© 2005 Pearson Education Computer Graphics with OpenGL 3e

© 2005 Pearson Education© 2005 Pearson Education

Computer Graphicswith OpenGL 3e

© 2005 Pearson Education© 2005 Pearson Education

Chapter 5Geometric

Transformations

© 2005 Pearson© 2005 Pearson EducationEducationComputer Graphics with OpenGL, Third Edition, by Donald Hearn and M.Pauline Baler.IBSN 0-12-0-153-90-7 @ 2004 Pearson Education, Inc., Upper Saddle River, NJ. All right reserved

Geometric Transformations

• Basic transformations:– Translation– Scaling– Rotation

• Purposes:– To move the position of objects– To alter the shape / size of objects– To change the orientation of objects


Basic two-dimensional geometric transformations (1/1)

• Two-Dimensional translation– One of rigid-body transformation, which move objects without

deformation– Translate an object by Adding offsets to coordinates to generate

new coordinates positions

– Set tx,ty be the translation distance, we have

– In matrix format, where T is the translation vector

xtx'x yty'y

y

xP

y

x

t

tT

'y

'x'P

TP'P


– We could translate an object by applying the equation to every point of an object.

• Because each line in an object is made up of an infinite set of points, however, this process would take an infinitely long time.

• Fortunately we can translate all the points on a line by translating only the line’s endpoints and drawing a new line between the endpoints.

• This figure translates the “house” by (3, -4)



Translation Example

y

x0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(1, 1) (3, 1)

(2, 3)



• Two-Dimensional rotation– Rotation axis and angle are specified for rotation– Convert coordinates into polar form for calculation

– Example, to rotation an object with angle a• The new position coordinates

• In matrix format

• Rotation about a point (xr, yr)

cosrx sinyy

cossin

sincosR PR'P

cossinsinsinsincos)sin('

sincossinsincoscos)cos('

yxrrry

yxrrrx

cos)(sin)('

sin)(cos)('

rrr

rrr

yyxxyy

yyxxxx

r

r


– This figure shows the rotation of the house by 45 degrees.

• Positive angles are measured counterclockwise (from x towards y)

• For negative angles, you can use the identities:– cos(-) = cos() and sin(-)=-sin()


6

y

x 0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6


Rotation Example

y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(3, 1) (5, 1)

(4, 3)



• Two-Dimensional scaling– To alter the size of an object by multiplying the coordinates

with scaling factor sx and sy

– In matrix format, where S is a 2by2 scaling matrix

– Choosing a fix point (xf, yf) as its centroid to perform scaling

xsx'x ysyy

y

x

s0

0s

'y

'x

y

x PS'P

)s1(ysy'y

)s1(xsx'x

yfy

xfx


– In this figure, the house is scaled by 1/2 in x and 1/4 in y• Notice that the scaling is about the origin:

– The house is smaller and closer to the origin



Scaling

Note: House shifts position relative to origin

y

x 0

1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

1

2

1

3

3

6

3

9

– If the scale factor had been greater than 1, it would be larger and farther away.

WATCH OUT: Objects grow and move!


Scaling Example

y

0 1

1

2

2

3 4 5 6 7 8 9 10

3

4

5

6

(1, 1) (3, 1)

(2, 3)


Homogeneous Coordinates

• A point (x, y) can be re-written in homogeneous

coordinates as (xh, yh, h)

• The homogeneous parameter h is a non-zero value such that:

• We can then write any point (x, y) as (hx, hy, h)

• We can conveniently choose h = 1 so that

(x, y) becomes (x, y, 1)

h

xx h

h

yy h


Why Homogeneous Coordinates?

• Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations • We will see in a moment that all of the transformations we discussed previously can be represented as 3*3 matrices• Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!


Homogenous Coordinates

• Combine the geometric transformation into a single matrix with 3by3 matrices

• Expand each 2D coordinate to 3D coordinate with homogenous parameter

• Two-Dimensional translation matrix

• Two-Dimensional rotation matrix

• Two-Dimensional scaling matrix

1

y

x

100

t10

t01

1

'y

'x

y

x

1

y

x

100

0cossin

0coscos

1

'y

'x

1

y

x

100

0s0

00s

1

'y

'x

y

x


Inverse transformations

• Inverse translation matrix



100

t10

t01

T y

x1

100

0cossin

0sincos

R 1

100

0s

10

00s

1

Sx

x

1









Basic 2D Transformations

• Basic 2D transformations as 3x3 Matrices

1100

10

01

1

y

x

ty

tx

y

x

1100

0cossin

0sincos

1

y

x

y

x

1100

00

00

1

y

x

sy

sx

y

x

1100

01

01

1

y

x

shy

shx

y

x

Translate

Shear

Scale

Rotate


Geometric transformations in three-dimensional space (1)

• Extend from two-dimensional transformation by includingconsiderations for the z coordinates

• Translation and scaling are similar to two-dimension, include the three Cartesian coordinates

• Rotation method is less straight forward• Representation

– Four-element column vector for homogenous coordinates

– Geometric transformation described 4by4 matrix



• Three-dimensional translation– A point P (x,y,z) in three-dimensional space translate to new

location with the translation distance T (tx, ty, tz)

– In matrix formatxtx'x yty'y

1

z

y

x

1000

t100

t010

t001

1

'z

'y

'x

z

y

x

PT'P

ztz'z



• Three-dimensional scaling– Relative to the coordinate origin, just include the parameter

for z coordinate scaling in the transformation matrix

– Relative to a fixed point (xf, yf zf)

• Perform a translate-scaling-translate composite transformation

1000

z)s1(s00

y)s1(0s0

x)s1(00s

)z,y,x(T)s,s,s(S)z,y,x(t

fzz

fyy

fxx

fffzyxfff

PS'P

1

z

y

x

1000

0s00

00s0

000s

1

'z

'y

'x

z

y

x



• Three-dimensional rotation definition– Assume looking in the negative direction along the axis– Positive angle rotation produce counterclockwise

rotationsabout a coordinate axis



• Three-dimensional coordinate-axis rotation– Z-axis rotation equations

– Transformation equations for rotation about the other two coordinate axes can be obtained by a cyclic permutation

x y z x– X-axis rotation equations

x'x

coszsiny'z

sinzcosy'y

1

z

y

x

1000

0100

00cossin

00sincos

1

'z

'y

'x

z'z

cosysinx'y

sinycosx'x

Rz

1000

0cossin0

0sincos0

0001

xx RR



• Three-dimensional coordinate-axis rotation– Y-axis rotation equations

– General Three-dimensional rotations• Translate object so that the rotation axis coincides with the parallel

coordinate axis• Perform the specified rotation about that axis• Translate object back to the original position

y'y

cosxsinz'x

sinxcosz'z

T)(RT)(R

PT)(RT'P

x1

x1

1000

0cos0sin

0010

0sin0cos

yy RR


• Inverse of a rotation matrix

RR 1

sinsin,coscos

1000

0100

00cossin

00sincos

1000

0100

00cossin

00sincos

1

zz RR

TRR 1 : orthogonal matrix



Concatenation of Transformations

• Concatenating– affine transformations by multiplying together– sequences of the basic transformations

define an arbitrary transformation directly– ex) three successive transformations

CBApApBCq

A B Cp q

CBAM

Mpq Mp q

CBA

p1 = App2 = Bp1q = Cp2

q = CBp1q = CBAp

p1 p2


Matrix Concatenation Properties

• Associative properties

• Transformation is not commutative (CopyCD!)– Order of transformation may affect transformation

position

321321321 M)MM()MM(MMMM


Two-dimensional composite transformation (1)

• Composite transformation– A sequence of transformations

– Calculate composite transformation matrix rather than applying individual transformations

• Composite two-dimensional translations– Apply two successive translations, T1 and T2

– Composite transformation matrix in coordinate form

PM'P

PMM'P 12

PTTPTTP

ttTT

ttTT

yx

yx

)()('

),(

),(

1212

222

111

)tt,tt(T)t,t(T)t,t(T y2y1x2x1y1x1y2x2



• Composite two-dimensional rotations– Two successive rotations, R1 and R2 into a point P

– Multiply two rotation matrices to get composite transformation matrix

• Composite two-dimensional scaling

P)}(R)(R{'P

}P)(R{)(R'P

21

21

P)ss,ss(S'P

)ss,ss(S)s,s(S)s,s(S

y2y1x2x1

y2y1x2x1y2x2y1x1

P)(R'P

)(R)(R)(R

21

2121



• General two-dimensional Pivot-point rotation– Graphics package provide only origin rotation– Perform a translate-rotate-translate sequence

• Translate the object to move pivot-point position to origin

• Rotate the object

• Translate the object back to the original position

– Composite matrix in coordinates form),y,x(R)y,x(T)(R)y,x(T rrrrrr



• Example of pivot-point rotation


Pivot-Point Rotation

100

sin)cos1(cossin

sin)cos1(sincos

100

10

01

100

0cossin

0sincos

100

10

01

rr

rr

r

r

r

r

xy

yx

y

x

y

x

,,,, rrrrrr yxRyxTRyxT

Translate Rotate Translate

(xr,yr)

(xr,yr)

(xr,yr)

(xr,yr)



• General two-dimensional Fixed-point scaling– Perform a translate-scaling-translate sequence

• Translate the object to move fixed-point position to origin

• Scale the object wrt. the coordinate origin

• Use inverse of translation in step 1 to return the object back to the original position

– Composite matrix in coordinates form)s,s,y,x(S)y,x(T)s,s(S)y,x(T yxffffyxff



• Example of fixed-point scaling


General Fixed-Point Scaling

100

)1(0

)1(0

100

10

01

100

00

00

100

10

01

yfy

xfx

f

fx

f

f

sys

sxs

y

x

s

s

y

x

y

Translate Scale Translate

(xr,y

r)(xr,yr)

(xr,yr)

(xr,yr)

yxffffyxff ssyxSyxTssSyxT ,,,, ,,



• General two-dimensional scaling directions– Perform a rotate-scaling-rotate sequence– Composite matrix in coordinates form

100

0cosssinssincos)ss(

0sincos)ss(sinscoss

)(R)s,s(S)(R

22

2112

122

22

1

211

s1

s2


General Scaling Directions

• Converted to a parallelogram

100

0cossinsincos)(

0sincos)(sincos

)(),()( 22

2112

122

22

1

211

ssss

ssss

RssSR

x

y

(0,0)

(3/2,1/2)

(1/2,3/2)

(2,2)

x

y

(0,0) (1,0)

(0,1) (1,1)

x

y s2

s1

Scale


General Rotation (1/2)

• Three successive rotations about the three axes

rotation of a cube about the z axis rotation of a cube about the y axis

rotation of a cube about the x axis

?


General Rotation (2/2)

zyx RRRR

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001


Instance Transformation (1/2)

• Instance of an object’s prototype– occurrence of that object in

the scene

• Instance transformation– applying an affine

transformation to the prototype to obtain desired size, orientation, and location

?

instance transformation


Instance Transformation (2/2)

TRSM

1000

000

000

000

1000

0100

00cossin

00sincos

1000

100

010

001

z

y

x

z

y

x

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© 2005 Pearson Education Computer Graphics with OpenGL 3e