39

I GEWMETRIA ston R3 - PRWTOBAJMIA SUSTHMATA

I EUJEIES ston QWRO

Mia eujeÐa kajorÐzetai:

1. apì dÔo shmeÐa

2. apì 1 shmeÐo kai th dieÔjunsh

3. wc TOMH dÔo EPIPEDWN (sto q¸ro)

PERIPTWSEIS (1) kai (2)

Epilègoume mia ARQH O sto q¸ro kai

orÐzoume ORJOKANONIKO SUSTH-

sq ma MA {0, ⃗i, ⃗j, ⃗κ}

−→
A ≡ OA = ⃗rA diˆnusma jèshc
−→
OA(xA , ψA , zA )

−−→ −→ −−→
M OM = OA+ AM
tuqaÐo shmeÐo tìte
−−→ −→
ìpou AM = tAB, t ∈ R
sq ma
−−→ −→ −→ −→
OM = OA + tAB ìpou tAB = t⃗u
−−→ −→
OM = OA + t⃗u
'Ara ⃗rM = ⃗rA + t⃗u DIANUSMATIKH EXISWSH

⃗rM = ⃗rA + t⃗u, t ∈ R

−→
⃗u = AB = ⃗rB − ⃗rA = (xB − xA , ψB − ψA , zB − zA ) ≡ (α, β, γ)

(x, ψ, z) = (xA , ψA , zA ) + t(α, β, γ)



 x = xA + tα
(Σ) ψ = ψA + tβ PARAMETRIKES EXISWSEIS


z = zA + tγ
An upojèsoume ìti ta α, β, γ ̸= 0 lÔnoume wc proc t

x − xA ψ − ψA z − zA EXISWSH EUJEIAS sto QWRO

= =
α β γ (SUMMETRIKES EXISWSEIS)

H eujeÐa dièrqetai apì to (xA , ψA , zA ) kai eÐnai parˆllhlh proc to ⃗u(α, β, γ).
PARADEIGMA Na brejeÐ h exÐswsh thc eujeÐac pou dièrqetai apì to A(1, 1, 1) kai eÐnai pa-

rˆllhlh sto ⃗u(1, 1, 1).

x−1 ψ−1 z−1
= = ⇔ (x = ψ = z)
1 1 1
{
dièrqetai apì to B(0, 0, 0)
Paristˆnei eujeÐa sto q¸ro (ε)
∥ sto ⃗u(1, 1, 1)
40

I PORISMATA
(ε1 ) (A, ⃗u)
(ε2 ) (B, w)
⃗

• (ε1 ) ∥ (ε2 ) ⇔ ⃗u, w

⃗ SUGGRAMMIKA ⇔ ⃗u × w
⃗ = ⃗0

• (ε1 ) ⊥ (ε2 ) ⇔< ⃗u, w

⃗ >= 0

I ASKHSH: Na brejeÐ h sqetik jèsh twn eujei¸n

(ε1 ) x = ψ = z
x−1 ψ−2 z−3
(ε2 ) = =
1 3 3
Sqetik jèsh 2 eujei¸n:

parˆllhlec (ε1 ) ∥ (ε2 )

temnìmenec (ε1 ) ∩ (ε2 ) ̸= ∅
asÔmbatec

• PARALLHLIA
}
(ε1 ) ∥ (1, 1, 1)
grammikˆ anexˆrthta ⇒ (ε1 ) ∦ (ε2 )
(ε2 ) ∥ (1, 3, 4)

• 
SHMEIO TOMHS 'Estw M ∈ (ε1 ) ∩ (ε2 )
 t=x=ψ=z
Yˆqnw x−1 ψ−2 z−3
 s= = =
1 3 4
ProkÔptei sÔsthma 5 exis¸sewn me 5 agn¸stouc.

}
x=t s= t−1
t−1 t−2 1
tìte
1
t−2
⇒ = ⇒ 3t − 3 = t − 2 ⇔ t =
ψ=t s= 3
1 3 2
1 1
gia t= prokÔptei s = −
2 2
z−3
s= ATOPO (z = t =
1
2
, s = − 12 ).
4
'Ara oi eujeÐec (ε1 ) kai (ε2 ) eÐnai ASUMBATES.

B' TROPOS: TUPOS pou dÐnei thn APOSTASH DUO EUJEIWN STO QWRO

I ASKHSH
{ 1
x+ψ+z =1
(Σ) Na deÐxete ìti (Σ) paristˆnei eujeÐa kai na brejeÐ h kanonik morf .
2x + 3ψ + 4z = 0
{ {
x + ψ = 1 − z (−2) −2x − 2ψ = −2 + 2z (+)
⇒ ⇒ −ψ = 2z + 2
2x + 3ψ = −4z 2x + 3ψ = −4z
 41

ψ = −2(z + 1) , x = z + 3, z ∈ R
{
ψ = −2(z + 1)
'Ara z∈R
x=z+3

 ψ =z+1 x−2 ψ z+1
−2 'Ara = =
 x−2=z+1 1 −2 1
'Ara h (ε) dièrqetai apì to A(−2, 0, 1) kai eÐnai parˆllhlh sto ⃗u(1, −2, 1).

I ASKHSH 2

2x + 3 2ψ 4z − 1
= = Na tejeÐ se kanonik morf
5 1 8
x+ 3
2 ψ z− 1
4
5 = 1 =
2 2
2
'Ara dièrqetai apì to shmeÐo B( 32 , 0, − 14 ) kai eÐnai ∥ ⃗u( 25 , 12 , 2)

I EPIPEDA sto QWRO

(ε) ⊥ (Π)
H (ε) eÐnai ∥ sth ⃗ℓ : (ε) ∥ ⃗ℓ.
dieÔjunsh

Opìte ⃗ℓ ⊥ (Π). JewroÔme A stajerì

sq ma shmeÐo tou epipèdou kai M tuqaÐo sh-

meÐo tou epipèdou.

−−→
AM ∈ sto epÐpedo (Π) kai isqÔei
−−→ ⃗
AM ⊥ ℓ
−−→
AM ⊥ ⃗ℓ

−−→ −−→ −→ −−→ −→

isqÔei < AM , ⃗ℓ >= 0 ⇔ < OM − OA, ⃗ℓ >= 0 < OM , ⃗ℓ >=< OA, ⃗ℓ >= p, ìpou M
tuqaÐo shmeÐo kai A gnwstì kai stajerì shmeÐo.

−−→
< OM , ⃗ℓ >= p paristˆnei epÐpedo sto q¸ro

ìpou ⃗ℓ(α, β, γ), −−→

OM (x, ψ, z), p = stajerˆ

< (x, ψ, z), (α, β, γ) >= p ⇔ αx + βψ + γz + δ = 0 (δ = −p)

EXISWSH EPIPEDOU αx + βψ + γz + δ = 0 me |α| + |β| + |γ| ̸= 0

Paristˆnei epÐpedo ⊥ sto ⃗ℓ(α, β, γ).

I PARADEIGMA 1
ZhteÐtai to (epÐpedo) p pou dièrqetai apì to shmeÐo A(1, 1, 1) kai eÐnai kˆjeto sto diˆnusma

⃗ℓ(1, 1, 1).
Apì thn exÐswsh : 1·x+1·ψ+1·z+δ =0
42

BrÐskoume to δ : 1 · 1 + 1 · 1 + 1 · 1 + δ = 0 ⇒ δ = −3
Opìte EXISWSH EPIPEDOU: x + ψ + z = −3

I PARADEIGMA 2
paristˆnei epÐpedo p ⊥ ⃗ℓ(1, 2, 3)
kai anazhtoÔme to shmeÐo apì to opoÐo

x + 2ψ + 3z + 4 = 0 dièrqetai dhl. (x, ψ, z) pou na

triˆda

ikanopoieÐ thn exÐswsh. Gia x = ψ = 0

4
prokÔptei z = −
3
−4
'Ara dièrqetai apì to A(0, 0, 3
).
sq ma

I EXISWSH EPIPEDOU POU ORIZOUN 3 MH SUGGRAMMIKA SHMEIA

Zhtˆme thn exÐswsh tou epipèdou pou orÐzoun ta


sq ma A(xA , ψA , zA ) 

B(xB , ψB , zB ) A(xA , ψA , zA )⃗ℓ =?(⊥ (Π))


Γ(xΓ , ψΓ , zΓ )
BrÐskoume ⃗ℓ ⊥ (Π) : ⃗ℓ = (−→ ←−
AB) × (AΓ)

−−→
< ⃗ℓ, AM >= 0 ìpou M tuqaÐo shmeÐo tou epipèdou, M (x, ψ, z)

PARADEIGMA
−→
AB(−1, 1, 0)
A(1, 0, 0) −→
AΓ(−1, 0, 1)
B(0, 1, 0)
Γ(0, 0, 1) −→ −→
AB × AΓ = (1, 1, 1) ⋆ = ⃗ℓ
i j κ
⋆ prokÔptei apì thn orÐzousa anaptussìmenh wc proc thn 1 h
gramm −1 1 0
−1 0 1
−−→
< ⃗ℓ, AM >=< (1, 1, 1), (x−1, ψ−0, z−0) >= 0 ⇒ x + ψ + z = 1 EXISWSH EPIPEDOU

−−→
I EKFRAZOUME KALUTERA ton tÔpo < ⃗ℓ, AM >= 0

−→
AB = (xB − xA , ψB − ψA , zB − zA )
−→
AΓ = (xΓ − xA , ψΓ − ψA , zΓ − zA )
⃗ℓ = − → −→ ( )
AB × AΓ = Dψz , −Dxz , Dxψ
−−→
AM (x − xA , ψ − ψA , z − zA )
−−→
< ⃗ℓ, AM >= (x − xA )Dψz − (ψ − ψA )Dxz + (z − zA )Dxψ = 0
 43

x − xA ψ − ψA z − zA
⃗ −−→
< ℓ, AM >= xB − xA ψB − ψA zB − zA = 0
xΓ − xA ψΓ − ψA zΓ − zA
EXISWSH EPIPEDOU sto QWRO

x ψ z 1
xA ψA zA 1
=0
xB ψB zB 1
xΓ ψΓ zΓ 1

I SQETIKH JESH 2 EUJEIWN sto QWRO

(Π1 ) ←→ (A, ⃗ℓ1 )

(Π2 ) ←→ (B, ⃗ℓ2 )
• ⃗ℓ1 ∥ ⃗ℓ2 ⇒ (Π1 ) ∥ (Π2 ) PARALLHLA TAUTIZONTAI

• ⃗ℓ1 × ⃗ℓ2 ̸= 0 ⇒ (Π1 ) ∩ (Π2 ) ̸= 0 sq ma

(ε) = (Π1 ) ∩ (Π2 ) ∥ (⃗ℓ1 × ⃗ℓ2 )

−−→ }
(Π1 ) ←→< AM , ⃗ℓ1 >= 0
−−→ SUSTHMA POU KAJORIZEI ta shmeÐa tom c.
(Π2 ) ←→< BM , ⃗ℓ2 >= 0
44

I EPIPEDA sto QWRO

}
αx + βψ + γz + δ = 0
(Π) : PARISTANEI EPIPEDO ⊥ ⃗ℓ
⃗ℓ = (α, β, γ) ̸= 0


 GnwrÐzw èna shmeÐo tou epipèdou

“PROTUPO EPIPEDO” kai èna diˆnusma sto opoÐo to



epÐpedo eÐnai KAJETO

I PARADEIGMA 1: TOMH EPIPEDWN

Na brejeÐ h exÐswsh thc tom c twn epipèdwn

(Π1 ) x + ψ + z = 1
sq ma
(Π2 ) 2x + 3ψ + z = 4
Dhlad zhteÐtai h exÐswsh thc eujeÐac (ε).
oc
1 TROPOS: GEWMETRIKOS
x − x0 ψ − ψ0 z − z0
H tom eÐnai eujeÐa thc morf c = = ìpou (κ, λ, κ) ∥ (ε) kai
κ λ µ
(x0 , ψ0 , z0 ) ∈ (ε).

'Estw
⃗ℓ1 ⊥ (Π1 )
⇒ (⃗ℓ1 × ⃗ℓ2 ) ∥ (Π1 ) ∩ (Π2 )
⃗ℓ2 ⊥ (Π2 ) | {z }
(κ, λ, µ)
}
⃗ℓ1 = (1, 1, 1)
⃗ℓ1 × ⃗ℓ2 = (−2, 1, 1) = (κ, λ, µ)
⃗ℓ2 = (2, 3, 1)

BrÐskoume shmeÐo tom c thc (ε)

ψ0 = 0
x0 = 3
x0 + z0 = 1
z0 = −2
2x0 + z0 = 4

'Ara to shmeÐo (x0 , ψ0 , z0 ) eÐnai to (3, 0, −2). x−3 ψ z+2

= =
'Ara h eujeÐa (ε) eÐnai ((ε) = (Π1 ) ∩ (Π2 )): −2 1 1
oc
2 TROPOS

BrÐskoume 2 shmeÐa thc tom c, èstw ta (x0 , ψ0 , z0 ) = (3, 0, −2)

−1
gia x1 = 0 tìte (x1 , ψ1 , z1 ) = (0, 23 , −2
).

3
Tìte (κ, λ, µ) ∥ (x0 − x1 , ψ0 − ψ1 , z0 − z1 ) = (3, − 32 , − −3
2
) = − (−2, 1, 1)
2
3
(κ, λ, µ) ∥ − (−2, 1, 1)
2
 45

I PARADEIGMA 2
Na brejeÐ h exÐswsh tou epipèdou pou dièrqetai apì to (1, 1, 1) kai eÐnai ⊥ sta (Π1 ), (Π2 )
{ }
x+ψ+z =1
(Π) :
2x + 3ψ + z = 4

AfoÔ (Π) ⊥ (Π1 ), (Π2 ) ⇒ (Π) ⊥ (ε) = (Π1 ) ∩ (Π2 ) ìpou (ε) ∥ (κ, λ, µ) ⇒ (Π) ⊥ (κ, λ, µ)
(Π) : κx + λψ + µz + ν = 0 kai epeid (1, 1, 1) ∈ (Π) prokÔptei ìti ν = κ + λ + µ
Opìte (Π) : κx + λψ + µz + (κ + λ + µ) = 0

I ASKHSH

x−1 ψ−2 z−3

DÐnetai to epÐpedo (Π): 2x + 3ψ + z = 4 kai h eujeÐa (ε): = =
1 2 3

Na brejeÐ mÐa eujeÐa tou (Π) ⊥ sthn (ε) kai na thn tèmnei. Pìsec tètoiec eujeÐec upˆrqoun

oc
1 TROPOS

x − x0 ψ − ψ0 z − z0
= = ìpou (x0 , ψ0 , z0 ) ∈ (ε), (κ, λ, µ) ∥ (ε)
κ λ µ

(x0 , ψ0 , z0 ) = (Π) ∩ (ε). Jètoume touc Ðsouc lìgouc t

x−1 ψ−2 z−3
t= = =
1 2 3
sq ma x=t+1   7
ψ = 2t + 2 2(t + 1) + 3(2t + 2) + 3t + 3 = 4 ⇒ · · ·t = −

 11
z = 3t + 3

x0 = 1 − 117 4
= 11 

7 8 4 8 12
'Ara ψ0 = 2(− 11 ) + 2 = 11 (x0 , ψ0 , z0 ) = ( , , ).
7 12

 11 11 11
z0 = 3(− 11 ) + 3 = 11
Oi eujeÐec pou eÐnai kˆjetec sthn (ε) kai dièrqontai apì to A. BrÐskontai se èna epÐpedo (Π1 ).
′
To (Π1 ) eÐnai ⊥ sthn (ε) (ìpou (ε) ∥ ⃗ℓ1 = (1, 2, 3)). H zhtoÔmenh eujeÐa (ε ) eÐnai h tom twn
(Π) kai (Π1 ). (ε′ ) = (Π) ∩ (Π1 ). 'Omwc (Π) ⊥ ⃗ℓ2 = (2, 3, 1). 'Ara (ε′ ) ∥ (⃗ℓ1 × ⃗ℓ2 ) = (κ, λ, mu).
'Ara (κ, λ, µ) = (⃗ ℓ1 × ⃗ℓ2 ìpou ⃗ℓ1 = (2, 3, 1) ⊥ (Π), ⃗ℓ2 = (1, 2, 3) ∥ (ε).

I APOSTASH SHMEIOU apo EPIPEDO

PROBLHMA DÐnetai epÐpedo (Π): αx + βψ + γz + δ = 0 kai èna shmeÐo P (x0 , ψ0 , z0 ) ektìc

epipèdou. ZhteÐtai h apìstash tou P apì to (Π) dhl. d(P, (Π))

46

APOSTASH: upˆrqei eujeÐa (ε)

pou dièrqetai apì to (P )
sq ma kai eÐnai kˆjeth sto (Π)
to opoÐo tèmnei sto shmeÐo P ′.
P ′ =probol tou P sto (Π).

'Estw A tuqaÐo shmeÐo tou (Π).

A ∈ (Π)
(P P ′ ) ⊥ (Π)
(P P ′ ) ∥ ⃗ℓ(α, β, γ) giatÐ ⃗ℓ ⊥ (Π).
′
Sto ORJOGWNIO TRIGWNO AP P

−→ −→
−−→′ −→ < AP , ⃗ℓ > ⃗ < AP , ⃗ℓ > ⃗
∥P P ∥ = ∥prob⃗ℓAP ∥ = ∥ ℓ∥ = ∥ ℓ∥
< ⃗ℓ, ⃗ℓ > ∥⃗ℓ∥2
−→
−→ ⃗ ∥⃗ℓ∥ | < AP , ⃗ℓ > |
= | < AP , ℓ > | =
∥⃗ℓ∥2 ∥⃗ℓ∥
Dhlad
−→
−−→′ | < AP , ⃗ℓ > |
∥P P ∥ = (2)
∥⃗ℓ∥
Ekfrˆzoume to A me paramètrouc. 'Estw A(x1 , ψ1 , z1 ) ∈ (Π).


 A(x1 , ψ1 , z1 )
P (x0 , ψ0 , z0 ) Apì thn sqèsh (2) prokÔptei:


ℓ(α, β, γ)

−−→ < (x0 − x1 , ψ0 − ψ1 , z0 − z1 ), (α, β, γ) >

∥P P ′ ∥ = √
α2 + β 2 + γ 2
∈(Π) ˆra −δ
isoÔtai me
z }| {
αx0 + βψ0 + γz0 − (αx1 + βψ1 + γz1 )
=| √ |
α2 + β 2 + γ 2
−−→ |αx0 + βψ0 + γz0 + δ|
∥P P ′ ∥ = √
α2 + β 2 + γ 2

APOSTASH SHMEIOU apo EPIPEDO

 47

EFARMOGES sto QWRO R2

I APOSTASH SHMEIOU apo EUJEIA ston R2

EujeÐa ston R2 (αx + βψ + γ = 0, |α| + |β| ̸= 0) (ε) ⊥ ⃗u ìpou ⃗u(α, β)

αx0 + βψ0 + γ
sq ma d(P, (ε)) = √
α2 + β 2

I PORISMA: EMBADON TRIGWNOU sto R2

A(x1 , ψ1 )
B(x2 , ψ2 ) E(ABΓ) =?
Γ(x3 , ψ3 )
1oc
TROPOS: EPIPEDO ⊂ R3

(x1 , ψ1 ) ≡ (x1 , ψ1 , 0) 
 1 −→ −→
(x2 , ψ2 ) ≡ (x2 , ψ2 , 0) opìte E(ABΓ) = ∥AB × AΓ∥

 2
(x3 , ψ3 ) ≡ (x3 , ψ3 , 0)
oc
2 TROPOS:

{ −→
BΓ(x3 − x2 , ψ3 − ψ2 )
⃗u(−(ψ3 − ψ2 ), (x3 − x2 ))
sq ma
−→
⃗u ⊥ BΓ = èqoun eswterikì ginìmeno

mhdèn.

x − x3 ψ − ψ3
BrÐskoume thn (ε) = 0 ⇔ (ψ2 −ψ3 )x+(x2 −x3 )ψ−x3 (ψ2 −ψ3 )+ψ3 (x2 −x3 ) = 0
x2 − x3 ψ2 − ψ3
1
IsqÔei o tÔpoc: E(ABΓ) = ∥BASH∥ · ∥UYOS∥
2
−→
BASH= ∥BΓ∥

BASH= ∥d(A, BΓ)∥

1 −→
E(ABΓ) = ∥BΓ∥∥d(A, BΓ)∥ =
2
((
1√ (((( 2 |(ψ2 − ψ3 )x1 − (x2 − x3 )ψ1 + ψ3 (x2 − x3 ) − x3 (ψ2 − ψ3 )|
((ψ
2(+
= (x −
( (
x( ) − ψ ) √ ((
2((( (((( 2
3 2 3 2
(x −( ((ψ
x()2(+ −ψ )
((((
3 2 3 2

x1 ψ1 1
1 x1 − x3 ψ1 − ψ3 1
= = x2 ψ2 1
2 x2 − x3 ψ2 − ψ3 2
x3 ψ3 1
48

x1 ψ1 1
1
'Ara E(ABΓ) = x2 ψ2 1
2
x3 ψ3 1
I PORISMA: TrÐa shmeÐa eÐnai SUGGRAMMIKA sto EPIPEDO

x1 ψ1 1
an kai mìno an x2 ψ2 1 = 0
x3 ψ3 1
−→ −→
Dhlad AB × AΓ = ⃗ 0
Sto epÐpedo dÐnetai mÐa eujeÐa (ε) h opoÐa qwrÐzei to epÐpedo se dÔo tm mata (hmiepÐpeda)

⃗u0 ⊥ (Π) ∩ (ε)

sq ma
−⃗u0 ⊥ (Π) ∩ (ε)

f (x, ψ) = αx + βψ + γ f : R2 → R
f (x, ψ) = 0 ⇔ A(x, ψ) ∈ (ε)
JewroÔme

H1 = {(x, ψ)|f (x, ψ) > 0}

H2 = {(x, ψ)|f (x, ψ) > 0}
 49

EPIPEDO-SHMEIO-EUJEIA ston R3

I APOSTASH DUO ASUMBATWN EUJEIWN

d((ε1 ), (ε2 )) =m koc koinoÔ kˆjetou tm matoc

sq ma 'Eqoume apodeÐxei ìti upˆrqei to

koinì kˆjeto tm ma

DEDOMENA

(ε1 ) : ⃗r1 = ⃗rA1 + t⃗u1 sq ma

(ε2 ) : ⃗r2 = ⃗rA2 + t⃗u2

−−−−→ (ε1 ) ∥ ⃗u1
(M1 M2 ) ⊥ (ε1 ), (ε2 ) ìmwc
(ε2 ) ∥ ⃗u2
−−−−→
'Ara (M1 M2 ) ∥ (⃗u1 × ⃗u2 )
−−−→
−−−→ < A1 A2 , ⃗u1 × ⃗u2 >
d(M1 , M2 ) = ∥prob⃗u1 ×⃗u2 A1 A2 ∥ = ⃗u1 × ⃗u2
< ⃗u1 × ⃗u2 , ⃗u1 × ⃗u2 >
−−−→ −−−→
| < A1 A2 , ⃗u1 × ⃗u2 > |  
 | < A1 A2 , ⃗u1 × ⃗u2 > |
= ∥⃗u
 1× ⃗u2 ∥ =
∥⃗u1 × ⃗u2 ∥
2 ∥⃗u1 × ⃗u2 ∥
−−−→
ìpou A1 A2 = ⃗r2 − ⃗r1 = ⃗rA2 − ⃗rA1

I APOSTASH SHMEIOU apo EUJEIA sto QWRO

(i) PRAKTIKH ANTIMETWPISH

DÐnontai ta A kai (ε).

Fèrw apì to A èna epÐpedo

sq ma
kˆjeto sthn (ε)

Opìte orÐzetai to shmeÐo A′ .

′
−
−→ ′
Zhtˆme to d(AA ) = ∥AA ∥

(ii) B' TROPOS

'Estw P tuqaÐo shmeÐo thc (ε)

sq ma
′ −→
tìte ∥P A ∥ = ∥probAA′ AP ∥ =
−→ −→
∥AP ∥ cos φ = ∥AP ∥ sin θ
H eujeÐa (ε) ∥ ⃗u tìte:

−→ −→
−→ ∥AP ∥∥⃗u∥ sin θ ∥AP × ⃗u∥ ∥(⃗rP − ⃗rA ) × ⃗u∥
∥AP ∥ sin θ = = =
∥⃗u∥ ∥⃗u∥ ∥⃗u∥

−−→ ∥(⃗rP − ⃗rA ) × ⃗u∥

'Ara ∥P A ′ =
∥⃗u∥
50

GEWMETRIA se R2 , R3 − DEUTEROBAJMIA SUSTHMATA

x − x0 ψ − ψ0 z − z0
ExÐswsh eujeÐac ssto q¸ro: = =
κ λ µ

EÐnai SUSTHMA PRWTOBAJMIWN POLUWNUMIKWN EXISWSEWN

ExÐswsh epipèdou sto q¸ro Ax + Bψ + Γz + ∆ = 0

PRWTOBAJMIA POLUWNUMIKH EXISWSH (me treic metablhtèc)

EUEJEIA kai EPIPEDO: GRAMMIKA ANTIKEIMENA

I KUKLOS STO EPIPEDO

PROBLHMA:

Ta shmeÐa twn opoÐwn h apìstash apì to dojèn shmeÐo eÐnai gnwst kai stajer

A: gnwstì shmeÐo
M : tuqaÐo shmeÐo. Jèloume d(A, M ) = R (gnwstì
sq ma kai stajerì).
−−→
∥AM ∥ = R
∥⃗rM − ⃗rA ∥ = R DIANUSMATIKH EXISWSH

Epilègw ORJOKANONIKO SUSTHMA SUNTETAGMENWN

{Oxψz} ⃗rA (xA , ψA , zA ), ⃗rM (x, ψ, z)

∥⃗rM − ⃗rA ∥ = R2
(x − xA )2 + (ψ − ψA )2 + (z − zA )2 = R2 SFAIRA

Sto epÐpedo z=0 {Oxψ} tìte (x − xA )2 + (ψ − ψA )2 = R2 KUKLOS

I Gia SUSTHMA me arq to A:

{A, ψ, z} (xA , ψA , zA ) → (0, 0, 0) tìte
x2 + ψ 2 + z 2 = R2 SFAIRA
ou
poluwnumik 2 BAJMOU sto q¸ro (3 metablht¸n)

I MELETH tou KUKLOU sto EPIPEDO

• DIANUSMATIKH EXISWSH: ∥⃗rA − ⃗rM ∥ = R ìpou A kèntro tou kÔklou kai M shmeÐo tou

kÔklou < ⃗rA −⃗rM , ⃗rA −⃗rM >= R2

• ANALUTIKH EXISWSH
(x − xA )2 + (ψ − ψA )2 = R2 −−→ −−→ −→
sq ma OM = AM + OA
2 2 2
X +Ψ =R
 51

−→
Ta sust mata {O, x, ψ} kai {A, X, Ψ} diafèroun katˆ mÐa metaforˆ wc proc to diˆnusma OA
touOxψ .
X = x − xA x = X + xA
Opìte
Ψ = ψ − ψA ψ = Ψ + ψA
Analutik exÐswsh kÔklou

se SUSTHMA ANAFORAS X 2 + Ψ2 = R 2
me ARQH to KENTRO tou KUKLOU

• PARAMETRIKES EXISWSEIS tou KUKLOU

Perigraf tou kÔklou qrhsimopoi¸ntac mÐa parˆmetro p.q. kÔkloc wc troqiˆ kinhtoÔ sto qrìno

x = x(t)
tìte M (x(t), ψ(t))
ψ = ψ(t)
x = R cos φ
ψ = R sin φ
sq ma sq ma
ìpou φ = ωt, ω h gwniak

suqnìthta.

X(t) = R cos ωt
Parˆmetroc eÐnai to t, gia ω = 1 eÐnai
Ψ(t) = R sin ωt
{ }
x = R cos t
perigrˆfei ta shmeÐa tou kÔklou me mÐa parˆmetro t. O kÔkloc sundèetai me
ψ = R sin t
mÐa periodikìthta.

• Upˆrqei mÐa diadikasÐa, h APALOIFH, pou apì tic parametrikèc exis¸seic odhgeÐ sthn analutik exÐswsh

(mÐa sqèsh qwrÐc thn parˆmetro)


cos t = Rx 
 X2 Ψ2
ψ
sin t = R + = 1 ⇔ X 2 + Ψ2 = R 2

 R R
sin2 t + cos2 t = 1

I 
SQETIKH JESH KUKLOU kai EUJEIAS sto EPIPEDO


 ∅
(C) ∩ (ε) 1 shmeÐo


2 shmeÐa
{
(C) : x2 + ψ 2 = R2
(ε) : αx + βψ + γ = 0, |α| + |β| ̸= 0
Apì th lÔsh tou sust matoc ja prokÔyoun ta “koinˆ shmeÐa” (an upˆrqoun). Epeid |α|+|β| ̸= 0
tìte èna apì ta dÔo eÐnai mh mhdenikì. 'Estw ìti eÐnai to B tìte: ψ = κx + λ kai anagìmaste

sto sÔsthma

{ ψ 2 = κ2 x2 + 2κλx + λ2
x2 + ψ 2 = R2
⇒ x2 + ψ 2 = (1 + κ2 )x2 + (2κλ)x + λ2 = R2
ψ = κx + λ | {z }
tri¸numo

• An ∆=0 tìte upˆrqei mÐa lÔsh: ENA KOINO SHMEIO

52

∆ = 4κλ2 − 4(1 + κ2 )(λ2 − R2 ) = 0

⇕
|λ| λ2
d(O(0, 0), ψ = κx + λ) = R ⇔ √ =R⇔ = R2
1 + κ2 1 + κ2
d(0, (ε)) = R : APOSTASH KENTROU tou KUKLOU apo thn EUJEIA Ðsh me AKTINA.

Epomènwc h lÔsh eÐnai h apìstash tou KENTROU tou KUKLOU apì thn EUJEIA.

sq ma

• An ∆>0 tìte 2 koinˆ shmeÐa

• An ∆<0 tìte kanèna koinì shmeÐo

I EFAPTOMENH KUKLOU

Upˆrqei èna mìno koinì shmeÐo metaxÔ (C) kai (ε) dhlad d(0, (ε)) = R.
EÐnai ORIAKH JESH thc TEMNOUSAS.
exÐswsh pou ikanopoieÐtai an
sq ma xx0 + ψψ0 = R2
h (ε) efˆptetai ston (C).
 53

exÐswsh kÔklou sto epÐpedo (KEN-

2 2 2
x +ψ =R TRO kÔklou h ARQH SUNTETAG-

MENWN)

(x − x0 )2 + (ψ − ψ0 )2 = R2 (KENTRO to shmeÐo (x0 , ψ0 ))

EFAPTOMENH KUKLOU: eujeÐa →akrib¸c èna koinì shmeÐo me ton kÔklo

I EXISWSH EFAPTOMENHS

H aktÐna pou antistoiqeÐ sto (x⋆ , ψ ⋆ )

prèpei na eÐnai kˆjeth sth zhtoÔmenh eujeÐa
sq ma
aktÐna →diˆnusma
⃗ℓ = (x⋆ , ψ ⋆ ) ⊥ (ε)

−−→ −→ −−→ −−→ ⃗ −−→

OM = OA + AM AM ⊥ ℓ dhlad < AM , ⃗ℓ >= 0 (⋆)
−−→
AM (x − x⋆ , ψ − ψ ⋆ )
⃗ℓ(x⋆ , ψ ⋆ )
(⋆) ⇒< (x − x⋆ , ψ − ψ ⋆ ), (x⋆ , ψ ⋆ ) >= 0 ⇔

x⋆ (x−x⋆ )+ψ ⋆ (ψ −ψ ⋆ ) = 0 ⇒ x⋆ x+ψ ⋆ ψ = (x⋆ )2 +(ψ ⋆ )2 = R2 (shmeÐo tou kÔklou isoÔtai me R2 )

(ε) ⇒ x⋆ x+ψ ⋆ ψ = R2

EXISWSH EFAPTOMENHS ston KUKLO pou dièrqetai apì to shmeÐo A(x⋆ , ψ ⋆ ) tou kÔklou

I SQETIKH JESH DUO KUKLWN

}
(C1 ) : x2 + ψ 2 = R12
(Σ)
(C2 ) : (x − α)2 + ψ 2 = R22
sq ma

α −→ DIAKENTROS

EpilÔoume to sÔsthma

I GWNIA DUO KUKLWN −→ gwnÐa efaptomènwn sto shmeÐo tom c

−−→ −−→
gwnÐa (O1 A, O2 A) = φ
sq ma
−−→ −−→
< O1 A, O2 A >
cos φ = −−→ −−→
∥O1 A∥ ∥O2 A∥
ASKHSH: SUNJHKH ORJOGWNIOTHTAS 2 KUKLWN

−−→ −−→
R1 + R2 = δ 2 < O1 A, O2 A >= 0
I ASKHSH: Na brejeÐ o G. T. twn mèswn parˆllhlwn qord¸n.

sq ma } oikìgeneia parˆllhlwn qord¸n

54

G.T.: DIAMETROS KAJETH sthn OIKOGENEIA PARALLHLWN QORDWN

SUSTHMA SUNTETAGMENWN pou tairiˆzei sto prìblhma. H arq twn axìnwn O(0, 0)
eÐnai to kèntro tou kÔklou.

Opìte exÐswsh kÔklou: (C) : x2 + ψ 2 = R2

Oikogèneia parˆllhlwn qord¸n (ε)⋆ αx + βψ + µ = 0, µ ∈ R, |α| + |β| ̸= 0

⋆ eujeÐec kèjetec sto diˆnusma (α, β) (ˆra metaxÔ touc parˆllhlec)
ψ = λx + µ⋆ , λ = stajerì, µ⋆ ∈ R (kai kˆjetec ston xx′ )
{
x2 + ψ 2 = R2 M1 (x1 , ψ1 ) x1 + x2 ψ1 + ψ2
⋆
⇒ ⇒ M = mèson(M1 , M2 ) M ( , )
ψ = λx + µ M2 (x2 , ψ2 ) 2 2

GEWMETRIKOS TOPOS tou M M (xM , ψM ) ψM = f (xM ) qwrÐc µ⋆

ìpou

ProkÔptei ψM = OxM EUJEIA apì thn ARQH O(0, 0). 'Ara DIAMETROS.

I ASKHSH: JewroÔme qordèc Ðsou m kouc (stajeroÔ). ZhteÐtai o G.T. twn mèswn twn qor-

d¸n.

sq ma

I SFAIRA

R3 
 ZhteÐtai o G.T. twn shmeÐwn M:
Ks meio tou q¸rou d(K, M ) = R


R>0 SFAIRA
−−→
Sto sÔsthma {K, xψz} ∥OM ∥ = R

x2 + ψ 2 + z 2 = R 2 EXISWSH SFAIRAS me KENTRO thn ARQH TWN AXONWN.

(x − x0 )2 + (ψ − ψ0 )2 + (z − z0 )2 = R2 KENTRO to (x0 , ψ0 , z0 )


 >R @ koinˆ shmeÐa
sq ma
d(K, (Π)) = R ∃ èna koinì shmeÐo


<R h tom eÐnai kÔkloc

Gia na deÐxoume ìti h TOMH eÐnai KUKLOS: Prèpei na brw to KENTRO kai thn AKTINA tou.

FèrwKK ′ kˆjeth sto epÐpedo kai

sq ma èstw M shmeÐo thc tom c. Sto
′
orjog¸nio KK M
}
KK ′ = d < R
⇒ (K ′ M )2 = R2 − d2 > 0
KM = R

'Ara ta shmeÐa thc TOMHS apèqoun apì stajerì shmeÐo stajer apìstash. 'Ara h TOMH

eÐnai KUKLOS.
 55

I EXISWSH EFAPTOMENOU EPIPEDOU

(S) x2 + ψ 2 + z 2 = R2 ZhteÐtai to (Π) pou efˆptetai

(x0 , ψ0 , z0 ) sthn S sto shmeÐo A.

−→ −−→
< OA, AM >= 0 ∀M shmeÐo tou zhtoÔmenou (Π)
x0 x + ψ0 ψ + z0 z = R2
⃗ℓ(x0 , ψ0 , z0 ) ⊥ (Π)

I PARAMETRIKH PERIGRAFH EPIPEDOU k' SFAIRAS


 ⃗r = ⃗r0 + t⃗u, t ∈ R  


eujeÐa
 
  x = x0 + tα 
EISAGWGH ψ = ψ0 + tβ t∈R

 x(t) = R cos φ 
 


 KÔkloc z = z0 + tγ
ψ(t) = R sin φ, φ ∈ R
• EpÐpedo

−−→
⃗r = ⃗r0 + AM
−−→
AM = t⃗u + sw ⃗ , grammikoÐ sunduasmoÐ
sq ma
⃗ t, s ∈ R
⃗u, w,
⃗ t, s ∈ R
⃗r = ⃗r0 + t⃗u + sw,

(x, ψ, z) = (x0 , ψ0 , z0 ) + t(α1 , α2 , α3 ) + s(β1 , β2 , β3 )



 (x − x0 ) − α1 t − sβ1 = 0
(ψ − ψ0 ) − α2 t − sβ2 = 0 Apaloif twn t, s.


(z − z0 ) − α3 t − sβ3 = 0

x − x0 ψ − ψ0 z − z0
α1 α2 α3 = 0 ⇔ αx + βψ + γz + δ = 0
β1 β2 β3

(α, β, γ) ∥ ⃗u × w
⃗

POLIKES SUNTETAGMENES sto QWRO

SFAIRIKES SUNTETAGMENES sto QWRO {⃗i, ⃗j, ⃗κ}

−−→
|OM | = ⃗r
−→ −−→
|OΓ| = |OM | | cos θ| = r| cos θ|
sq ma
−→ }
OΓ = (r · cos θ)⃗κ
−→ z = r · cos θ θ ∈ (0, π) (− π2 , π2 )
OΓ = z
56

−−→ −−→
|OM ′ | = |ΓM | = r · sin θ

sq ma x = (OM ′ )· = r · cos φ sin θ

φ = (OM ′ ) · sin φ = r sin φ sin θ

φ ∈ (0, 2π)
(φ, θ) −→ (r cos φ sin θ, r sin φ cos θ, r cos θ)
x, ψ, z
x2 + ψ 2 + z 2 = r2
φ = gewgrafikì plˆtoc φ ∈ (0, 2π)
θ = gewgrafikì m koc θ ∈ (0, π)
 57

I H DEUTEROBAJMIA EXISWSH sto EPIPEDO

E : αx2 + βxψ + γψ 2 + δx + εψ + z = 0
{O, xψ} : EPIPEDO,
{(x, ψ)|(E)} = C : zeÔgh pou ikanopoÐoun thn (E) kai C : KAMPULH EPIPEDOU

EIDIKES PERIPTWSEIS

• δx + εψ + z = 0. An |δ| + |ε| ̸= 0 tìte paristˆnei eujeÐa. An δ =ε=0 kai z ̸= 0 tìte h

exÐswsh den paristˆnei kˆti. Tèloc an δ=ε=z=0 paristˆnei ìlo to epÐpedo.

• αx + γψ + z = 0
2 2

{
KUKLOS (exartˆtai apì touc suntelestèc)
}
α=γ=1=z
x2 + ψ 2 + 1 = 0 adÔnato giatÐ den upˆrqei kÔkloc fantastik c aktÐnac

I MELETH thc (E)

Mèsw thc melèthc eidik¸n deuterobˆjmiwn kampul¸n pou prokÔptoun apì gewmetrikˆ probl -

mata.

I PROBLHMA: Sto epÐpedo dÐnetai mia eujeÐa (δ) kai èna shmeÐo (E). ZhteÐtai o G.T. twn
d(M, E)
shmeÐwn M tou epipèdou gia ta opÐa = stajerì > 0. SumbolÐzoume thn ekkentrìthta
d(M, δ)
me e.
ISTORIA tou PROBLHMATOS:

sundèetai me tomèc epipèdou kai k¸nou.

sq ma.
sq ma
Oi kampÔlec pou prokÔptoun apì thn

tom tou epipèdou kai k¸nou eÐnai o G.T.

tou probl matoc.

I MELETH tou PROBLHMATOS

(i) Epilog katˆllhlou sust matoc suntetagmènwn.
(ii) “Metagraf ” tou probl matoc sto epilegmèno sÔsthma.
(dedomèna (δ) eujeÐa, E shmeÐo).

Fèrw apì to E thn KAJETH sto (δ).

E −→ ESTIA
δ −→ dieujetoÔsa
sq ma
M : tuqaÐo shmeÐo
r = (M E): ESTIAKH APOSTASH
To E paÐzei to rìlo tou POLOU

EB =stajer apìstash= p
2 2
(M E) = r
(M Γ)2 = (EB − r cos θ)2
(M E)2
= e2
(M Γ)2
58

(M E)2
= e2 ⇒ r2 = e2 (p − r cos θ)2 (3)
(M Γ)2
EXISWSH tou G.T. me qr sh POLIKWN SUNTETAGMENWN

ˆrq mètrhshc twn gwni¸n, O:

pìloc

x = r cos θ
sq ma
ψ = r sin θ
x2 + ψ 2 = r2
ψ
arctan = θ
x
•1 h
PERIPTWSH: e=1
r
Prèpei p > r cos θ. Opìte apì thn (3) prokÔptei =1
p − r cos θ
•2 h
PERIPTWSH: e ̸= 1
r
Apì thn (3) prokÔptei = ±e ⇒ r = f (θ)
{ p − r cos θ
e>1 upˆrqoun shmeÐa kai dexiˆ thc δ pou ikanopoioÔn th sunj kh

e<1 ìla ta shmeÐa pou ikanopoioÔn th sunj kh eÐnai aristerˆ thc δ

sq ma (gia ε>1 shmeÐo dexiˆ thc δ)

Parabˆlloume to e me th monˆda kai prokÔptei

e = 1 −→ PARABOLH
e < 1 −→ ELLEIYH
e > 1 −→ UPERBOLH

r2 = e2 (p − r cos θ)2
x = r cos θ
ψ = r sin θ
(O, xψ)
r 2 = x2 + ψ 2
x2 + ψ 2 =
sq ma
{z θ}) ⇒
e2 (p2 − 2pr cos θ + r|2 cos2

x2
x2 + ψ 2 =
e2 p2 − 2pe2 r cos θ + e2 x2 ⇒

(1 − e2 )x2 + ψ 2 + 2pe2 x − e2 p2 = 0

(1 − e2 )x2 + ψ 2 + 2pe2 x − e2 p2 = 0
• gia e=1
ψ 2 = e2 p2 − 2e2 x ⇒ ψ 2 = p2 − 2px ⇔
p
ψ 2 = 2p( − x) (4)
2
 59

Jètoume

} Y 2 = −2pX
X =x− p
(2)
2
⇒ sq ma
Y =ψ
PARABOLH sto SUSTHMA ↗
}
X ⋆ = −X
Sto SUSTHMA ⇒ Ψ⋆ = 2pX ⋆
Ψ⋆ = Ψ

• e1 ̸= 1
2pe2
(1 − e2 )[x2 + x] + ψ 2 = e2 p2 prospajoÔme na to kˆnoume tèleio tetrˆgwno me katˆllhlec
1 − e2
pe2 p2 e2
prosjafairèseic (1 − e )(x + −
2 2
) + ψ =0
1 − e2 1 − e2
pe2
X = (x + )
Jètw 1 − e2 →
Ψ=ψ
e2 p2 e2 p2
(1 − e2 )X 2 + Y 2 = ⇒ diairoÔme me
1 − e2 1 − e2
X2 Y2
+ =1
e2 p2 e2 p2
1 − e2 1 − e2
DiakrÐnoume tic peript¸seic

(i) 1 − e2 > 0 ⇔ e < 1.

2 e2 p2 2 e2 p2
Jètoume α = , β =
(1 − e2 )2 1 − e2
X2 Y2
+ = 1
α2 β2

(ii) 1 − e2 < 0 ⇔ e > 1

e2 p2 e2 p2
Jètoume α =
2
, β 2
= −
(1 − e2 )2 1 − e2
X2 Y2
− 2 = 1
α2 β

TELIKH APANTHSH: to prìblhma tou G.T. se katˆllhlo sÔsthma suntetagmènwn eÐnai

2
ψ = 2px (e = 1) PARABOLH
2 2
x ψ
2
+ 2 = 1 (e < 1) ELLEIYH
α β
x2 ψ2
− = 1 (e > 1) UPERBOLH
α2 β 2
Gia thn EKKENTROTHTA

Jètw (1 − e2 )α2 = β 2
β2
1 − e2 = 2 ìpou α > β P = d(E, δ)
α
60

β2 α2 − β 2
e2 = 1 − = = e2
α2 α2

I SQOLIA

− H exÐswsh pou prokÔptei eÐnai eidik morf deuterobˆjmiac exÐswshc dÔo metablht¸n

αx2 + 2βxψ + γψ 2 + δx + εψ + z = 0. (E)

− H (E) faÐnetai na eÐnai genikìterh giatÐ me katˆllhlouc suntelestèc mporeÐ na ekfrˆsei eu-

jeÐec kai kÔklouc

− ProkÔptei to er¸thma ti paristˆnei h (E). Melèth gewmetrik kˆje antikeimènou pou prokÔ-

ptei!
 61

EPIPEDO sto QWRO

Me apaloif paramètrwn thc Ax + Bψ + Γz + ∆ = 0

f (x, ψ, z) = 0
z = f (x, ψ) EPIFANEIA
 sthn èkfrash Mouge
x = f1 (u, w) 

ψ = f2 (u, w) Gauss


z = f3 (u, w)
EUJEIA sto QWRO
−→ −→
Ax = λAB
−−→ −→ −−→ −→
OX − OA = λ(OB − OA ⇒
−−→ −→ −−→
sq ma
OX = (1 − λ)OA + λOB
DIANUSMATIKH EXISWSH

 EUJEIAS sto QWRO

x = (1 − λ)x1 + λx2 

ψ = (1 − λ)ψ1 + λψ2 PARAMETRIKES EXISWSEIS


z = (1 − λ)z1 + λz2

x − x1 ψ − ψ1 z − z1
= = =λ
x2 − x1 ψ2 − ψ1 z2 − z1
DIEUJUNSH EUEJEIAS

−→ ⃗u · ⃗i = cos α (α : gwnÐa pou sqhmatÐzei to⃗ume tonOx)

AB
⃗u = −−→ ⃗u · ⃗j = cos β (β : gwnÐa pou sqhmatÐzei to⃗ume tonOx)
|AB| ⃗u · ⃗κ = cos γ (γ : gwnÐa pou sqhmatÐzei to⃗ume tonOx)
ASKHSH EpÐpedo pou pernˆei apì to shmeÐo A(4, −2, 1) kai ⊥ sthn eujeÐa ⃗u(7, 2, −3).

(x − x1 )α1 + (ψ − ψ1 )α2 + (z − z1 )α3 = 0

7(x − 4) + z(ψ + 2) − 3(z − 1) = 0
7x − 28 + 2ψ + 4 − 3z + 3 = 0 sq ma

7x + 2ψ − 3z = 21 (5)

EUEJEIA tou ⃗u pou dièrqetai apì to (1, 1, 1) kai èqei dieÔjunsh ⃗u

x−1 ψ−1 z−1

= = =λ
7 2 −3
SHMEIO TOMHS EUJEIAS kai EPIPEDOU


x = 7λ + 1 

ψ = 2λ + 1 Jètw sthn (5) x, ψ, z


z = −3λ + 1
I APOSTASH ASUMBATWN EUJEIWN
62

−→
AΓ = λ⃗u1
−→ −→
OΓ − OA = λ⃗u1
−→ −→
OΓ = OA + λ⃗u1
sq ma ···············
x1 = α1 + λu11
ψ1 = α2 + λu12
z1 = α3 + λu13

−−→ −−→
∆(x2 , ψ2 , z2 ) O∆ = OB + µ⃗u2
⃗u1 (u11 , u12 , u13 ) ···············
−→
OA(α1 , α2 , α3 ) x2 = β1 + µu21
−−→
OB(β1 , β2 , β3 ) ψ2 = β2 + µu22
⃗u2 (u21 , u22 , u23 ) z1 = β3 + µu23

x2 − x1 = β1 − α1 + µu21 − λu11
ψ2 − ψ1 = β2 − α2 + µu22 − λu12
z2 − z1 = β3 − α3 + µu23 − λu13

(x2 − x1 )u11 + (ψ2 − ψ1 )u12 + (z2 − z1 )u13 = 0

(x2 − x1 )u21 + (ψ2 − ψ1 )u22 + (z2 − z1 )u23 = 0

STROFH AXONWN
(OA1 ) = ρ, (OA2 ) = ρ
ρ cos φ2 = x2
EPIPEDO sq ma
ρ sin φ2 = ψ2
θ = φ2 − φ1 ⇒ φ2 = θ + φ1
( ) ( ) sin2 θ + cos2 θ = 1
x2 ( ) x1
= X cos2 θ − sin2 θ = cos 2θ
ψ2 ψ1
2 sin θ cos θ = sin 2θ

x2 = ρ cos(θ + φ1 ) (6)

ψ2 = ρ sin(θ + φ1 ) (7)

(6)⇒ x2 = ρ cos θ cos φ1 − ρ sin θ sin φ1 = x1 cos θ − ψ1 sin θ

(7)⇒ ψ2 = ρ sin θ cos φ1 − ρ sin θ cos φ1 = x1 cos θ + ψ1 sin θ

( )
cos θ − sin θ
X= ORJOGWNIOS METASQHMATISMOS
sin θ cos θ

QWROS
 63

 
cos θ − sin θ 0
 
sq ma  sin θ cos θ 0 Strof sto xψ EPIPEDO

0 0 1
 
1 0 0
 
0  strof sto ψz EPIPEDO

0
64

GEWMETRIKH MELETH KWNIKWN TOMWN

I ENIAIOS TROPOS ORISMOU

d(M, E)
{M | = e stajerì}
d(M, δ)

Katal xame stic peript¸seic

sq ma ψ 2 = 2px se katˆllhlo sÔsthma suntetagmènwn

x2 ψ 2
+ =1
α2 β 2
x2 ψ2
− =1
α2 β 2

I PARABOLH ψ 2 = 2px

• Upˆrqei ˆxonac summetrÐac, o xx′

(x0 , ψ0 ) ∈ C ⇒ (x0 , −ψ0 ) ∈ C
(DEN EQEI KENTRO SUMME-

TRIAS)

sq ma
• ExÐswsh efaptomènhc

Oriak jèsh thc tèmnousac

(èna koinì shmeÐo, MH PARALLHLH

me ton ˆxona summetrÐac)

dψ p
ψ 2 = 2px ⇒ 2ψdψ = 2pdx ⇒ =
dx ψ
• ExÐswsh Efaptomènhc thc Parabol c

ψ = f (x), 'Estw M (x0 , ψ0 ) tìte:

ψ − ψ0 = f ′ (x0 ) (x − x0 )
| {z }
klÐsh thc efaptomènhc

{ √
2px, ψ ≥ 0
ψ= √ (kai x ≥ 0)
− 2px, ψ < 0
√ 2p p p
Tìte gia f (x) =2px ⇒ f ′ (x) = √ =√ = , ψ ̸= 0
2 2px 2px ψ
√ p
OmoÐwc an f (x) = − 2px tìte f ′ (x) =
ψ
p
H klÐsh thc efaptomènhc sto (x0 , ψ0 ) eÐnai gia x0 ̸= 0
ψ0
′
gia x0 = 0 ⇒ o ˆxonac ψψ eÐnai efaptomènh giatÐ eÐnai oriak jèsh tèmnousac

p
ψ − ψ0 = (x − x0 ), (x0 , ψ0 ) ∈ C
ψ

ψψ0 − ψ02 = px − px0 ⇒ ψψ0 = px + 2px0 − px0 ⇒ ψψ0 = p(x+x0 ) x0 ̸= 0

 65

EXISWSH EFAPTOMENHS thc PARABOLHS ψ 2 = 2px

2oc
TROPOS: M (x0 , ψ0 ) { }
ìlec oi eujeÐec ektìc apì tic parˆllhlec ston xx′ (ˆxonac summe-

ψ − ψ0 = λ(x − x0 )
trÐac). Ftiˆqnoume to sÔsthma kai apaitoÔme lÔsh.
ψ 2 = 2px
p
Prèpei ∆ = 0 opìte prokÔptei λ = .
ψ0
′
An M (0, 0) tìte efaptomènh: ψ ψ .

• Basik Idiìthta thc Parabol c

Katoptrik Idiìthta

− MÐa fwteÐnh aktÐna parˆllhlh me ton (kÔrio) ˆxona anakl¸menh dièrqetai apì thn (kÔria)

estÐa.

− ANTISTROFA: mÐa fwtein phg topojethmènh sthn kÔria estÐa prokaleÐ fwtein dèsmh

parˆllhlwn ∥ x′ x aktÐnwn.

Prèpei na deÐxoume ìti eÐnai

sq ma DIQOTOMOS

sq ma

Dhlad prèpei na deÐxoume ìti:

< ⃗u, w
⃗> < w,
⃗ ⃗v >
=
∥⃗u∥ ∥w∥
⃗ ∥w∥
⃗ ∥⃗v ∥

'Ara prèpei na broÔme poia dianÔsmata emplèkontai

sq ma

• Basik Idiìthta

p
H dieujetoÔsa ( x=− ) èqei thn ex c idiìthta:
2
EÐnai o G.T. twn shmeÐwn apì ta opoÐa ˆgontai KAJETES EFAPTOMENES

x2 ψ2
ELLEIYH + = 1
α2 β2

H exÐswsh isqÔei se katˆllhlo sÔsthma

d(M, E)
=e<1
d(M, δ)
sq ma
e2 p2 e2 p2
α2 = β 2
=
(1 − e2 )2 (1 − e2 )
α2 1
ParathroÔme ìti: = kai ìti α 2 − β 2 = α 2 e2 .
β 2 1 − e2
66

Eisˆgoume nèa metablht :

c2 = α 2 − β 2
c2
e2 =
α2
c = eα
α2 α
=
c e
x2 ψ 2
+ = 1 (C)
α2 β 2

• Upˆrqei kèntro summetrÐac O(0, 0) h arq twn axìnwn. An (x0 , ψ0 ) ∈ (C) ⇒ (−x0 , −ψ0 ) ∈
(C)
• Upˆrqoun dÔo ˆxonec summetrÐac
EÐnai to KENTRO SUMMETRIAS

d(M, E)
=e<1
d(M, δ)
sq ma
d(M, E ′ )
=e
d(M, δ ′ )
• IsqÔei ìti d(M, E) + d(M, E ′ ) = STAJERO = 2α

d(M, E) = ed(M, δ)
d(M, E ′ ) = ed(M, δ ′ ) opìte
2α
d(M, E) + d(M, E ) = e[d(M, δ) + d(M, δ ′ )] =
′

′
e
'Ara d(M, E) + d(M, E ) = 2α

• Efaptomènh thc èlleiyhc

β 2 x2 + α2 ψ 2 = α2 β 2
dψ β2 x
2β 2 xdx + 2α2 ψdψ = 0 ⇔ =− 2 (klÐsh efaptomènhc sto shmeÐo M (x, ψ)
dx α ψ

x=α x = −α efaptomènec gia ψ=0

• Katoptrik Idiìthta 'Elleiyhc

h efaptomènh diqotomeÐ thn gwnÐa

pou sqhmatÐzoun oi estiakèc aktÐ-

sq ma nec.

TROPOS KATASKEUHS

efaptomènhc
 67

• IDIOTHTA: Ta mèsa twn parˆllhlwn qord¸n

parˆllhlec qordèc

sq ma
{ = λx + µ, λ =stajerì, µ ∈ R
ψ
β 2 x2 + α2 ψ 2 = α2 β 2
ψ = λx + µ

Apì to sÔsthma prokÔptoun dÔo shmeÐa (x1 , ψ1 ), (x2 , ψ2 ).

x1 + x2 ψ1 + ψ2
xM = , ψM = ψM = f (xM )
2 2

B' TROPOS

}
β 2 x21 + α2 ψ12 = α2 β 2
(−)
β 2 x22 + α2 ψ22 = α2 β 2
β 2 (x21 − x22 )α2 (ψ12 − ψ22 ) = 0 ⇔ diaforˆ tetrag¸nwn

ψ1 + ψ2
ψ12 − ψ22 β2 ψ1 − ψ2 2 β2
= − ⇔ · = −
x21 − x22 α2 x1 − x2 x1 + x2 α2
2 }
| {z
ψM
xM
ψM β2
λ =− 2 EujeÐa pou pernˆei apì to kèntro.
xM α

(Gia thn parabol prokÔptei eujeÐa parˆllhlh me ton ˆxona summetrÐac).

68

I MELETH thc B' BAJMIAS EXISWSHS sto EPIPEDO

(E) : αx2 + 2βxψ + γψ 2 + 2δx + 2εψ + z = 0

− GEWMETRIKO ANTIKEIMENO (eÐnai h (E) GEWMETRIKO ANTIKEIMENO?)

EpÐpedo → GewmetrÐa Epipèdou ≡ Orjokanonikˆ sust mata. Dhlad (E) anafèretai se èna
h

orjokanonikì sÔsthma {Oxψ}. Se mia allag sust matoc h (E) (E ′ ) ìpou (E ′ ) prèpei na
eÐnai Ðdiac morf c me thn (E).

− SUMPERIFORA thc (E) stic ALLAGES SUSTHMATWN

− UPARQEI SUSTHMA pou APLOPOIEI thn (E) ?

ALLAGH SUNTETAGMENWN sta ORJOKANONIKA SUSTHMATA

x = cos θx′ − sin θψ ′ + x0

ψ = sin θx′ + cos θψ ′ + ψ0
} }
cos θx′ − sin θψ ′ x0
ìpou strof kai metaforˆ.
sin θx′ + cos θψ ′ ψ0

αx2 + 2βxψ + γψ 2 (b'bˆjmio tm ma thc (E))

2δx + 2εψ + z (a'bˆjmio tm ma thc (E))

H metaforˆ ephreˆzei to a'bˆjmio tm ma thc (E)
}
x = x′ + x0
den allˆzei to b'bˆjmio αx2 + 2βxψ + γψ 2
ψ = ψ ′ + ψ0

{
Meletˆme to b'bˆjmio tm ma se sqèsh me th strof

x = cos θx′ − sin θψ ′

ψ = sin θx′ + cos θψ ′

Upˆrqei gwnÐa θ ¸ste sto {O′ x′ ψ ′ } na mhn parousiˆzetai o ìroc x′ ψ ′ .

x2 = cos2 θ(x′ )2 + sin2 θ(ψ ′ )2 − 2 sin θ cos θ(x′ ψ ′ )

x2 = sin2 θ(x′ )2 + cos2 θ(ψ ′ )2 + 2 sin θ cos θ(x′ ψ ′ )

xψ = sin θ cos θ(x′ )2 − sin θ cos θ(ψ ′ )2 + (cos2 θ − sin2 θ)x′ ψ
Tìte

( )
αx2 + γψ 2 = α cos2 θ + γ sin2 θ + 2β sin θ cos θ)(x′ )2 + α sin2 θ + γ cos2 θ − 2β sin θ cos θ)(ψ ′ )2

[−α 2| sin{z θ cos θ} +2β(cos2 θ − sin2 θ)]x′ ψ ′

θ cos θ} +γ 2| sin{z
sin 2θ cos 2θ

= [−α sin 2θ + γ sin 2θ + 2β cos 2θ]x′ ψ ′

 69

=[ (γ − α) sin 2θ + 2β cos 2θ ]
| {z }
An eÐnai Ðso me mhdèn tìte sto {O′ x′ ψ ′ } den upˆrqei x′ ψ ′

(α − γ) sin 2θ = 2β cos 2θ
α−γ cos 2θ
= = cot 2θ
2β sin 2θ
α−γ
An β ̸= 0 ∃θ : (cot 2θ = ) ¸ste sto sÔsthma {O′ x′ ψ ′ } na mhn upˆrqei o ìroc x′ ψ ′ .
2β
′
H (E) anˆgetai sthn (E )

(E ′ ) : α′ (x′ )2 + (γ ′ )(ψ ′ )2 + 2δ ′ (x′ ) + 2ε′ (ψ ′ ) + z ′ = 0

•1 h
PERIPTWSH α′ , γ ′ ̸= 0
•2 h
PERIPTWSH α′ γ ′ = 0.
Se kˆje mÐa apì tic parapˆnw peript¸seic qrhsimopoioÔme tic metaforèc.

′′ ′′ ′′ ′′ ′′
1. Tìte α (x )2 + (γ )(ψ )2 + z = 0 paristˆnei kwnik tom

kÔkloc

èlleiyh

uperbol
{
x=ψ
temnìmenec eujeÐec p.q. x2 − ψ 2 = 0 ⇔
x = −ψ
fantastikìc kÔkloc x2 + ψ 2 + 1 = 0
fantastik uperbol x2 − ψ 2 + 1 = 0.

2. α′ ̸= 0, γ ′ = 0
α′ (x′ )2 + 2δ ′ (x′ ) + 2ε′ ψ ′ + z ′ = 0 metaforˆ
′′ ′′ ′′ ′′
α (x )2 + ε ψ = 0
parabol

An cot 2θ = κ cos θ, sin θ?

2 tan θ 1
tan 2θ = 2
2
, 1 + tan θ = ⇒ sin2 θ
1 + tan θ cos2 θ

EFARMOGH

(E) : αx2 + 2βxψ + γψ 2 + 2δx + 2εψ + z = 0


xψ = 1 
 { {


xψ − 1 = 0 α−γ π π
cot 2θ = = 0 ⇒ 2θ = 2
⇒θ= 4
α=γ=δ=ε=0 
 1 3π 3π

 2 4
2β = 1
{ √
x = x′ cos π4 − ψ ′ sin π4 x = (x′ − ψ ′ ) 22
Opìte ⇒ √
ψ = x′ sin π4 + ψ ′ cos π4 ψ = (x′ + ψ ′ ) 22
70

{ ′′ ′′ ′′ ′′ √
x = x cos 3π − ψ sin 3π
x = (−x − ψ ) 2
⇒ ′′ √
4 4
′′ 3π ′′ 3π ′′
ψ = x sin 4 + ψ cos 4 ψ = (x + ψ ) 2

 (x′ )2 − (ψ ′ )2
 (E ′ ) : =1⇒ (x′ )2 − (ψ ′ )2 = 2
(E) ⇒ ′′ 2
2 ′′ 2
 (E ′′ ) : − (x ) − (ψ ) = 1 ⇒ (x′′ )2 − (ψ ′′ )2 = 2

2

sq ma ISOSKELHS UPERBOLH

I GRAMMIKH MELETH thc (E) - ANALLOIWTES

(E) : αx2 + 2βxψ + γψ 2 + 2δx + 2εψ + z = 0

I ANAGWGH se GINOMENO PINAKWN

( )( ) ( )
α β x x
(x, ψ) +2 (δ, ε) + |{z}
z =0
β γ ψ | {z } ψ
| {z } | {z } B Γ
A X

opìte X t AX + 2BX + Γ = 0 ìpou X t AX : b'bˆjmio kai 2BX prwtobˆjmio.

A =SUMMETRIKOS= At

I ALLAGH SUSTHMATOS SUNTETAGMENWN

{ }
PPt = I
X ↔ X⋆ X = P X ⋆ + X0
ìpou
det P > 0


 A
⋆

(E) → (E ⋆ ) → B⋆

 ⋆
Γ
PARADEIGMA

'Estw X = P X ⋆ tìte X t = X ⋆t P t = X ⋆t P −1
X t AX = (X ⋆ )t P −1 AP X ⋆ ⇒
−1
A→ P | {zAP}
A
ìmoioc me ton

'Emeinan analloÐwta:

1. H MORFH thc EXISWSHS

2. det(A) = det(A⋆ ) dhlad det(A) = αγ − β 2 = det(P −1 AP )

tr(A) = tr(P −1 AP ) =
{α+γ
αγ − β 2
analloÐwta thc (E):
α+γ
 71

3.    
α β δ x
   
(E) : (x, ψ, 1) β γ ε ψ  = 0
δ ε z z
| {z }
det(M )analloÐwto

TRIA ANALLOIWTA thc E

I Prìtash Stic diˆforec allagèc suntetagmènwn upˆrqoun treÐc posìthtec

j1 = α + γ
j2 = αγ − β 2
j3 = det(M )
pou mènoun analloÐwtec.

Anˆloga me ta prìshma twn j 1 , j2 , j3 kajorÐzetai to eÐdoc thc kampÔlhc.

x2 + ψ 2 = 1 j1 = 2, j2 = 1, j3 = −1 < 0
x2 − ψ 2 = 1 j1 = 0, j2 = −1, j3 = 1 > 0
p.q. ψ 2 − 2x = 1 j1 = 1, j2 = 0, j3 = −1 < 0
x+ψ+1=0 j1 = 0, j2 = 0, j3 = 0
x2 + ψ 2 + 1 = 0 j1 = 2, j2 = 1, j3 = 1
SUMPERASMA

1. 'Otan j3 = 0 den upˆrqei B'BAJMIA KAMPULH (pragmatik fantastik ) ekfulismìc

se eujeÐa!!
 {

 ELLEIYH (KUKLOS)
 ̸= 0 kèntro summetrÐac tìte
2. j2 UPERBOLH (pragmatik /fantastik )
{


 = 0 den upˆrqei kèntro summetrÐac, upˆrqei ˆxonac summetrÐac PARABOLH

{
<0 UPERBOLH
3. j2
>0 ELLEIYH
72

I Melèth thc b'bˆjmiac exÐswshc sto epÐpedo

(E) : αx2 + 2βxψ + γψ 2 + 2δx + 2εψ + z = 0



 ELLEIYH (pragmatik /fantastik )

 UPERBOLH (pragmatik /fantastik )
(E)

 PARABOLH


TEMNOMENES EUJEIES

I Se katˆllhlo sÔsthma suntetagmènwn {O′ x′ ψ ′ } (E) → (E ⋆ )

{Oxψ} → {O′ x′ ψ ′ } sundèontai mèsw strof¸n kai metafor¸n.
Me th STROFH exafanÐzetai o ìroc xψ .

Me th METAFORA rujmÐzontai oi prwtobˆjmioi ìroi.

α−γ { }
cot 2θ = x = x′ cos θ − ψ ′ sin θ
2θ
gwnÐa strof c ψ = x′ sin θ + ψ ′ cos θ

Ti mènei ANALLOIWTO

Br kame treÐc posìthtec j1 j2 j3

An j2 ̸= 0 upˆrqei KENTRO SUMMETRIAS

EURESH KENTROU SUMMETRIAS

{
2αx0 + βψ0 + δ = 0
(Σ) ⇒ (x0 , ψ0 ) KENTRO SUMMETRIAS
βx0 + 2γψ0 + ε = 0

EFARMOGES

I x2 − 4xψ + ψ 2 + 10x − 8ψ + 7 = 0

Ti paristˆnei kai se poio sÔsthma paÐrnei aploÔsterh dunat morf

1 2 5
j1 = 2, j2 = −3, j3 = 2 1 −4 = −142
5 −4 7
̸ 0
j3 = kampÔlh

j2 ≠ 0 upˆrqei kèntro summetrÐac

j2 < 0 UPERBOLH

upˆrqei sÔsthma ìpou h (E) èqei morf α ′ X 2 + γ ′ Ψ2 + J ′ = 0

x2 − Sx + P ìpou S = x1 + x2 , P = x1 x2
}
j1′ = α′ + γ ′ = 2
t2 − 2t − 3 = 0 opìte α′ , γ ′ = 3, −1
j2′ = α′ γ ′ = −3
j3 −142 142
j3′ = α′ γ ′ z ′ = −142 ⇒ z ′ = = ⇒ z′ =
j2 −3 3
 73

h
1 perÐptwsh:

142
3x2 − 1ψ 2 + =0 (8)
3
h
2 perÐptwsh:

142
−x2 + 3ψ 2 + =0 (9)
3
3x2 ψ2
(8) ⇒ − 142 = 1
− 142
3
− 3

EURESH tou SUSTHMATOS SUNTETAGMENWN

α−γ x2 ψ2
cot 2θ = , − (√ ) + (√ ) = 1
2β 142 2 142 2
3 3

π π
cot 2θ = 0 ⇒ 2θ = ⇒θ =
2 4

Allag suntetagmènwn

x′ − ψ ′
} x= √
x = cos π4 x′ − sin π4 ψ ′ 2
⇒
ψ = sin π4 x′ + cos π4 ψ ′ x′ + ψ ′
ψ= √
2
x′2 − 2x′ ψ ′ + ψ ′2
x2 =
2
x′2 − ψ ′2
xψ = −4 = −2x′2 + 2ψ ′2
2
x′2 + 2x′ ψ ′ + ψ ′2
ψ2 =
2
x2 + xψ + ψ 2 = x′2 + ψ ′2 − 2x′2 + 2ψ ′2 = −x′2 + 3ψ ′2
b'bˆjmio tm ma
 
 10x′ − 10ψ ′ 

 10x = √ 


 2 

′
− ′ 2x′ − 18ψ ′
b'bˆjmio tm ma 8x
√
8ψ + 10x + 8ψ + 7 = √ +7


8ψ = 
 2

 2 

 
7
′ ′2 ′2 2x′ − 18ψ ′
opìte (E ) : −x + 3ψ + √ +7=0⇒
2
2 18
−2x′2 + 6ψ ′2 + √ x′ − √ ψ ′ + 14 = 0
2 2
√ ′ 2 √ ′ √ √ 9 √ √ 9 9
(− 2x ) + 2( 2x ) − 1 + 1 + ( 2 3ψ ′ )2 − 2 √ ( 2 3ψ ′ ) + ( √ )2 − ( √ )2 + 14 = 0
3 3 3
√ ′ √ √ ′ 9 2 9 2
−( 2x − 1) + 1 + ( 2 3ψ − √ ) + 1??( √ ) = 0
2
3 3
−X 2 + Ψ2 + K = 0
x⋆
z }| {
√ √ 1
X = 2x′ − 1 = 2 (x′ − √ )
2
74

ψ⋆
z
}| √ {
√ ′ 9 √ 1 √ 3 2
Ψ = 6ψ − √ = 6(ψ ′ − √ = 6 (ψ ′ − )
3 6·3 2
81
−2x⋆ + 5ψ ⋆ + (15 − ) = 0
2 2

I 2x2 − xψ − 15ψ 2 + 5x − 3

j1 = α + γ = −13

α β 2 − 12 1 30
j2 = = 1 = −30 =
β γ − 2 −15 4 4
2 − 12 5
2
−15 0 1 −12 0 5 − 21 −15
j3 = − 12 −15 0 = 2 + +
0 −3 2 52 −3 2 52 0
5
2
0 −3
3 5 75 360 + 3 + 375
= 90 + + ( ) = ̸= 0
4 2 2 4

j3 ̸= 0, j2 < 0 ˆra UPEERBOLH

17
cot 2φ =
−1

I (2x + 3ψ − 1)(x + 7ψ + 4) = 0 ⇒ TEMNOMENES EUJEIES

epimeristik · · · · · · prokÔptei

2x2 + 17xψ + 21ψ 2 + 7x + 5ψ − 4 = 0

j1 = 23
1 − 289
j2 = <0
4
j3 = 0

√ √ √
I x2 − 2 3xψ + 3ψ 2 − 4(1 + 2 3)x + 4(2 − 3)ψ + 20 = 0

j1 = 4 }
j2 = 0 → DEN UPARQEI KENTRO
PARABOLH
j3 = −32 → KAMPULH
Poia eÐnai h aploÔsterh morf

ψ 2 = 2px
x2 = 2pψ
ψ 2 = κx + λ ⇒ ψ 2 = κ(x + λκ ) = 2 κ2 (x⋆ ) = 2px⋆
γ ′ ψ ′2 − 2px′ = 0
0x′2 + 0x′ ψ ′ + γ ′ ψ ′2 − 2px′ + 0ψ ′ + 0 = 0
j1 = γ ′
j2 = 0
 75

0 0 −p
j3 = 0 0 0 = −p2 γ ′ prèpei γ ′ = 4 kai p2 γ ′ = −32
−p 0 0
{
γ′ = 4 √
2 ′
⇒ p2 = 8 ⇒ p = ±2 2
p γ = −32
√
√ 2 ′
(i) perÐptwsh 4ψ ′2 − 2 · 2 2x′ = 0 ⇒ ψ ′2 = 2 x PARABOLH
2√
√ 2 ′
(ii) perÐptwsh 4ψ ′2 + 2 · 2 2x′ = 0 ⇒ ψ ′2 = −2 x PARABOLH
2

Pwc sundèontai oi duo parabolèc

( )
x → −x′
katoptrismìc wc proc ψ
ψ→ψ
sq ma ( )
−1 0
PÐnakac katoptrismoÔ wc proc ψ
0 1

x2 + ψ 2 + 1 = 0

j1 = 2

x2 + ψ 2 = −1 ⇒ x2 + ψ 2 = i2 KÔkloc
1 0
j2 = =1 me kèntro (0, 0) kai aktÐna i
0 1
Fantastikìc kÔkloc: x + ψ − i = 0
2 2 2

0 0 0
j3 = 0 1 0 = 1
0 0 1
j1 · j3 > 0 fantastikìc kÔkloc èlleiyh

j1 · j3 < 0 pragmatik èlleiyh.