nnr:2;STIGATIONS INTO SOME OF THE
iRO?Ef~TIES
OF THE:;ONIO G1CTIJR.3 HI TErmE DIAEH0JONS
by
R.icLard Dieberich
3enior Honors Thesis
ID 499
Ad vi so r :
,J:) (.,LCv~ f.. J:) Q.~,
June 5, 1966
(
PREFACE
The author's
mathe~atics
stratio:l
str~ngej
orl~in
Nh~ci
~otiv3tion
should be presented for the
is to follow.
In a classroom demon-
iveE dlE'ing the author's stude::lt teaching, a
~odel
of a cone
con~c
of the
~as
being used to illustrate the
sections.
A student asked
~hether
the focus of a parabola lay on the axis of the cone from
llhich it is
~rojuced.
He was answered pos:tively, but
the author, being in a somewtat subordinate pOSition,
disa:;reed quietly.
';[bile a tter:lpting to prove th3 t this
\las not senera2.1y tile case--tha t the focus of a parabola
nee cl no t 1 ie on
t~le
axis of the cone from \·;hl.c h i t is
produced--the aut lor's interEst in and appreci3tion for
the ::lany aspects of tb3 t f.<:illily of curves knOim as the
conic sections increased.
J11ile at te.r.p tin~ to solve tile lJro ble:J :leal iYl E:
the focus of
:)a:::c'abola, the s)atial aspects-·-the
!l
~·.i th
l)hysic~=tl
intersection of plaLles ::,nd cones--ire;{ the clutLor's
tion.
attel~-
The rotation of ficures in space led to an attempt
to describe SOI:le tl1ree-dimensional interpret:;,tions of the
aSY::lptotes of
2.tte~,:pt
plane
8.
l:lJPc:::c'bola.
This led to
8.
sUCCEssful
to iescrioe, Given a hyperbola, a ccne and a
~hic~
can produce this
hy}erbol~.
The nExt obvious
a
involvi~g
cOD}letel~'
lifferent cJmbination of plane
~ifte
this existence
~2~
proved,
sever~l
un~uccessful
r
atte~pts
\'rere :n:;:1e to dis»3.Y :em infini te m;liluer of combi.nations
of ccne and Jlane which would all
~lypcrbola
~
ment~l
ti~e-out
s~ction
uni1ue
The
thE
s~~e
.
was
t~ken,
and a proof thnt
J
spe-
mlY be 9TJduced f'ro:n. a -=iver: cone 'oy a
~_Yi!erbola
cific
duplic~te
wa~
prc~lem
~orged.
referre~
to above--that of
d~s]lHying
inf'in: i~e nUI:10er of cornbln;'l tions of cone 'tn 1,1:"ne -,;i1ic11
not Deen 2:Jlved.
would produce the
algebr~
T~e
sild
ti~e.
bacic refereilce for this paper has been the
tflird edi tj,on of' .Jor'j.n Cell's ll.nalytic Geo::uetl:;'L.
book
Inc.,
w~s
ill.
jublishel in Gcw
1:';"50.
obta1ned tt.ere.
Yor~
by
Joh~
Iiley
~~j
Sons,
an
1.
The locus of the foci of all parabolas produc~d
on a specific cone
. 1
2.
A 2;Eneral SOl).rce for a h:'peroola •
3.
Three-dimenslon~l
Jnter~retations
13
of the
16
clsy::nptotes of a hyperbola
4.
The non-uniqueness of the source of a
rlYp E!rb01a
h
..J.
The unLjueness of l1Yi)erbo1a
22
~Jrojuction
fixed cone • •
r-
0.
.?ro~!
ecti()l1S of further ;\or2: t'J
in a
.....
be jone . . . . .
37
1.
The locus of the
of all
f~ci
parabol~s
produced on
s;ecific cone.
~
COLsider the
of a
equ3tto~
ri~ht
circular cone
(1.01)
z-axis.
Oonsider the fanily of
z
(1.02)
)l~nes
- ax + lea.
~
In ordEr to produce a )8r8bola, a Jlane must intersect
3.
C~ille
:'ind be
:.:-)ar~111cl
tJ one of the elemen tf: of the cone.
EaC:l 'uember of thE; :'>:l,;nily (1.02) is obviously IY3.r3.1lel to
,u
-
ax, y
-
1)
w~ich ~s
an element of the cone (1.01).
intersection of eich
(1.01)
In
=0
~ill
~ember
of thlE
Therefore, the
fa2~ly
of )lanes with
Jroluce a par3bola.
Fi~ure
1,
(1. 04)
~la~e
3CJE represents
z - -
~lX
+ 23.
b
which is a
rne~ber
of the
f3.~ily
(1.02), passes through
2
(b,O,a), and is parallel to (1.03).
the
)1a~e
The intersection of
(1.04) with the cone (1.01) produces ttc para-
bola "._l.ch '.fill be considered tf-;.roushout the rera:':lin::ier
of tois section.
T.iis parabola's vertex is at (b,O,a)
and its axis is
z
(1.05)
:b~rClm
= -1ax) +
FiGure 1, the
y
2a,
:l.~stance
= o.
fro)::! the vertex (b,O,a)
to tile z-axis slOEg (1.05), the axis of t.e ?ar.'?.'oola, is
Va 2 +
b~~,
or c, fro.'Il the Pythagorean relaticmship.
If it
can be ::,nOhrll tna t the d:i.st3.nce fro:n the vertex of this
parabola to its focus is 8lso c, then the locus of the
foci of all parabolas
obt~ined
will be the axis of that cone.
by
sectionin~
a fixed cone
If this locus is not the
axis of the cone, the proper locus will be displayed •
.,...
.-'
c
\
"7
L
Ot-
b
o
B
y
e:
\
3
In oriel' to deter line the
bola's vertex to its
se~iuej}ce
,{;~ere
t~iis
~lll
dE:3ired.H~3t'.,ilce
the
inspection of the coefficients
formed equ8tion.
by
tho parabola
foc~s,
iron:.
para-
undergo a
of traD.;cforrJ'.3. ti·)l1s ,;hicb 'dill place it in a co-
ordins. te plane
by
dist~lnce
The vertex
~ill
o~
:.iayoe obtaiaed
tte parabola's trans-
be )laced at
t~e
origin
the fo IlOllin.; transh; ti on:
=x ,
z =z y' = y.
X,
(1.06)
b
a
Under the equations of transformation (1.06), the
cone (1.01) becomes
(1.07)
Under
Cl:l8
S:l:rre equations (1.06), Fcc plane 0_.04) becomes
(1.08)
Z I
The parabola is now in
=-
th~
ax I
•
1)"
situation described in Fisure 2.
}!:.'
u x.
b
i
r -
4
The parabol::-J.:1ill
be pl::.iced in the x"ytt-plane
nO·,i
by s rot3tion of the aX82 aro~nj the y'-axis.
T~e
equa-
tions of trsnsformatlon are:
x'
= ::(ltcos(130o
+ e) - z"sln(1800 + e)
Zl
= x";::in(1300
+
y'
= y".
The trisonometric
e)
of 8 can be
fu~ctioTIS
Xl
-
-
+ zllcos(1800 +
bx lt
C
(1.09)
z'
= axil
-- bzC
C
e)
deter~ined
from
az tl
C
tt
y' = ylt.
After 3)plyL1C s1uations (1.09) 8.nc1 ;IlUCllS.l[jeiJrai.c
( .,.L. -:..'0)
(0 4 - ,./+)zIl2 + (2J.3bc - 2ab 3 c)z" +
4i 2 b 2 cx tt
-
a 2 c 2 ylf2 - (2ab 3 + 2o. 3 b)x"z"
(loll)
z"
The parabola
situation wnere it
bein~
~2Y
= o.
discussed
be
= o.
~bt_ined
hR8
been saved to a
by intersect:ng (1.10)
with (1.11).
If t~is is performed ulBebraicllly, the
eQuation of the J1rqbJln
(
-
~il]
be
obt~inei,
~n0
~il1
be
y,,2
1r»
-L • ..LL
'rhe paraoolio
i[3
n'JJ in the
:)0[:1
tion illustr:?tc;:i i.n
i.
I,oo~=
at
]!'j
i;nre 1., Pi,;urc 2, and F 'ure :5 in
a::e3 11"-ve moved.
As
5
tcw~'
result, FiGure 3 is very
Figure :5 i2 redr8:ffi frO:11 '1 be tter
order.
~nolear.
:Dint of vlei-l in
f/
6
FrO::ii Cell
'8
~)q;::e
Ans.l;rtj_c Geo:netry,
t?1e distance
e~? U
if the
;:3.~'&bol8.
(one
t :j
C) :(1
e-
the fJllo·Jinc.:: form:
:)i:'
. '1
V2.rlJ.
0 e
:~:'"le-i':n";rtr~
3ut
1.s
.:'.1
)2 =
of t~1e cOE.,ffic~e~lt
GE
the xl! tl::l'::~ in (1.12)
is
( l.13 )
(1.13) ex)resses the distance
the focus '* of the pard DO J ::1.
of
("'
-',i.L:l
't' ("~.. (4)
, 0')
1.
fro.~l
.L •
the; vertex
j'
O:~
•
o"et
~~
fro~
the vertex to
l'oduc ed OJ the ill tcrsec tl on
('~..!._.~L./
17,)
erur"
J
'o.~
'~'J
v
i"'lC
,JI_
tills .pD.r:::.bolcl to the ::L:is of
(1.01),
if'
and insJect the equality.
''"30th of thesl: d.i~t::;,tlcesJere":e3"sured Ions tilE scome
line, the 3~=is ;y;' the )".:lr:'loola, ~o "GIlBY CO:.ild ')e properly
co .:;p'lred.
7
(1.1 4 )
a
=0
0'0.1;",
In genera], then, ( ..,-L •
i ty.
In
0
theriTords,
'001'3.8 obt.ine:l fr;;;:.
cone.
t~ll
~'\
..L4,
.
If::;
2,:;1
locus of the foci of the
fi:ced cone is ,.ot tllt:;
2.
lwt
'"xi~!
equal~:)ara-
of thet
The true locus 11il1 not: be dis.,layecl.
r2he inter-
section of each
a
par~bo~a
w~ose
~ember
focu~
of this family with (1.01) defines
is in the xz-coordinate plane.
The locus of the foci in the xz-plane will be one
e]e~ent
of the locus of tie focl of all par3bolas obtained from
(l.Ol).
If this first lacus--in the xz-coordinate plane--
is found
~nd
r)ta~ej
around the z-axis, the locus of the
foci of all parabulaE: obt:J Ined fro;:!
describe1.
Consider :pI igure _.
z;c.
'-1 \
i
ux
..
.: C
t:~is
cone
~'li
J_1 be
8
12r~er
The
by
par~boLa,
whose vertex is
~oint
~,
~roduced
is
-':",ite intersect:m of (1.01) '.vith
.:. = -
(1.15 )
ax
+ 21a,
b
section.
3 and A arc the
b -
par:~_ 'oJ.:~s
]
res;:ec t"lve_y.
see if tbey lie
FiGure
~
o~
Inspect
foc~
of
~oints
t~cse
0,
!,
two
~nd
3
to
a straioht line.
13 )artially
rcdra~n
in ?i;ure
J
iOI- cJ.arity.
1:.
o
c
ftLb.O,
La)
E(b~O. a)
o
y ,
'
Tri'J.ugles OEJ aua. OFD are :::iiliilar.
If it
C -;,;1
be
sho;m ti:l8. t
(1.16 )
the 11
~~ 0
in t sO, A ,
a straight line.
'ill d
B
~.[ i
11 ha v e bee n s h o"l"n to::' i e
:J n
9
Fron the previous
jeve1o~~Ent,
=
EA
(1.17)
b
2
c
and
Jhen the res~lt2 in (~.17) ~nd (1.13) ~re subst~tuted
02
a2
-
as ,:ere EA :lEd EO.
+ b
]'fjj
2
3y t2e
= FD
.
fytha~orean
relationship,
(l.19 )
-- Lc •
:lfter these equat'_ons have been trGi.lsl'orm.ed by (L06)
e1uat~on
and (1.09), insJeatioL of the
of this
par~bo1a
ilill yield
(1.20)
-c
.'Then the results fro:n (1.17), (l.le), (1.19), 8.rid (1.20)
3re
~3ubstituted
into (l.l~),
(l.lo) becomes tilE: fo:2.10,rins:
10
n.
EC
(1. 21 )
0
c
0
c
Therefore, 0, A,
~ni
2
?
2
2
--
FE
FD
2_
b .w
= -c
=-
0
C
•
1
Lc
2
•
2
B do lie on a stral=ht line.
~y
If the equation a
this
l~ne
is
of
ictpr~ined
~rd
pl~nes
(1.02)
rot2ted
FLl1d pOint ;L
ii~t~nce
The
:::: to A is (1.1.:,).
-V(b
3y t>:ll list'lnce formula,
=EA
-Va
(1.22)
,2
=
D
2
+ 1)2
from
-,
.,
.J.. .J..
this v:J.lue Illto t~18 e1U·-1t,i.J~l nf tile Jine,
z
(1.05).
=
b
z
a 3 + 2a 3 + 2ab
8.
(1. 23)
z
= a(a 2
a
2
2
+ b
2
2
+ 2b 2 )
+
~
0
2
(1.23) is the z-coorlinqte of 90int A.
(1.24 )
(1.25)
The~efore,
,
thi.rJ element,
Jl:Wl1
\f
l
~
,•.r·c:::::,
:..:.ccord inc: to Fi c)J:::'t::: 7,
)
,
is rotated
('1
COlle
~Iill
~rou~d
be ob
the
t~~ilJed.
~'"
=(a 2 :b2b2)X
z = z (3 2:b 2
)X
Z
2b
= (a2:0 2'0 2)2 . .
_ (?TI)2 = '{2 +
~2
il.
z2
<"
..J.
2~,"\2
!,,/
The locus of tIE =·Jcl. .:,f '3.11
(1
(,"'1
..L,\
.J- ........
)'::J.r:~bo:'..:lS
obt~~ined
freD
'L'
- S
..
t~e
z-axis.
a st3.ndari rot-'ttion :~;lC:C:l3.~lisLl.
It CJ:r;,cs from
Cell's iI.n3. 1.ytic Geometry, _,)3.;;e 271, Clnd Tjl=~ be u~:;ed
*Th:i.8 i::;
i:l tll2 2:c:n::~,~11d2r" of tl~J.i~-: ~j:Jf;er.
The
:lcre b2C '.u:.c, L--lt,il(' ~lE t ~,82S
tCl.lS Ll(;C:l' ..'.::is;-u i;::- Lt:':;f:l ii'j. tl:.c'· ~~"o=._2_~;'ri!"_~
~~c:";',
S8\rer;?. . 1
4JOl""'e
is
('e~'erel1ce
(.-:'~c.r'e
~rl
_,I
t~::;jl'~'Lll'_~
(J',~ .. :t~:: t
ti~ne~
::;e:1~~:JIE1
::~
t
.1"_
I
~~ ~:_ :}w~·
f~~~:)~:.:."2
:-:: 1.~,;11'· .. ~'"
-c~:,
._\) ':.r L. 2_ __
(~i. ~~
Fj .. J1.l~C '-r
. . L .;
L -<.8._."
~~Q
1_
.<::"lj
"1 (: .:.
c~-~e
~_(~\le~i..-
13
2.
A general
lor a hyperbola.
sour~e
i:jonsider a 'e.lcrc:!.l LYl!erbo1a of the follOlrU::LS form:
;t. = 1.
,..,
z2
(2.1)
a
The transverse
its center
~2
of
~XLS
the
of LIe can jusa te
'"
t::.
ot::.
t~ls
ori~tn.
of
::lxis
The
ie tte Z-a21G,
~yperb~li
(b,O)
(-t,O) are ena-points
hy{Je rbo la, ;:-JLG.l and -a are
ttl}. s
~raph
a~d
~lld
of (2.1)
,2
o •
(O,c) ::Lld (O,-c),
a,·~)
(O:~
.7=.,
l~A.
A hyperbola
of tIle C,Jne.
~ay
be
J.~.
tl'"
~roJuced
\ \\.
'~
by intersecttn
s
rl~ht
14
may oe
~roduced
i~terEectins
by
(2.3)
Hi th
x _. b.
(2.4)
Proof.
~erforD
ttc
intersect~an
b 2z
2
a 2 (b)2 + a 2 y'-?
1
z2
=
~l
Proof.
Con~lder
~~c
intE!rSect::'on of (;:.j)
c
=W + 0 2 ,
a1gebraicGlly.
2
r.. .
r>
b
2
hyperbola (2.1) produced by tie
~il.tLl
/"8'e c L:;
(2.4).
,E'rom t'il-,ure 8,
U·"(;list,-~ncc
::CrJjl t::,i:
Driz;in
t:J tr:.C :focus.
i~~
lts }roper tilrce-dirnensioll3.1 posit'.:Jl1 in £,i.;ure 9.
'/
I
b",,\.)
.\
_r:l
,i. .... c::
+
n~
I
(2.6)
8.
2
-Ja 2
Fo 2
+ b
+ b
:::
Ao 2
2 -
_2
2
Ao
:::
2
AO ::: c
:::
,r;
.
16
3.
'rhree-dinens:J~1:3.1
lnterpretatlo:1s of the ':lsymptotes
Jf a hyp e rbola.
Consider the hyperbolq
-L = 1,
z2
(3.01)
?
b
a~
x = b,
2
The asymptotes of (3.01) are y = bz and
a
The proof of this consists of solving
y= -
bz.
a
(3.01) for y and
sholving that, as z. increases Hithout lisit, yapproClches
±.!2£.
ThG,)ro:)f l.s c:),nqleted by SilO .ling that tlJ.8 perpena
dicular 1istqnce from a jeneral point on the hyperbola
to Y = :t bz appro'.lches zero a::: z geL3 infinitely large. *
a
1'",'~) three-di:icm~i::)Yl)l interpret'3.tiJ!lS for the asympo~
totes
.1
hyperbola will
no~
be 1i8918yed.
Thl? :~yperbola (3.01) is proluced by intLrsectL'lG
(3.02)
2 2 = a 2 x~ + a 2 yC0
b z
0
x-b.
*c e.,-.!..,
-I -,
'
Ana l
ytlC
8-eome t ry, page
115.
17
T~e
3sy~~totes
of (3.01) are
(3.04)
y
=±
bz, x-b.
a
~,s
Observe that (3.01) and (3.04) are ;'lritten
of
equatl~ns.
(3.04)
e~ch
The
re3~on
for this is thnt (3.01) and
describe a line (or lines) in space.
three-li:D(:l1Si)n9.1 cO·Jrdin:.lte
i~l
8.
8.~j
ttlt; inte1'2ection of t.'fO surfaces.
of two
3urf'~ces
r~rametric
:!ll..u:t be described
dhen t:-H:: eClu':1.tions
e~uatloll
1 -':
1
.L
fr~~
~
.Ille
•
1n
b8th
L -- 1
r.
a
A line
2
z2
(3.05)
~ysT,em
are written :lown to describe
ae .. ove tie second
jairs
b2
c.
and
(3.06)
y
= :t
bz.
a
(3.05) 'ud (3.06) are cy:~illders, and (3.0~), :1;Yl~erbo1ic
cY:~lnder,
in the
a)prOaCL8s t>:.e cylir:ders described 'oy (3.06)
:3 .. :U6
illi:.:.uner as the ilyperbo1a (3.01) a,,;:rJ:ches
its 'J.sy:nptotes (3.04).
FiGure 10.
(3.05) 3.:1d (3.06) :3.rs
Jrc:th'L
in
18
Ct
F, S
u.t'"
e
--
lO
~,
,
"
J...
d
~
I
"
The f::i tuation described in Fi2~cre 10 2,~; til,;; first
a hyperbola.
The second
inte~~r0t~tion
Jil~
nOJ be
displqyed.
Consider ti1e tr'Jces of (3.05) and (3.06) in the yzThe yz-tr ce of (3.05) has
plqTIe.
metric
tt~
follolng para-
equ}t~Qns:
z2
r
d
L
2
b
2
2
= 1,
= O.
x
The YZ-Lrace of (3.06) 1s describ6d by
(3.08)
Y
= = EE.,
a
T 'r'le
:'~;ra' ".)" '1"1S
;,0'
7'
0~
(~). ()"()
:=1.~
'.1
--'~,
x
::=
o.
(-J". I',.")
.~,.
v -'
are:1rcn·-.rn ln
l!'lc;ure 11
.
.J-
•
"
1..
_ t__,__ .:
."
.j..__
l; .
v
I
,,-
I
".
(l
[1 -
q'~
19
1/
'.J
I
(3 .U"'7)
•
-t~1.otawe
circ~lar
c'i.lld-
("7.J. u·'~)
'3.roun:l tae z-axis.
0
hyperboloid of tuo steets,
z2
is obtained.
ThE hyperboloi.d of t\·ro sheets (3.0S') is nO'rl " res ting tl
inside the
c~ne
(3.02), 3nd has Duch thE same relationship
to the cone as a typerbol3 has to its aEymptotes.
::L
------------------~~--------------~~
r.
~
The f9.c t
(3. (9) may be
I
, , \,
'.
that thE: cone (:5.02) is an :}.SY:=lP tote to
E'Jroven
in the same ,my in
1{h'LCh
Y
=±
OZ
a
20
were proven to be asymptotes for (3.01).
=k
z
(3.10)
thr:JUgh the cone
res1.~lts
the hyperboloid of
'l:1d
Pass a 01ane
The
~)heets.
t\i'O
of thL: :rLll be the equations of td8 circles.
Obtain the racli.us of
E:,~ch
Df
L18S(;
circlesmi
SihLi
that,
ss k approaches infinity, thE difference betaeen the
rad~i
of the two circles a)proaches zero.
I-I 1 th
Intersect (3.09)
,,2
X
~'\.
a
2 2
b k
2
b
(3.10).
2
~
2
b2
x -
aC 2
-
X
= 1
a 2 y2 = a 2 b 2
+ y2 = b2~k2
2
a
,..,
c ')
j_
a2
(3.11)
r.
np =
-../x 2
+ y2 = b..yk2
a
a2
Intersect (3.02) with (3.10) •
. 2.2
o
K
= a 2x 2
o 2lK 2 -
~
a
(3.12)
rc
X
"
C.
2
+ a y2
r.
+ yc.
2
=bk =-ix 2
+ y2
9.
rc 1s the radius of
t~e
circle of intersection of
(3.10) .ritb tile c)l1e, Ivhile r hp is the r:.;t.iil1~; of
of intersection of (3.10) Jith the hyuerboloid.
tLlE:
circle
21
(3.13 )
== blc
a
The
CJrr8
(3.0~~)
does
'etC";:,
~lS
a thre8-di::ler:.::~imal
asynptote fJr the hyplrbolold of two sheets (3.C9).
22
4.
nJ~-uniqueness
The
thc,~r
so~rce
of the
~
of
hnJerbola.
i:1tersectio:1S. *
describ2d i.
~
Fi~u~e
13a.
-----t-::?'t""'---I-___---t--..-
~;'It
is
C3ses
(C)
(h)
(a)
possible to dis _,l:~:{,r~'llc.
','i:ill
.t~'~'")od~-1ce
:J.
tIle
tr':. viu.~
S~.:l~C
1111
ini t.e tJ.~lLlber o i.'
i~2IT~Cl'lb:)1~~.
~';l)=:'
ex:a~-;lple,
rotate (2.h) arour~d t:1G Z-~;i:{~_r:;.
-~~,(; irlt~':;cct~;):_ Jf (2.4)
-,Ti +~,
(~., -),) "T~
'I ':
Y'1-"l"C"
(2_ • 1) , . ,"-,,
''','' "<>-,~ "i-le'c"C" .. ", "')\
u.o-':"
,...l .......
_"_ .....
J .... ,_".......
... ....;
1. _
C .• _
trle tl'lC
\;2 . .l. .l. G~::
~~ ~l~le s •
~_
co~;binitJ')ll:c:Tt'
..-:tt·~Jr·::~-:tt.~\E~~
,;-:_'"':--
~ ~-"o.
l':lnely:.5
''':~~t~'
cc:~~es
"\J
',..I ' - ' .....
C;~~lC~,:,
d.,oVt:·
•
~ ..
:)'"
.....
Id
\
,_. •
,.
-!- l~
:1.i:i-'J..<·t:d
l:_i'j:'e:"c~Lt
"V'Cl.,tE'~{
:< : '
1
-- -
.L) 0
)
•
Consio er tile cone
(4.01)
anJ
t~cc
l::l.lle
c. \J
( .),..,. •.n'·
z
~,
Iilf:;.~;I~ct
t11c
i'oJ_l~) IiLJ,,-
x(tafl e) + H.
di·~..:ra:n
,..
•
.
t-
.\
"
I
I-I
24
Solve for H.
:r-''Ji.2Jt -:3 haf; coorcUnates (J{b,O,C).
.c,
= x(tan
e) +
H
0
= lCb{tan
e)
+ H
~
(4.03)
H = -Kb(tan
(4.02)
(4.04)
illD.y
be re 1.Irittcn
z
= x(tan
The equ3. ti.ons of tr.e
::~o,;
~::.s
e)
f'ollo~TS:
8) - Kb(tan 9).
~13.!Le
ani cone :)e ing COYLsi dered are
now (4.0~) and (4.01).
Th2 anGle of rotati'JD may be
Let al1i;le XEA be 8.
ro~nd
by
ins~ectlng
TlleD:..'.n;;lE CED if: ::11:;:.0 8.
In a rcLltio:..'1, :C:o,;ever, the axes .:love--not
ttll;;
Construct
:":';-cure.
25
The x-axis
;i~11,;ove tL~'cu:.::h3.Ll
chl:le of (8 .- 90°)
*
in
ThETei'cre, the
'10°)
:;;
(4.05)
.
o
,I
I
(a)
cO':lside:::'cd
'o~cute
t~!
bE
tlJ8 ne;;sL_ve
8.L,::.;lc :le'\2U1'e1 fro,] the X-'~Xi2.
is a )o::..:itlve acute :1n18.
tion of e is twofold.
In
In F:gu1'c :;6b,
J1'~)osltive
F:i,~ure
e
loa, e
i::: :, nes:itive
First, consideration of 8 and tan
.. ~ ,-nh
,~ "-, 1'0021.I.o1.0n
~
~.
i ~.LeJ.eL
'-" -~ 1 t 1
. f B. ..
" .,.;. e """1
3.H,_, e U1.
;,o 1..·._:.<.l'1 -08 ,..- __
.:erl..::",
ne._,ativc acute an::;le.
i.fo.i8ver, the ;:::i.tuat:lou i;{iel'e e
is a posi ti ve :o;.cute 3..11[1e i[o the only one '\orh:~cb !':(·ed.s
to )e c011:.:.1.1e1'ed.
Ti1is ,\ii11 be:.ltslJLi~Ted fc:nrtJy.
.L"
....
.'
1 '
e
26
~econ~11~r
1t
_
.....
-,,'I',,',l -;
__ ...L
"oe
'.,,'.',I,(V"'I'-,(,
_
'-"
_
t.'lot
c_
1'l<.~,
.JL!'(...
y'r'eC+l" -,',('
"I+e
.............
-0
lJ,..\.,...,)
\... . . . .
(II
OJ~'.
()l)
' j . ' - .....
T,d-+-h
1~ ...... I., .. .1.
(i;..OLL) 1."iJl yield the :::"ome resu_It as the intej'sscting of
(4.0:) t·;ith
z = x [tan (-6)] - Kb [tun (-8)].
(4.06)
This 'dill ll:rri t the
(4.07) Lemma:
C~IE!,)i.1er3.bly.
of con:: :Lie-::1tion
SP"Hl
The Lame hyperbola
if:
obt~lined
from intersc:ctin;
(4.01) ~ith elt~er (4.0 4 ) or (4.06).
i'he
;~eco::ld
8i tU::I_tio;~
i:~
I':erely
t~J.c
first 81. tua tion
tlle :Call 0 .ring:
y' (cos 'J..O"0 0
)
-
z - Y I (sin H30 0
)
+ z' (co s
y
(4.08)
--
Z
I (
.
100°)
Sln
000 )
l
-'- '-,
-
yI
=
z'•
If the cone (4.01) is rotatej tt~0u~h an anGle of 18e o
ro t:l ted ar::nmd the x-s_:d s us ing
e1u3tlollb of rotation (4.03), the result is (4.06).
z -. x (tan e) - 1(b (tan e)
(4.09)
-z'
= x'(tan
9) - Kb(tan e)
z' - -x'(tan e) + Kb(tan e)
z, - x' [tun (-8)] -[Kb t'ocn
(-9)].
,
27
z == x(tan 6) - ICt:( t3n
Fro'] the :l::"Sc'Bs:on
rotation
o:~·
t:i8
e), 90 0
e (arctan ~.
(
:ceceJin; (1.+.05), the ::nc1e of
l':c order to ;lacc them
a:C3S
~Ln
re1:J__tj,OI:'S~liiJ to
the
~)roper
re1cd,ionsi.lip
Jicturec1 in FiGure 13'0) is (6 - :;'0°).
T~le ef}.uations of
ro ta tio:'l are
:{
== XI[COS
~~
--
(4.11)
(e
-
9 00 )]
-
ZI
[Sin
(6
-
9 00 )1 + ZI[COS (e
X'[Sin (9
_~ 0° )]
JO o )]
,7
•
:Then severa]. ;·;re11-k::1cnfll trisonom8 tric i den ti tL es are
applied,
(4.11) beCCllles
x == x'(sin e) + z'(cos e)
(4.12)
z - -x'(cos e) + zl(sin e).
,U::::ebraically
inter~;ect
o
== a"-z
(,'~.C;l)
';;ith (,+.04).
2
~
tan
(4.13)
c
, 2
=
0
(
e
-
- (2atan20K)Z
e
28
-;::.)
(4 e--.""
at a
i0
..........
'1
(_
"V"C'T'OJlie c',.rlinder
.l.J..~
e-2n~led
...J V
-
...
section.
the y-cL:is usi.n ..
"
T~ls
cO~1tainil1r:,
lr,.'-,";)e:cbola
_t~H~
__
-
thE:: equ'lt.iJns of
x,
(co~
Fi~u~e
is drm1D in
rot~,:tj.Ol1
12
e) + z I (sin e)
(-4.1~?).
-
(4.14) is a hyperbolic c;:lnder containing the
in a section parallel to the
contains the
hy~crbola
in
~
zy-~12ne.
s~ction
17.
hy~erbola
Since (4.14)
parallel
t~
thE
29
2
Z
-=-)f
2
(4.15)
- a2 s
tan
ta 2
n
'(Si
8)]2 - 2a2bK(COB 8)2' '" O12b2[2 + a y2
2
2
8
[b ( n 8) - 8 ] (co$2 )2 ,2 - 2a2bE(cos 8)z' " ,,2b K2 + a y2
2
[b2(Sin2s)
82(C0028)]z'2 - 2n2bK(cos 8)z' _ a2b2K2 + a 2y2
2
FiSu:re
-------T----~~---
z" '" z' -
(4.16)
~.1
,,2bK(cos EI)
02CSin2s) _
y" .:::- yr.
___1 ____
8
2
(coS
8)
2
30
Translate (4.1~) ~ith equations of tr~nslution (4.16) •
z .. 2
(4.17)
.2.. (.
)2
ab KSln e)
(
(4.17) is a 2yperbola of form
z2
"~i
a
By Theorem (2.2),
t~~e
cone
2
a (O)S
(4.19)
th
o
(4.17) ~ay be vroiuoei by int8rsect~ng
a)
\"li
c-
"
- L = 1.
2
ttlC::
plane
2
~
e)
~.
c:.
n
ZC
31
Gone:
(1)
21ane:
z
= x(t3n
e) -
Kb(t~n
8)
(4.20)
Cone:
(4.13)
(2 )
2h;ne:
The~;e
tJO si tlFlti ms
'irE:
(!-t.19)
obv~~)uslyHstinct.
cone froll1 (1+. 2C ) {1.) is :2
verte:: J.rl:::.lE
:)1-
t,w
C:J112
t'1l b, -.,:'-:.1 'e the
'5:'
frOin (4. c'C) (?) is
'3.:CC
3.
.____-+_---::*""''--'II~'----~x
""
~-----
(Q)
The verte::
(b)
33
_0
',- 'D ~L l'.t i.
r'
')
a C( CrJS c.;.::,)
v
(
J,
r'
-~
\
'+.c.))
a
--2/--~~
,,'
D 2J.:.2
!
;D
'"'
'2'-'2
!Jl1.J::::t
2
no . ; De :i"ound.
re 190.
(L~.24)
r
tan
e
+ KO,
r.
Z
tJ the
',8
1ar
r,
J.
, __ He ot'
;'"
'" 1.1 .
'T ~1 e
8..;.-
P ::' i) -,~ ~_l·a·_I. t 1. :J 11
-
J~: L: ~~
-t.,O
l.:C
, t
-?
) ,~: - ,.
2( . 2 ~ )
bSln
')
-
'~l
'--
r £;e ts
~,
'-( C'J'':''''
,:',..,\
I
~
~
(
,
t)(;tteI'
~'.. 4
( :.~' • '.')
(4.26)
y ;::
;1;
(4.26)
~.
9.nd
of z ::: x( -;:;,3.::1 e) - Lb (t}ll 8)
~;ect~n
x
:=
i .",
± Oz,/;,'.
~iiC
«)rj'Y'l"l':
en
:jI
tCe
c~
.V.'.
' ) e f':;·.·1d,:l, tll the
~oj~
::'.nc(
(4.27 )
(4.28)
D"c.
:=
(
E 2 -.-
ta~
e
( re
\.X r-....l.,
t.~.'
+ Kb, 0,
+
l'~o
r)
( 4-.2 ',) )
,
..L.; .. 1'1
G..:tl. ...
(4.29)
J:", ..
L
-
- ,_ L 0 c~ ,( c:.l.ll. e~
c... - 0 ( tS.n .)
T'"
[
r
.J
,
1
a
(4.30)
.,
,
J,.-",
c:
0,
-
__~I~~b~~_·~.(~t~a_n_2_e~)~_, 0, -
(4.)1)
* 'r'~ncsc}rDceiure2
c'-'. tj.oru: of
.
•
1~VQ1Ve
jurpose if thef::e
v8ry
lon~
~nd
:.:;1:11.;:),-11:1. ti'Jtls.
It
;:erE: re)ro~luc8d l1i;l~C.
c~10e i)l":;.i
c
e~tcnsve
n.
t::..
35
Dc:'"'}~?_ i
S
2~J~: I'O: Ll~_\ t (: j_ y
r
.... - "c
L
v311
.~ 2'
'-~ild.
2
~
l_L
-- li:n
r-+oo
'J":'
t ("e y- c c)() r l i 11, te
e
+
te r:c,s 1. n
::.t.llj
C Gll S
teI~2l~3
,:L~,
~~!l
T'
t
c~.ll ts
1"1
C::;i:lst::.nts
J
1
-: lim
1:'+00
+ I
1
2
2e
r + const2nts
r + const2uts
1
.
= ~J..lfn
a~ 3 .t n 2n
~
~--~--
1
+ 1
t,~lll "
ee
1 + tan
(4.32)
__ b
2 (tan 2 8) - a 2
a
222
+ a (taLl e)
the -::::roof
(i+. 33 )
2
e
Consi~er
the follo{ing.
~)
s'
-- 0
')
b~(sL1'-8)
b 2 (;-0 iJ} ;~e
)
')
~
r
Y12 e) - b~(tan~e)(l - sln~e)
.,2(Cl·
\. '-' AJ.l
c.....-_
-
a
2
_ b~(sln2e) + b2(sin28)(ta~2e)
(4.34)
a 2 b 2 (tan 2 e)
_,/+
a 2 (cos 2 e) - a 2 (sin 2 e)
= a2[b~:(~in2e) + b 2 (sin 2 e) (t·:m 2 S)
r'
a~(cos
b 2 (tan 2 e) _
0.
2
= b 2 (sin 2e)
2
e) -
-.2
?J
?
a-(Ein-8~
- 32(cc~28)
2
(1 - cos e)
32
19b is
3.
reLiresent~ ti.on of
If it
(4.20)(:1).
oe
c~n
sho~n
thJ. t
(4.21)
(
~?)
4.c~
I
)e t:-le e L~
::.>:.:~
be
t~e
~~Tl1is
;.T('u-"l
~L"
liU
','[81"'8
-2;-2
/:1)., C,
•
t 10 :0 1.:8 -'1
This
~.L...L
~s
e2Gily seen to
fallowIng:
l.:::;
allJ~;:~,ble.
1"'·\"8
1l..'_'
~:l.e
List'~L~lCC
~l~ebr:-lic
TlIE. :Jere
"U-·Y>CC"",+.-cr'
'-"..L,-... I. t.J\",~",_ J.
..
(.)
;:)<:~.._ne
S~l,;~pe,
Qct.,-E.:211
+'1'+
'v'l ..~ U
or
si~rlil_:r
ver~tic(::j~..
~'-~_11j_'i0_J '~.til,--~ns
1~re8ervatiolJ. of (4.21)
"",",'8,.'00'.",<=:
..... ~:,
_
'-".,j·_~I""""
;,'iJ:).lJ
(-L')
.,
'"r'O'1
_" ... J'__
ll~r)t;1"Ool~_}.8.
(')1
l __
(
~ir~cc
-.2;:-:0 "besTL
):re~3erv-ed,
~;.:Q~1
t~le
tl1~-.;.t
t~e
~·ome
f:CC21ltl~ici
t:;r
37
5.
The uniqueness of hy)crbola .Jroductlon
lIT
a
COa8.
bola obt9ined by sectioning the cone
2
r)
a xC:
+
(5.01)
_~lane
z
(5.02)
= x(tan
e) - K1b(tan e)
can not be obtainej by any other section of (5.01),
exclu1iD~
the trivial
ca[~
where (5.02) is merely
roLlted.
The procedure will be as follows.
iug
pla~e
(5.03)
Another section-
will be chosen.
z
= x(tan ¢) -
K
2
b(tan
¢)
The dl s t::.nce be tIft en t:'18 vert i ce s of the rlype rbola .9rojuced by (5.01) and (5.02)
~ill
be forced to be ttc sqme
produced by (5.01) and (5.03).
~''J:rm
DEIDC,
be found
as used in the Jreceling
for eact hy}erbola.
It will then be demonstrated that
38
the onl;1
H'3.y
in .lLich these ra tiDS
IW<.y
be equal (and
hence, the on\y \[a,/ in '..'lich the hyperbolas r:;)y be the
S'<iTIE:)
of
1;:;
lor
e
to De eql..1=\l to ¢.
L!.Ytje:roola~)roluctio::l
Therefore, unL~ueness
in ,:;. fi::81 c)ne,riLL be )roven.
OO;J.sider I:'isure 20.
If :~
=±
3.x/b o.re i::"llgebraic::;.lly intcr;3ected..,i th (5.02),
tie coordinates of A,J. and
3~
..1
~re
found to be
(5.04)
The Pythagorean relationship then yields
(5.06)
, 2
+ 0
39
CODsider Figure 21.
The coordin:l tes of A2 and 3
2
may be found.Ti th the same
procedure that was used to find the coordinates of
A~
.L
and
(5.03)
Another a:9plication of
t~e
Pytha~orean
rel:ltio!lShip yields
(5.09)
Notice from Fizure 20 and ?igure 21 the functions
of Kl and
1(2"
'They are t:."lC para.lleters '.ihich, if -:::1.de
lar2;er or smaller, cE'.ve tIle .abil':' ty to 81:1.de the planes
40
in a direction perpendicularly away from or toward the
zy-plc.ne.
Set T'"B-, equal to An 3,..., , and find O~lt HLat K,) must
~
~
~
be in order to insure
~
the distances
tha~
:rJ3,
~
~L
ya
2 + b2
2aKI b( tan e)
a 2 _ b 2 (t9.r- 2 e)
L
be~;een
the
= A2B2
rYa 2
= 2aK,;!b(tanr;tJ
+ b
2
a 2 _ b 2 (tan 2¢)
=
a
2
2
2
b (tsn e)
(5.10)
,.
e ~..::.)..) .<~',
-;
( 5 •1. 0) .Lc
. . . 1.1 t·1:: . . 1 ...Ly
{". . piece of "mathematical :1lachinery" ,
r'
~'lhic;:l
C" C'
r,
'0
"\T1.1l, upon (Jur choicE of
all 2n;~le
f; for a sectioning
plane, slide this ;lane to the proper position--the posi-
-
tion where AIBl
If the
= A-2 B2 •
~yperbola
obtained at the inters:ction of
(5.01) and (5.02) is cO:'1s:'d.ered t:J be fi:::ed., 8.nd if t:'le
intersection of (5.01)
~itt
a fa~ily of hyperbolas since
follo~in~
situ~tion
(5.03) is cansiicred to be
¢
is arbitrary, then the
exists.
*Observe that, if e = f;, t~en Kl
= K2 •
41
Cone:
(5.11)
2 2
Plane:
Cone:
= a 2x 2
b z
z
b
= x(tan
2 2
z
2 2
+ a y
e) - Klb(tan e)
= a 2x 2
(5.12)
2 2
+ a y
ISl = [a 2
H he re
2
- b ( tan 2('~nilil (tan e)
,<\~1[8.?
[(tun
(5.11) defines
,
2
2
b (tan
e)]
~ sih~le hy~erjola.
(5.12) defines
Gy~erbolas
from (5.11) and
a family of hyperbolas.
- ''"') .jjve ttc'
( :::;.JC::
..
All
clistance~
oet . ;een t:leir vertices
-2/-2
FiLd tbe
rltios D2 E 2 I 'D 2 C2
~;
in the jrecedinc section.
~~d
~,.:.~
---2-?
Dl.;"l!
~
ID 1'-'1
n
- 38 used
',et -'-',-'e
Cll'b"cr-i-,+
"}," correspond
G ...
t.· v
........
...L....;.:),,}..;
From (11-.32),
2
(5.13)
a •
-2
Dr-.C
c: 2
? + a 2(t an 2~,)
at;
Again from (4.32),
(5.14)
In:3p ect tLe follo',ring equ3.li ty.
-2
D2E2_
-2
D2 C2
e~ual.
........
42
of ::111 of' tLe h;n.Jerbo13s -;)8in:.;
It JU.l no,' be
Given a
cow::L~E'red
Hill be equal.
br a cOl1siJcI''J.tion of (5.15), tb:,t,
::'-~lolm,
praluced by a 8-anLled section and a
h~9crbo12
::; ;~c ticnin,.::.;
the
Sl~E
cone and havinc the distances
rc.s~;ectlve
(5.16)
their
betwee~
vertices equal,
b
2
2
~
(tan c 8) -::t
a 2 + a:::(tan 2 e)
2
2
[tan e + (tan e) (tan 2¢n)] '0
2
-
(t;).n 2 0n )a
2
= [tan 2¢n + (tan 2¢n)( tan 28 ~ b
r,
2
'""l
(tanCe)ac.:
~!-.
Equ9_tin:j coefficients;' yields
c"
tan ,0 n
(5.17)
¢n
= tan 2 e
*i(
= 8.
t:lL reeder that
a unique
~ectl~n.
oj ..
A corollary ai' t:le FUl1(L.,nc Yl t
this to be done.
i~*
Recall
th~t
these
positive acute
~nglcs
~nGles.
j
1 -;"heorem of _U:;e bra al10'''{::3
h3ve been restricteJ to being
43
6.
:?ro j ections of fllrtb.er
A General SClUrce for :".
Section 2.
:;'nis sJurce
~i:JS
~r:Jl"'k
to be .jane.
~cy;:,erbol::.
lus been found in
)roved not to be
un:L~ue
in
Sectlon 4, and the existence of at least two sources for
every "yptrbola
vl3.S
displayed.
In Section 5,
i~
was proven
specific hyperbo12 may be obtained from a specific
th~t
2
cone
~ith
~he
a unique section.
next question is whether or not an infinite
number (If co:nbilLitioJ.1s of _"lane and cone "lIhicll produce
a
spec:fic hYgerbcla may be iisplayed.
certain tjst
all of
~h0
t~lc
co~ic
situation daes exist.
f8cti)ns.
nalul tl~ll t.
the rest )f the; conic sections.