C' curves such that the union of each a, with the line segment from a,(0) to
a,(L) is a conver curve. Suppose that for almost all t e [0, L], K, (t) is well
defined, e.g., a, is piecewise C², and
Ki(t) 2 K2(t)
for almost all t e (0, L]. Then
(n1(0), an(1) < dist (02(0), a{(1).
dist
az(L)
a(0)
a,(L)
az0)
az(6)
a(to)
Proof. After a rigid motion we may assume that the segment a1 (0)a (L) is
parallel to the r-axis and a is rotating counterclockwise, see the picture
below. Then there exists to € [0, L) such that a (to) is horizontal. After a
rigid motion, we may assume that a, (to) is horizontal as well. Now let 6, be
the angle that a makes with the positive direction of the r-axis measured
counterclockwise. Then 6, € [-x, 7] (for 0 this follows from convexity of
a1, and for 02, this follows from the assumption that k2 < K1). Further note
that
10,(t)| = |0,(t) – 0,(to)| =
Thus |61(t)| > |62(t)|, and, since |0,(t)| € [0, #], it follows that
cos (61(t)| < cos |02(t)|-
Finally note that, if we set e := (1,0), then
||a1(L) – a1(0)|| = (a1(L) – a1(0), e1)
cos |0,(t)| dt
cos |02(t)| dt
(az(L) – az(0), e,)< ||a2(L) – az(0)|l-
Exercise 8. Is Shur's arm lemma true for nonconvex arcs?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
