Are length and curvature
§ 13.3
Definition
The
.
be
Fit
Let
length of
arc
vector
a
The
function Fltkc
from
curve
t=a
to
f- High has
,
t=b
is
t.fi#ttHdt=fabf'HitgH2th'Hidt
Example
<
TH
=
1.
Find
Cott
,
The
sint
Remark Compare
:
,
"
length of the are of
t >
11
from the point
distance
The displacement
with
"
tea
from
)
Feb
hah
-
Marc
length
displacement
"
the
,
t=b
toll
0,0 )
displacement
to
helix
is
'
'
.
,
0.4N
.
.
Remark
than
be represented by more
but The are length is independent
curve
function
vector
The
A
:
,
parameterization
Definition The
SHK
,
Back
to
Recall
the
the
lull
F
fat
du
'
defined
is
at
.
length formula for
direction of
t
increasing
Ftthcott
are
geometry
tangent vector TTH
direction of the
curve
The
To
The
of
.
length function
are
.
Example 2. Find
from 11,0 o ) to
one
can
unit
sint
.
tfttft,
=
,
indicates
.
how
measure
we
Remark
introduce
:
fait
the
curvature
Curvature
Intuitively straight
,
Circlet have constant
curve
,
.
measures
lines
change direction
The sharpness
have zero
curvature
at
The
of
curvature
all
points
.
.
he
curve
,
t
>
a
sharp
name j
The
to
curvature
The
arc
length
rate
The
is
s
.
change
of F
with
rapeet
.
curvature
Definition The
of
curve
a
of
curve
is
k=ld¥l
Theorem
.
function
The
Fit
curvature
is
k=
of
particular
form Y
.
=
fix
,
if
the
curve
#I
th
taxIF #
th
It )
13
IF
is
curve
in 1122
then
Iflx )
ttfixpy
"
1
K
=
given
by
a
vector
A
IFHH
=
In
a
3k
(2)
and
is
of
The
Example 3. Find
Example
4.
the
Find the
curvature
curvature
of
of
Fltkccott
rttkct
,
E
,
,
t3
sintit
>
.
>
The
frame
TNB
Definition
.
Given The
Tttl
tangent vector
unit
define
unit
the,
vector
normal
we
,
# its
NTTK Filth
bi normal
unit
and the
vector
Bltktlttxnttt
These vectors
they form
a
are
Fnenet
*
muatually perpendicular
-
frame
§
To
and
.
unfaded
-
T
niz
normal
plane
ovulating
plane
s
>
N