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1
PROJECTIONSAND
CYLINDRIC EXTENSIONS
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2
PROJECTIONS
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3
Beginning with an example
There are three et!
X"#$%&"'&X(#$%&"'&X3#$%&"&('
Carteian pr)*+,t!
i321 XXXX
3Ni=
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Ea,h element x in the et
i ,alle* a 3.t+ple e/+en,e0
iX3Ni
>==e,ti)n
Cal,+late R"(9x"&x(:
9.
}2.,9.,!.max{
)}2,,(),1,,(),,,(max{
),(),(
,
122112
21
=
=
=
=
==
RRR
RxxR
xxif
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""
Pr)>e,ti)n RR of13
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"(
Cal,+lati)n )6 the pr)>e,ti)n
Cal,+late R"39x"&x3:
.1
}.1,1.max{
)}2,1,1(),2,,1(max{
)2,1(),(
,2,1
133113
31
=
=
=
=
==
RR
RxxR
xxif
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"3
Pr)>e,ti)n RR of23
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"-
Cal,+lati)n )6 the pr)>e,ti)n
Cal,+late R(39x(&x3:
2.
}1.,2.max{
)}2,,1(),2,,(max{
)2,(),(
,2,
233223
32
=
=
=
=
==
RR
RxxR
xxif
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"1
Pr)>e,ti)n RR of1
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"
Cal,+lati)n )6 the pr)>e,ti)n
Cal,+late R"9x":
.1
}".,.,.1
,2.,9.,!.max{
)}2,1,(),1,1,(),,1,(
)2,,(),1,,(),,,(max{
)()(
111
1
=
=
=
=
=
RRR
RRR
RxR
xif
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";
Pr)>e,ti)n RR of2
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"e,ti)n
Cal,+late R(9x(:
.1
}.1,#.,.
,".,.,.1max{
)}2,1,1(),1,1,1(),,1,1(
)2,1,(),1,1,(),,1,(max{
)1()(
1
222
2
=
=
=
=
=
RRR
RRR
RxR
xif
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"=
Pr)>e,ti)n RR of3
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(%
Cal,+lati)n )6 the pr)>e,ti)n
Cal,+late R39x3:
.1
}.1,1.,".,2.max{
)}2,1,1(),2,,1(),2,1,(),2,,(max{
)2()(
2
333
3
=
=
=
=
=
RRRR
RxR
xif
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CYLINDRIC EXTENSIONS
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((
Pr)2lem De,ripti)n
Bae* )n partial relati)n&we
,)n>e,t+re the wh)le relati)n0
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(3
N)tati)n De,ripti)n
I6 X#$X"&X(&X3'&Y#$X"&X3'
then X.Y#$X('
.s$c% t%at
)()]([
yxxeachfor
yRxR
=
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(-
An example )6 ,8lin*ri, exteni)n
96)ll)wing pre5i)+ example:
T) in6er R9x"&x(&x3:6r)m R"(9x"&x(:
)()}]({[)]([ 12312 yRxXRxR ==
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(1
An example )6 ,8lin*ri, exteni)n
96)ll)wing pre5i)+ example:
I6 x" #%& x( #%& R"(9%&%:#%0=0
All the 3.ar8 e/+en,e9with x"#x( #%:
ha5e the ame in6erre* relati)n! R9%&%&%:#%0=
R9%&%&":#%0=
R9%&%&(:#%0=
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(
An example )6 ,8lin*ri, exteni)n
96)ll)wing pre5i)+ example:
T) in6er R9x"&x(&x3:6r)m R"39x"&x3:
)()}]({[)]([ 13213 yRxXRxR ==
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(;
An example )6 ,8lin*ri, exteni)n
96)ll)wing pre5i)+ example:
I6 x" #%& x3 #(& R"39%&(:#%0
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(e,ti)n R"(-&R"3&an*
R-0
92: C)mp+te the ,8lin*ri, exteni)n
9,: C)mp+te the ,8lin*ri, ,l)+re 6r)m the
three ,8lin*ri, exteni)n in 92:0
}],,{[}],,{[}],{[ 321!!213312! XXXRXXRXR
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39
T) expre R9X"&X(&X3&X-:
in mem2erhip matrix
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-%
R9X"&X(&X3&X-:#%0-@2&t&8&i
%0@a&&x&i %0=@2&&8&i
"@2&&8&> %0@a&t&8&>
%0(@,&&8&iI e/+al t)!
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-"
Pr)>e,ti)n
R
12!R
12!R
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-(
Pr)>e,ti)n
13R
R
13R
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-3
Pr)>e,ti)n
!R
R
!R
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--
The ,8lin*ri, extenti)n )6 12!R
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-1
The ,8lin*ri, extenti)n )6 13R
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The ,8lin*ri, extenti)n )6 !R
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-;
The ,8lin*ri, ,l)+re )6!1312! ,, RRR
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48
ENDO
PROJECTIONS AND EXTENSIONS
