Mnthhttweeky
lets
ho
see
to extort
Claim :
th
geoetnzkformth
fist
the
&
fdamhl tons
second
.
E¥Ey*
€yytp
-2
TPZ lies
on
side of
ae
Z
I.
llipk
TPE Cub
Z
hypohok
.
To build up lets think
f Ip
.
When
we
Ip
depict
meaning
In gene
So
M
Anti
A form Q
t
,
)
psitk
B
fm
B
definite
if
Qtii
) <
ddmik if it
E ]
EE
qndrkfm Ip
& the
positk definite
were
⇐>
-
.
.
all eiennkesf M
QIITI 10
B either
⇐st=o
o
4 QH 4
positk
or
negate definite
0
.
to
#
o
=
,
>
are
it
&
B
.
indefinite
if
F
.
t.ws
,
QK
&
o
>
o
.
,
new
EHIPKZ
is
p
a
)
)
OH
kk
.
> 0
*
Ip
At
difftwy
Ipliit
&
20
definite
negate
3
a
slightly
tht
Iplt
Read :
a
proud tut the mkrix
we
,
in
parabolic
pub
,
Ip
B
is
definite
semikfnite
.
in
meaning
either
D hyperbolic
p
a
Np ) <
kin
0
< o
*
Ip
tpk
,
it
20
B
ihdefnik
tv
or
IF
IT )
10
¥
t
.
:L
.
,
So
Pq Tf
:
If
¥
E
pic
.
.
fmlre
hear
new
,
3
p
Then
point
U
F nhd
,
of
,
by
,
dtpk
an
as
hyphok point then any nhdf
a
p=HQo )
Suppose
:
}
dain
the
lwhih
190,0 )
=
p
points
paths
pts
always
adhere by
can
we
all
st
on
.
in
he
U
both
the
m
TPZ
sides of
rotath
translator &
sole
same
ftp.z
.
)
.
than
Taylors
Ilu D=
In
,
ii
txvvtklxuu
;%oFi÷i=0
what
+2in
Xvii ) TRTU
D
wt
,
,
.
items
µ÷
÷
)
height
of
LI
D=
,
D
xtu
abmethe target plane is
given by
,
N
) Exu
's [
=
=
o
to A,
is
p
a
if p
Far smll
which D
a
:
kept [ D)
Ip
Ip
hyprhok , then
I un )
.
he
{
D
u
,
is
than >
u
13
et ]{ D
tg
+
( ki )
definite
B indefinite
.
tzkxunn ) ii
gv 2)
+
( so
cool
,
( like
2ft Iv 'Yg=-
particular
+
.
different qn
for
alt )
0
,
.
same sign
dyes sign
th pay
.ms#tTph,H=o then the
'
,
it
+
LR
,
B)
)
Ip ( 5 T )
asymptote direct
In 'v )
!
&
.
B) uvthxmn )
ain )
<
Iplniw ) by the
4
+2 Lin
+47,5 )
so
Bree the R tens
on
+
v
+2 fuut
Hei
+
edipk port then
gh
÷
aekt
,
to
0
=
If
D)
=
cane
tv
all
e E
TPE
.
.
the pgnlf
alt ) IUHI
=
,
as
.
VHH 3
an
asymptotic (
wu
iff
Et
frewhth
the tons
On
:
e=r
alt )
so
f=0
,
B
cosulatrcosu )
=
g
,
asymptk
an
¥ 1 un )
are
F
=
Whihnealdsohefrulv )
abut
Wht
,
}
a
line
a
,
of
't
u
,
curvature
a
az ,
an
usj
az
the Weingarten
)
u
aft ( at
-
(v
t
r
'
)
'
cosulatrosu )
=
O
or
equivalently
,
,
)
rcosu
:
,zv
11 )
(
,
'
n
)
't
Xu
"=
yi
(
dttpkttt)
'
HHNH
=
,
if
a
'Hl=lu' HHIHD then
,
(i)
'
, z )
'
multiplying 14 by
,
-
an
or
azz
)
'
uh
ta
and subtracting to get
u
,
'
za )
'
.
=
O
qnet.ms
(f
Eg :
=
Xu
=
tazzv
u
solely mukplyiy
-
or ,
'
#
'
'
We
(
⇐>
'
'
dicks ?
principal
Read alt )
:
tht
saw
we
,
E
-
e
F)
little
EG ) iv
-
tlgf
'
.
f G) ( if
0
On the tons of revolution ,
E=r2
so
we
F=
0
G
=
( at
rcosn
)
'
get
(
Pusulatrwsu )
-
rlatrwnp )
or
-
rlatrusw )
a
ulv
'
=
0
iv
'
=o
⇐>
rlatrasd
(
room
-
)
latrwsu)
uiui
=o
Bt
NI
then this
the
.
-
means
pat
key
coordinate
'=o
u
of
it
or
the
and
the above ablation
lines
lines
are
of ant
tht f-
was
we
F=o
⇐>f=F=o
arrtwe
of
lines
just the
are
Tuft
.
lines
coordinate
:
.
Exe : Surfaces of
III
revolt
EFG
To st the coetb
In
( 64
=
Iu=(
Iuv
=
In
=
'
6
In =/
Also
Sinn
-
-
(
(
Hun
64
-6%
,
,fg
,
Smu
s.nu
,
-614
"
0
,
)
}
E=
)
can
,
,
4 ldcosu , 6 Glenn ,
"
o
)
0
)
Tfu
F=
assure
Similarly
of
,
f-
&g=Y
0
casey
D
familiar
We
.
4
-
4 4
hwtht
D parametrized
=
then
by odgth
6=1
,
,
so
4
"
"
' '
6--1642+1442=12%112
0
Y ld )
!
UYYU
e=efT#t*×IxuD=tyf4e%s
'
a
feet
4
-
Bit
44
'
"
,
.im#
.
y %)
Smu
614
,
,
coyote
we
,
614cm
,
644
cosu
)
e
.
Iii
:*
"
cos
u
-
4
o
xi
Sinn
|= tyl
0
'
.
the
cnrtrf
His
gimj
-
Yyltsiiutyioiu
))
=
-
ai
kingfish 19.14*-144
Ntt
Sinn f-
F.
,
coordinate
the
knew
we
cures
lines
again
are
"
-46 "l
curvature
of
f#¥¥
Ago
So
:
k=¥¥m=
when the
K=0
#
4444811
'
parabolic
Y '=o
points
?
or
Y
'
6
telltale
=
' '
"
-
4 y'
"
=o
-4
"
a)
(the cavehsapontfzrocwatwe )
•⇐÷y
#
parabolic
Of
come
,
they
'
better
a
different
writ
.
expusinfr
:
'
K
:
2+4 2=1
'
v
yield
4 'll
so
-
k=
Ky
"
"f44"
'
+24
Y
Y
=
"
to
or
lick
"=
Applicant A swfeefrwdkhhsastat
:
Sdwthsftk
knit
ODE
are
a
Siple
4 'll
Gaussian
"
=
-94
-
"
,
4÷
curvature
#
-
4÷=K
.
or
:
"=t 9YhYaYTL" D
Ashe :
4
"
=
-
Kole
Keo
#
Y
"
=
0
⇐>
614
⇐t 4"=4
( ¢ Is the aniwpmnetrizthfr
.
=
avtb
µ
A
are
or
µ
;jjh¥gjY
all
the psadosphethntheobme
cylinder
y
& Y?
)
Panah
!