8. Transformacje 3-D

8.1. Transformacje w przestrzeni 3-D

8.1.1. Transformacje elementarne

•

• zmiana skali,

• !

y

x

z

→ [ ]

′ ′ ′ → [′ ′ ′ ]

 





[

′ ′ ′ ]

= [ ]



 





 

"









[

′ ′ ′ ]

= [ ]



 









[′ ′ ′ ] = [ ]⋅( )

Zmiana skali









[

′ ′ ′ ]

= [ ]















[′ ′ ′ ] = [ ]⋅ (

)

#

 Θ Θ







[

− Θ Θ

′ ′ ′ ]

= [ ]















[′ ′ ′ ] = [ ]⋅ (Θ)

# $









[

Θ Θ

′ ′ ′ ]

= [ ]







− Θ Θ











[′ ′ ′ ] = [ ]⋅ (Θ)

#

 Θ

− Θ







[

′ ′ ′ ]

= [ ]





 Θ

Θ











[′ ′ ′ ] = [ ]⋅ (Θ)

! "#Θ $

%& ' (

y

Θ

( ) x

( )

z

Dane:

• opis osi obrotu ( ), ( ),

• opis obiektu (np. siatka wieloboków),

• # Θ

%$&'()($ #

#Θ.

Rachunek wektorowy (przypomnienie):

wektor - = [ ]

*+', = + +

Operacje rachunku wektorowego:

= [ ] , = [ ]

1. Suma wektorów

+ = [ + + + ]

2. Iloczyn skalarny

Definicja 1

⋅ = +

+

Definicja 2

⋅ = φ

v2

φ

v1

3. Iloczyn wektorowy

Definicja 1

 





× =





 

y

uy

ux

x

uz

z

Definicja 2

× = ⋅ φ

v1x v2

v2

u

φ

v1

Reprezentacja osi obrotu:

y

Θ

( ) x

u

( )

z

+

punkcie ( ) #-

= [ ]

gdzie

−

−

−

=

, =

, =

= ( − ) + ( −

) + ( −

)

./& = .

(

1. " )

) !

2. # )

* + , np.z

+-!

, & 0!1,"/

,

3. # +Θ.

4. Transformacja odwrotna do wykonanej w kroku 2.

5. Transformacja odwrotna do wykonanej w kroku 1.

Krok 1: 2 /+

+

y

Θ

u

x

z

'









(− −

−

) = 











−

−

−



"+ ( )

do punktu (

)

.

Krok 2: /+

$#

2& 0!1,"

y

′

u

b

x

α

c

z

a

= [ ], #+

′ = [ ], !1,"

= [ ]

- wersor osi z

". /(x-z) jest

. ′ #α .

.'α , lub sinα i cosα ?

Z definicji iloczynu skalarnego

′ ⋅ = ⋅ +

⋅ + ⋅ =

(def.1)

′ ⋅ =

′ ⋅ =

′ ⋅ ⋅

α

(def.2)

′ = + = , =

′ ⋅ =

⋅ α

#

α =

Z definicji iloczynu wektorowego

 

′ × =





  = ⋅

(def.1)





′ × =

′ ⋅ ⋅ α =

⋅

⋅ α

(def.2)

#

α =

/ & 0

!1,"(&' 1#α.

Dokonano obrotu opisanego przez macierz













(α)

= 





−











2&

y

β

x

′

′

d

z

a

′ = [ ], &! ( "

0!1," /

′ = [ ], &! ( "

wektor u,

= [ ]

- wersor osi z.

.'β , lub sinβ i cosβ ?

Z definicji iloczynu skalarnego

′ ⋅ =

⋅ + ⋅ + ⋅ =

(def.1)

′ ⋅ =

′ ⋅ =

′ ⋅ ⋅

β

(def.2)

′ = + = + ( + ) = + + =

=

′ ⋅ =

⋅ α

#

β =

Z definicji iloczynu wektorowego

 

′ × =





= ⋅

(− )





(def.1)





′ × =

′ ⋅ ⋅ β =

⋅

β

(def.2)

#

β = −

2& (&'

#β.

Dokonano obrotu opisanego przez macierz













(β) = −











0 & '*

) + *+

macierze:

Krok 3. # +Θ.

 Θ

Θ



− Θ Θ







(Θ ) = 











Krok 4. Transformacja odwrotna do wykonanej w kroku 2

−

−

(β) ⋅ (α)

Krok 5. Transformacja odwrotna do wykonanej w kroku 1

−(− −

−

)

1 *+ & '*

* *'* 2

(Θ) = (− −

−

) ⋅ (α) ⋅ (β) ⋅ (Θ) ⋅

⋅ −

−

−

(β) ⋅ (α) ⋅

(− −

−

)

"

Dane:

#) (

" +



( ) = 



,



(



) = 



.



Obiekt:

Sfera o promieniu =

) (



) = 



.



3.24 +Θ

+ )++.

5 , x-y) przed dokonaniem

obrotów pokazuje rysunek.

" * 4!

5 ,$6-+

natomiast ich obraz perspektywiczny (bez osi)

*+!