3D Transformation 3D Transformation

3D Geometric

T f i

Transformations

Chapter 5 Chapter 5

Intro. to Computer Graphics Spring 2008, Y. G. Shin

3D Transformation 3D Transformation

Right-handed coordinate system

⎥ ⎤

⎢ ⎡

⎥ ⎤

⎢ ⎡

⎥ ⎤

⎢ ⎡ ' x x

x

x

t

x

y

⎥ ⎥

⎥

⎢ ⎢

⎢

⎥ ⎥

⎥

⎢ ⎢

= ⎢

⎥ ⎥

⎥

⎢ ⎢

⎢ '

'

3 2

1

3 2

1

z y t

z z

z

t y

y y

z

y

⎥ ⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎣ 0 0 0 1 1

3 2

1

z z t z

z h

z

z

x

z

3D Transformation 3D Transformation

⎤

⎡ 1 0 0

ƒ Translation

⎤

⎡

ƒ Scaling

⎥ ⎥

⎤

⎢ ⎢

⎡

0 1

0

0 0

1

t t

⎥ ⎥

⎤

⎢ ⎢

⎡

0 0

0

0 0

x

0

s s

⎥ ⎥

⎥

⎢ ⎢

⎢

1 0

0

0 1

0

z y

t t

⎥ ⎥

⎥

⎢ ⎢

⎢

0 0

0

0 0

0

z y

s s

⎥ ⎦

⎢ ⎣ 0 0 0 1

z

⎥

⎢ ⎦

⎣ 0 0 0 1

z

3D Rotation 3D Rotation

⎥⎥

⎤

⎢⎢

⎡ −

0 0 cos

sin

0 0 sin

cos

θ θ

θ θ

⎥⎤

⎢⎡

0 i

0

0 0

0 1

θ θ

⎥⎥

⎥

⎢ ⎦

⎢⎢

⎣

=

1 0 0

0

0 1 0

0

0 0 cos

) sin

( θ θ

z θ R

⎥⎥

⎥⎥

⎢⎢

⎢⎢ −

= 0 sin cos 0

0 sin

cos ) 0

( θ θ

θ θ θ

Rx

⎦

⎣ 0 0 0 1 ⎥

⎢ ⎦

⎣0 0 0 1

3D rotations do NOT commute!

3D rotations do NOT commute!

Rotation About Arbitrary Axis

Rotation About Arbitrary Axis

Rotation about an arbitrary axis Rotation about an arbitrary axis

1 Translation : rotation axis passes ^{⎡1 0 0 −x ⎤} 1. Translation : rotation axis passes

through the origin

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

−

= −

1 0 0 0

1 0 0

0 1 0

0 0 1

1 1 1

z y x T

2. Make the rotation axis on the z-axis

) ( )

(α R β

R ⋅

⎦

⎣0 0 0 1

) ( )

(α _y β

x R

R

3. Do rotation

) (θ

R_z(θ) ^(x1,y1,z1) R

4. Rotation & translation

) ( )

( ¹

1 ⁻1 β ⁻ α

− ⋅R ⋅R

T ⋅R_y (β)⋅R_x (α) T

Rotation about an arbitrary axis

T R

R R

R R

T

R(θ) = ⁻¹ ⋅ _x⁻¹(α)⋅ _y⁻¹(β)⋅ _z(θ)⋅ _y(β)⋅ _x(α)⋅

Rotation About Arbitrary Axis Rotation About Arbitrary Axis

R t t t th i

„ Rotate u onto the z-axis

Rotation About Arbitrary Axis Rotation About Arbitrary Axis

„ Rotate a unit vector u onto the z axis

„ Rotate a unit vector u onto the z-axis

„ u’ : Project u onto the yz-plane to compute angle α

„ u’’ : Rotate u about the x-axis by angleu : Rotate u about the x axis by angle αα

„ Rotate u’’ onto the z-asis

Rotation About Arbitrary Axis Rotation About Arbitrary Axis

„ Rotate u’ about the x-axis onto the z-axis

„ Let u=(a,b,c) and thus u’=(0,b,c)

„ Let u_z=(0,0,1)

⋅′u c u

y u’ u

2

cos 2

c b

c

z z

= +

= ′

u u

u α u

α x

z sin

x

z u u u

u

u′× = ′ α ^{z uz = (0,0,1)}

xb

= u

sin α = b = b

2

sin

c b

z

= +

= ′

u

α u

Rotation About Arbitrary Axis Rotation About Arbitrary Axis

R t t ’ b t th i t th i

„ Rotate u’ about the x-axis onto the z-axis

„ Since we know both cos α and sin α, the rotation matrix can be obtained

can be obtained

⎟⎟

⎞

⎜⎜

⎛ −

0 0

0 0

0

1 c b

y u’

⎟⎟

⎟⎟

⎜⎜

⎜⎜

+

= +

0 0

0 0

) (

2 2

2 2

2 2

2 2

b c b

b c b c

b

x α

R

α x u u

β

⎟⎟

⎜ ⎠

⎜

⎝

+ +

1 0

0 0

2 2

2

2 c b c

b ^x

z α

u’’

„ Rotate u’’ onto the z-asis

„ With the similar way, we can compute the angle β

u

Rotation about an arbitrary axis

i th l t i

using orthogonal matrix

„ Unit row vector of R rotates into the principle axes x, y, and z.

⎤

⎡ r

r

r

y

P3 y

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

=

x x

x

r r

r

R

3 2

1

P1

P2 3

P3

⎥⎦

⎢⎣ r

r

r

x z

x z P^{2 1}

P

z z

Initial position Final position

Rotation about an arbitrary axis

i th l t i

is the unit vector along P P that

R

using orthogonal matrix

is the unit vector along P₁P₂ that will rotate into the positive z axis

P P R

[ ]

2 1

2 1 3

2

1

P P

P r P

r r

R

=

=

Rx

y

P3

Ry

is perpendicular to the plane P P and P that will rotate into

R

Rz

P3 P¹

P2 3

P₁, P₂ and P₃ that will rotate into the positive x axis

P P P

P

x z ²

P P¹

[ ]

2 1 3

1

2 1 3

1 3

2

1

P P P P

P P P

r P r

r

R

×

= ×

z

=

Rotation about an arbitrary axis

i th l t i

using orthogonal matrix

Finally

y R Finally,

[

]

y

r r r R R

R =

= ×

Rx

P R

P3

Ry

Rz

P x P3 P¹

P2

P1

⎥ ⎤

⎢ ⎡ R

z ² P

⎥ ⎥

⎥

⎢ ⎢

= ⎢

⋅

=

R R R

R

R ( β ) ( α )

⎥⎦

⎢⎣ R

Reflection Reflection

„

Reflection about xy plane

⎥ ⎤

⎢ ⎡ 1 0 0 0

y

⎥ ⎥

⎥

⎢ ⎢

= ⎢

0 1

0 0

0 0

1 0

RF

⎥ ⎥

⎢ ⎦

⎢

⎣

−

1 0

0 0

0 1

0 0

z z

fl b b l

Reflection about an arbitrary plane

1. Translate a known point P, that lies in the reflection plane, to the origin of the coordinate system

of the coordinate system

2. Rotate the normal vector to the reflection plane at the origin until reflection plane at the origin until the plane lies on z=0 plane.

3. After also applying the above

P

3. After also applying the above

transformations to the object, reflect the object through z=0 coordinate plane.

4. Perform the inverse transformations.

Shear Shear

„

Shear in x-direction

⎤

⎡ + +

⎤

⎡

⎤

⎡1 a b 0 x x ay bz

⎥⎥

⎥⎤

⎢⎢

⎢⎡ + +

⎥ =

⎥⎥

⎤

⎢⎢

⎢⎡

⎥⎥

⎥⎤

⎢⎢

⎢⎡

= 0 0 1 0

0 0

1 0

0 1

z y

bz ay

x

z y x b

a SH_x

⎥⎥

⎢ ⎦

⎢

⎥ ⎣

⎥

⎢ ⎦

⎢

⎥⎣

⎥

⎢ ⎦

⎢

⎣0 0 0 1 1 1

0 1

0

0 z z

0

When b 0 =

When b =

Shearing along xy plane Shearing along xy-plane

Direction vector

y

⎥⎥

⎤

⎢⎢

⎡1 0 0

c a

(a, b, c)

⎥⎥

⎥

⎢⎢

= ⎢0 1 0

c bc SH_xy

⎥ z

⎥⎥

⎢ ⎦

⎢⎢

⎣0 0 0 1 0 1

0 0

⎞

⎛ a b

(0, 0, 1) is moved to

⎠

⎜ ⎞

⎝

⎛ , , 1

c b c

a

How we represent Rotations?

How we represent Rotations?

ƒ Rotation (direction cosine) matrix

ƒ Euler angles u e a g es

ƒ Angular Displacement

ƒ Unit quaternions

ƒ Unit quaternions

Euler Angles (Euler’s Theorem) Euler Angles (Euler’s Theorem)

Arbitrary rotation can be represented by three rotation along x,y,z axis.

⎤

⎡

=

0 C

S S

S C C

C

) ( ) ( )

( )

, , (

β β

β

γ β

α α

β γ

S S C

S C R

R R

R_{XYZ z y x}

⎥⎥

⎥⎤

⎢⎢

⎢⎡

− +

+

= 0

0 C

S - S S C C

C

β β

β

γ α γ

β α γ

α γ

β α β

α

γ α γ

β α γ

α γ

β α β

α

S C C

S S C

C S

S S C

S

S S C

S C

⎥⎥

⎢ ⎦

⎢

⎣

−

1 0

0 0

γ 0 β γ

β

β C S C C

S

↔ rotation matrix

) , , ( :

angle

Euler angle : R

( γ , β , α )

Euler R

γ β α

Not easy

Gimble

Gimble

Gimble Lock Gimble Lock

„

Rotation about three orthogonal axes

12 bi ti

„ 12 combinations

„ XYZ, XYX, XZY, XZX

„ YZX, YZY, YXZ, YXY

„ YZX, YZY, YXZ, YXY

„ ZXY, ZXZ, ZYX, ZYZ

„

Gimble lock

„ Coincidence of inner

t d t t

most and outmost gimbles’ rotation axes

„ Loss of degree of

„ Loss of degree of freedom

Gimble Lock Gimble Lock

bl l k b f

„ Gimble lock gives ambiguous representation of a rotation angle.

„ Two different Euler angles can represent the same

„ Two different Euler angles can represent the same orientation

π π

) 0 2 ,

1

,

(θ π

) ,r ,r (r

R =

= , )

, 2

2

_{= (} 0 π ₋ θ R

and

„ This ambiguity brings unexpected results of animation where frames are generated by interpolation

interpolation.

Gimble Lock Gimble Lock

Y축 Z축

Y축 Z축

X축 Y축

Z축

X축

Y축 Z축

Y축 Z축

X축 X축 X축 X축X축 X축

) 0 0 0 ) (

0 0 0

R(^(0,0,0) R(50,0,0) R(50 π 0) R(π π 0)

R ,0)

, 2 50

( π

R R(50,π,0) R(π,π,0)

Y축 Z축

Y축 Z축

Y축 Z축

Y축 Z축

Y축 Z축

X축 X축 X축 X축 X축

) 0 , 0 , 0

R( ,0)

, 2 0 ( π

R , 50)

, 2 0

( π −

R , )

, 2 0

( π π

R R(π,π,0)

Gimble Lock Gimble Lock

π

⎤

⎡ 0 0 − 01

y.

arbitraril and

set and

2 ,

Set θ

₌ π θ

θ

⎥⎥

⎥⎤

⎢⎢

⎢⎡

− +

+

= −

= 0 0

0 0

0 1 0

0

z x z

x z

x z

x z

y

x c c s s c s s c

c c s

s s

c c

) s , , (

R θ θ θ

⎥⎥

⎢ ⎦

⎢

⎣

+

1 0

0 0

0

z 0

x z

x z

x z

xc s s c s s c

c

⎤

⎡ 0 0 1 0

⎥⎥

⎥⎤

⎢⎢

⎢⎡

−

−

−

= cos( ) sin( ) 0 0

0 0

) cos(

) sin(

0 1 0

0

z x

z x

θ θ

θ θ

θ θ

θ θ

f l d d h d ff

⎥⎥

⎢ ⎦

⎢

⎣

−

−

−

1 0

0 0

0 0

) sin(

)

cos(θ_x θ_z θ_x θ_z Transformation only depends on the difference

We lost one DOF.

Angular Displacement Angular Displacement

Arbitrary rotation can be represented by one rotation

„ Arbitrary rotation can be represented by one rotation (by a scalar angle) around an axis(unit vector)

„ No Gimble lock but NOT smooth interpolation for

„ No Gimble lock but NOT smooth interpolation for animation

„ Supported by OpenGL

θ angle an

by

r unit vecto a

about

Rotating v r u r

θ vr

ur v r = ( u r ⋅ v r ) u r + cos θ ( v r − ( u r ⋅ v r ) u r )

θ ^u

) (

sin

) ) (

( cos )

(

v u

u v u v

u v u v

r× r +

⋅

− +

⋅

=

θ

θ

)

(

Angular Displacement Angular Displacement

) (

sin )

) (

( cos )

( u v u v u v u u v

v r

= r ⋅ r r + θ r − r ⋅ r r + θ r × r sin

cos v v

v r

+ cos θ v r

+ sin θ v r

v + θ + θ

=

vr

v

θ

vr

θ

3

Quaternions Quaternions

M lti li ti f t l b Ù

„

Multiplication of two complex numbers Ù Rotation in 2D space

imaginary

2 2

1 1

2

1

p = ( a + b i )( a + b i )

1

p

θ real

2

1 θ

θ +

) ( _{1 2}

2

1 θ θ θ

θ

=

= e

e

e

θ θ

θ i eⁱ bi

a+ = cos + sin =

„

Quaternions are 4D analogs of complex number

„

Multiplication of two quaternions Ù

„

Multiplication of two quaternions Ù

Rotation in 3D space

Quaternions Quaternions

„ Quaternions are defined using one real part and three imaginary quantities, i, j and k

k v j

v i

v s

v s

q = r ( , ) = +

+

+

j ik

ki i

kj jk

k ji

ij

k j

i

=

=

=

=

=

=

−

=

=

=

1 ,

2

j ik

ki i

kj jk

k ji

ij = − = , = − = , = − =

2 2

2

v

v v

s

q = s + v

+ v

+ v

q = + + +

Quaternions

Pr

Quaternions

A t ti f t i t

P ( x y z )

„ A rotation of a vector point about the unit vector by an angle can be computed using the quaternion

u θ ^{P = ( x , y , z )}

the quaternion

2 ) sin 2 ,

(cos )

, (

), ,

0

( = =

= p q s v u

P θ θ

2 2

−1

⋅

⋅

= q q

rotated

P

P where q

= ( s , − v )

) ,

(

and

2 1

1 2 2

1 2 1

2 1 2

1

2 2 2

1 1 1

v v

v s v

s v

v s

s q

q

) v , (s q

) v , (s

q r r r r r r

r r

× +

+

⋅

−

=

⋅

=

=

When

) ,

(

2

1

q s s v v s v s v v v

q + + ×

) ,

0

rotated

( p

P =

) (

) (

2 )

2

(

p v

v p

v v

p v p

p

= s + ⋅ + s × + × ×

Quaternions Quaternions

θ

Quaternions for Rotation: θ )

sin 2 2 ,

(cos )

, (

), , 0

( θ θ

u v

p

P = q = s =

) (

) (

2 )

2

(

) (

) (

2 )

2

(

p v

v p

v v

p v p

p

= s + ⋅ + s × + × ×

(

)

cosθ θ

=

= v

s

[Example] Rotation about z-axis

( )

sin 2 1 , 0 , 0

2 ,

cos =

= v

s

•pp_rotated

•

o p

z ur

Q t i R t ti i M t i F

Quaternion Rotation in a Matrix Form

Assuming that a unit quaternion has been created in the form:

Then the quaternion can then be converted into

( s , a , b , c )

Then the quaternion can then be converted into

a 4x4 rotation matrix using the following expression

⎤

⎡

⎥⎥

⎤

⎢⎢

⎡

−

−

− +

+

−

−

−

= ^{2 2}

2 2

2 2

2 2

1 2

2

2 2

2 2

2 2

1 )

( ab sc a c bc sa

sb ac

sc ab

c b

M_R θ

⎥⎥

⎢ ⎦

⎢

⎣ 2ac − 2sb 2bc + 2sa 1− 2a² − 2b²

I i f h i i i

It is often necessary to have a quaternion rotation in a matrix form, e.g., to load onto graphics hardware for hardware vertex transformations

hardware vertex transformations.

Quaternions Quaternions

„ Useful for animations

ƒ Independent definition of an axis of rotation and an angle

ƒ Smooth interpolation

ƒ No Gimble lock

Far more complicated to read and conceptualize than

„ Far more complicated to read and conceptualize than Eular angle

„ Interpolation can be expensive in practice

„ Interpolation can be expensive in practice

Modeling Transformation

Modeling Transformation

Modeling Transformation Modeling Transformation

?

?

Modeling Transformation Modeling Transformation

Modeling Transformation

y '

y y

y

A

r p B

x ' x

y _{( , ) ( , )}

'

'

Scale x y T x y M

= ⋅

y

p A B

r

xx

Modeling Transformation Modeling Transformation

y y

Translate by -r, bringing t th i i

p A B

x

pAB r

r to the origin.

pA

x

p A B

r

x

y y

)

Rotate ( A translate ( A )

p B

x x

x x

Modeling Transformation Modeling Transformation

translate both A and B by –p, bringing p to the origin.

y y

g g p g

p B ^p B

x

p B

x

y y

Rotate A&B Translate

y y

Rotate A&B

x

x

Modeling Transformation

Modeling Transformation

Wh t t!

What next!