CENG 477 Introduction to Computer...

CENG 477Introduction to Computer Graphics

Modeling Transformations

Modeling Transformations• Model coordinates to World coordinates:

World coordinates:All shapes with theirabsolute coordinates and sizes.

zworld

xworld

yworldModel coordinates:All shapes with theirlocal coordinates and sizes.

CENG 477 – Computer Graphics 2

Basic Geometric Transformations• Used for modeling, animation as well as viewing• What to transform?

– We typically transform the vertices (points) and vectors describing the shape (such as the surface normal)

• This works due to the linearity of transformations• Some, but not all, transformations may preserve attributes like

sizes, angles, ratios of the shape


Translation• Simply move the object to a relative position

TPP

PTP

+=

y'x'

=tt

=yx

=

t+y=y't+x=x'

y

x

yx

'

' ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

P

P'


Rotation• A rotation is defined by a rotation axis and a rotation angle• For 2D rotation, the parameters are rotation angle (θ) and the

rotation point (xr, yr)• We reposition the object in a circular path around the rotation

point (pivot point)

a

xr

yr


Rotation• When (xr, yr)=(0,0) we have:

P

P'

θφθφθφ

θφθφθφ

cossinsincos)sin(sinsincoscos)cos(

rrryrrrx

+=+=ʹ

−=+=ʹ

φ

φ

sincosryrx

=

=The original coordinates are:

θθ

θθ

cossinsincosyxyyxx

+=ʹ

−=ʹSubstituting them in the first equation we get:

In the matrix form we have: PRP ⋅=ʹ

⎥⎦

⎤⎢⎣

⎡ −=

θθ

θθ

cossinsincos

R


Rotation• Rotation around an arbitrary point (xr, yr)

• These equations can be written as matrix operations (we will see when we discuss homogeneous coordinates)

P

P'

θθ

θθ

cos)(sin)(sin)(cos)(

rrr

rrr

yyxxyyyyxxxx

−+−+=ʹ

−−−+=ʹ

(xr,yr)


Scaling• Changes the size of an object • Input: scaling factors (sx, sy)

PSP

S

' ⋅

⎥⎦

⎤⎢⎣

⎡

=

ss

=

ys=y'xs=x'

y

x

yx

00

PP'

non-uniform vs.uniform scaling


Homogenous Coordinates• Translation is additive, rotation and scaling is multiplicative

(and additive if you rotate around an arbitrary point or scale around a fixed point)

• Goal: Make all transformations as matrix operations• Solution: Add a third dimension

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

1yx

=hyhxh

=hyx

=Phy=y

hx=x h

hhh


Homogenous Coordinates• In HC, each point now becomes a line• The entire line represents the same point• The original (non-homogeneous) point resides on the w=1

plane

x

y

w

w = 1p1

p2


All points along this linein HC represent the samepoint, p1

Transformations in HC

• Translation:

• Rotation:

• Scaling:

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

1001001

where y

x

yxyx' t

t=t,tTPt,tT=P

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⋅

1000cossin0sincos

where θθθθ

=θRPθR=P'

( ) ( )⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅ʹ

1000000

where y

x

yxyx ss

=s,sSPs,sS=P


Transforming Vectors• Vectors can be rotated and scaled• But translating a vector does not change it! Why?

– A vector is a difference between two points– These two points translate the same way– So the vector remains the same

• Mathematically this can be achieved by setting the last coordinate of a vector to 0 (the last coordinate of points should be 1)


2D point 𝑥𝑦 in HC is equal to

𝑥𝑦1

2D vector 𝑥𝑦 in HC is equal to

𝑥𝑦0

Composite Transformations• Often, objects are transformed multiple times• Such transformations can be combined into a single composite

transformation• E.g. Application of a sequence of transformations to a point:

PMPMMP 12

⋅=

⋅⋅=ʹ


Composite Transformations• Composition of the same types of transformations is simple• E.g. translation:

( ) ( ) ( )yyxxyy

xx

y

x

y

x

yxyx t+t,t+tT=tttt

=tt

tt

=t,tTt,tT 212121

21

1

1

2

2

1122

1001001

1001001

1001001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

PTTPTTP⋅⋅=

⋅⋅=ʹ

)},(),({

}),({),(

1122

1122

yxyx

yxyx

tttttttt


( ) ( ) ( )yyxxyy

xx

y

x

y

x

yxyx ss,ss=ssss

=ss

ss

=s,ss,s 212121

21

1

1

2

2

1122

1000000

1000000

1000000

⋅⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅ SSS

( ) ( )

( )φ+θ=φ+θφ+θφ+θφ+θ

=φθ+φθφθ+φθφθφθφθφθ

=θθθθ

=φθ

R

RR

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −

⋅

1000)cos()sin(0)sin()cos(

1000coscossinsinsincoscossin0cossinsincossinsincoscos

1000cossin0sincos

1000cossin0sincos

ϕϕ

ϕϕ

Composite Transformations• Rotation and scaling are similar:


Rotation Around a Pivot Point• Step 1: Translate the object so that the pivot point moves to

the origin

( )rr y,x −−T

( )rr y,xM −−= T1


Rotation Around a Pivot Point• Step 2: Rotate around origin

( )θR

( )θR=2M


Rotation Around a Pivot Point• Step 3: Translate the object so that the pivot point is back to its

original position

( )rr y,xT

( )rr y,xM T=3


Rotation Around a Pivot Point• The composite transformation is equal to their successive

application:

( ) ( ) ( )rrrr y,xy,xMMMM −−== TRT θ123


Scaling w.r.t. a Fixed Point• The idea is the same:

– Translate to origin– Scale– Translate back

( ) ( ) ( )

( )( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−⋅⋅

1001010

1001001

1000000

1001001

yfy

xfx

f

f

y

x

f

f

ffyxff

syssxs

=yx

ss

yx

=y,xs,sy,x TST

( )ff y,x −−T

( )yx s,sS

( )ff y,xT


Order of matrix compositions• Matrix composition is not commutative. So, be careful when

applying a sequence of transformations.

Pivot

21

Rotation and translation

Pivot

12

Translation and rotation


Other Transformations• Reflection: special case of scaling

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

100

010

001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

100

010

001

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−

100

010

001


Other Transformations• Shear: Deform the shape like shifted slices (or deck of cards).

Can be in x or y direction

(1,1)(3,1)

(2,1)

x ' x shx y y ' y

1 shx 00 1 00 0 1

(0,1)


3D Transformations• Similar to 2D but with an extra z component• We assume a right handed coordinate system• With homogeneous coordinates we have 4 dimensions • Basic transformations: Translation, rotation, scaling

x

y

z

z

x

y

y

z

x

Equivalent ways of thinking about a right-handed CS


Translation• Move the object by some offset:

z

y

x

z

y

x

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

ʹ

ʹ

ʹ

11000100010001

1zyx

ttt

zyx

z

y

x

PTP ⋅=ʹ

P

Pʹ


Rotation• Rotation around the coordinate axes

• Positive angles represent counter-clockwise (CCW) rotation when looking along the positive half towards origin

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

x

y

x-axis y-axis z-axis


Rotation Around Major Axes• Around x:

• Around y:

• Around z:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

10000cossin00sincos00001

)(θθ

θθθxR PRP ⋅=ʹ )(θx

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

1000010000cossin00sincos

)(θθ

θθ

θzR PRP ⋅=ʹ )(θz

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

10000cos0sin00100sin0cos

)(θθ

θθ

θyR PRP ⋅=ʹ )(θy


Rotation Around a Parallel Axis• Rotating an object around a line parallel to one of the axes:

Translate to a major axis, rotate, translate back• E.g. rotate around a line parallel to x-axis:

z

y

x

z

y

x

z

y

x

z

y

x

Translate Rotate Translate back

PTRTP ⋅−−⋅⋅=ʹ ),,0()(),,0( ppxpp zyzy θ

(yp, zp)


Rotation Around an Arbitrary Axis• Step 1: Translate the object so that the rotation axis passes

though the origin• Step 2: Rotate the object so that the rotation axis is aligned

with one of the major axes• Step 3: Make the specified rotation• Step 4: Reverse the axis rotation• Step 5: Translate back

z

y

x


Rotation Around an Arbitrary Axis


Rotation Around an Arbitrary Axis• First determine the axis of rotation:

• u is the unit vector along v:

),,( 12121212 zzyyxx −−−=−= PPv

),,( cba==vvu


Rotation Around an Arbitrary Axis• Next translate P1 to origin:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

−

−

=

1000100010001

1

1

1

zyx

T


Rotation Around an Arbitrary Axis• Then align u with one of the major axis (x, y, or z)• This is a two-step process:

– Rotate around x to bring u onto xz plane (CCW)– Rotate around y to align the result with the z-axis (CW)

z

x

u

y

α

z

xu

y

β

We need cosine and sine of angles α and ß


Rotation Around an Arbitrary Axis• We need cosine and sine of angles α and ß:

z

x

uu'

uz

y

αux

uyu = ux + uy + uz = ux + u’

22z whereucos cbddc

+==ʹ

=u

α

db

=ʹ

=uyusinα

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

1000

00

000001

)(

dc

db

db

dc

x αR


Rotation Around an Arbitrary Axis• We need cosine and sine of angles α and ß:

z

x

y

ux

222

2222

coscba

cbuu zy

++

+=

+=

uβ

22zy uu +

β u

222sin

cbaaux

++==

uβ

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++

+

++−

++++

+

=

1000

00

0010

00

)(

222

22

222

222222

22

cbacb

cbaa

cbaa

cbacb

y βR


Note that 𝑎& + 𝑏& + 𝑐& = 1

-

+

Rotation Around an Arbitrary Axis• Putting it all together:

),,()()()()()(),,()( 111111 zyxzyx xyzyx −−−⋅⋅⋅⋅−⋅−⋅= TRRRRRTR αβθβαθ

This is the actual desiredrotation. Other terms arefor alignment and undoingthe alignment


-+

Alternative Method• Assume we want to rotate around the unit vector u:• We create an orthonormal basis (ONB) uvw:


z

x

u

y

z

x

u

y

v

w

1) To find v, set the smallestcomponent of u (in anabsolute sense) to zero andswap the other two whilenegating one:

E.g. if u = (a, b, c) with c being the smallest absolutevalue then v = (-b, a, 0)

Note that we are just finding one of the infinitely many solutions

This corresponds to projectingthe vector to the nearest major planeand rotating it 90° along the axisperpendicular to that plane

2) w = u x v

3) Normalize v and w

Alternative Method• Now rotate uvw such that it aligns with xyz: call this

transform M• Rotate around x (u is now x)• Undo the initial rotation: call this M-1

• Finding M-1 (rotating xyz to uvw) is trivial:


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=−

1000000

1

zzz

yyy

xxx

wvuwvuwvu

M

Verify that this matrix transforms x to u, y to v, and z to w

• How to transform x = [1 0 0 0]T

such that it turns into [ux uy uz 0]T

• Similar for the y and z axis

Alternative Method• Finding M is also trivial as M-1 is an orthonormal matrix (all

rows and columns are orthogonal unit vectors)• For such matrices, inverse is equal to transpose:


⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000000

zyx

zyx

zyx

wwwvvvuuu

M

Alternative Method• The final rotation transform is:

• We assumed that the origin of uvw is the same as the origin of xyz

• Otherwise, we should account for this difference:


𝑀,-𝑅/(𝜃)𝑀

𝑇,-𝑀,-𝑅/(𝜃)𝑀𝑇

Translate the originof uvw to xyz

Undo thetranslation

Scaling• Change the coordinates of the object by scaling factors

x 'y 'z '1

sx 0 0 00 s y 0 00 0 sz 00 0 0 1

xyz1

P' S P

z

y

x

z

y

x

PSP ⋅=ʹ

P

Pʹ


Scaling w.r.t. a Fixed Point• Translate to origin, scale, translate back

z

y

x

z

y x

z

y

x

z

y

x

Translate Scale Translate back

PTSTP ⋅−−−⋅⋅=ʹ ),,(),,( ffffff zyxzyx


Reflection• Reflection over the major planes:

z

y

x

z

y

x

z

y

x

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

1000010000100001

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

1000010000100001

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

1000010000100001

How about reflection over an arbitrary plane?


Transforming Normals• When we transform an object, what happens to its normals?• Do they get transformed by the same matrix or does it require

a different one?

x

y

1

1 x

y

1

3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

100010003

S

Scale by:

𝒏 =1/√21/√20

𝒏 =3/√21/√20


Transforming Normals• After the transformation the normal is no longer perpendicular

to the object• Also it is not a unit vector anymore

x

y

1

1 x

y

1

3

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

100010003

S

Scale by:

𝒏 =1/√21/√20

𝒏 =3/√21/√20


Transforming Normals• Rotation and translation has no problems• But, since all transformations are combined into a single

matrix M, we should consider the general case.

• We must have n.(b-a) = n.v = 0 and this relationship should be preserved after the transformation

n

a

bv


Transforming Normals• That is n.v = 0 and n'.v' = 0 where v' = Mv and n' = Zn• Z is the matrix we are looking for• How to compute Z?

n

a

bv


Transforming Normals• n.v = nTv = 0• n'.v' = n'Tv' = n'T Mv = nTZTMv = 0• If ZTM = I (identity) the relationship will be preserved• So Z = (M-1)T

• Note that this is equal to (MT)-1 as (M-1)T= (MT)-1 for a square (n by n) matrix M

n

a

bv


A Word on Notation• Until now, we performed transformations by multiplying our

points from the right:

• Another notation is to multiply from the left:


𝑥9𝑦9𝑧91

=

𝑎 𝑏 𝑐 𝑑𝑒 𝑓 𝑔 ℎ𝑖 𝑗 𝑘 𝑙0 0 0 1

𝑥𝑦𝑧1

𝑥9 𝑦9 𝑧9 1 = 𝑥 𝑦 𝑧 1

𝑎 𝑏 𝑐 𝑑𝑒 𝑓 𝑔 ℎ𝑖 𝑗 𝑘 𝑙0 0 0 1

D

Note that in this caseeverything is transposed

A Word on Terminology• Imagine a 2D rotation matrix such as:

• Transforming an object by this matrix will not change its shape• If we also add translation:

• The shape will remain intact


cos(𝜃) −sin(𝜃) 0sin(𝜃) cos(𝜃) 00 0 1

cos(𝜃) −sin(𝜃) 𝑡/sin(𝜃) cos(𝜃) 𝑡M0 0 1

A Word on Terminology• In general, an arbitrary sequence of rotation and translation

matrices will have the following form:

• Such transformations are called rigid-body transformations• A shape may be rotated and translated by its form is not altered

in any way


𝑟-- 𝑟-& 𝑡/𝑟&- 𝑟&& 𝑡M0 0 1

A Word on Terminology• Imagine also adding scaling• The matrix will now look like:

• Such transformations will not necessarily preserve lengths and angles, but parallel lines will remain parallel


𝑎 𝑏 𝑡/𝑐 𝑑 𝑡M0 0 1

where a, b, c, d contain the effect ofrotation and scaling combined

Original Rotation Rotation and scaling

A Word on Terminology• Such transformations are called affine transformations• An arbitrary sequence of rotation, translation, scaling, and

shearing will produce an affine transformation• Note that we still have some degrees of freedom left in the last

row of our matrix:

• By using this we can create projective transformations in which parallel lines may no longer be parallel


𝑎 𝑏 𝑡/𝑐 𝑑 𝑡M0 0 1

CENG 477 Introduction to Computer...

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