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Transcript of © 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
© 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
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
© 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
– 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)
Basic two-dimensional geometric transformations (1/2)
© 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
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)
© 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
Basic two-dimensional geometric transformations (2/1)
• 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
© 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
– 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()
Basic two-dimensional geometric transformations (2/2)
6
y
x 0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
© 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
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)
© 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
Basic two-dimensional geometric transformations (3/1)
• 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
© 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
– 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
Basic two-dimensional geometric transformations (3/2)
© 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
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!
© 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
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)
© 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
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
© 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
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!
© 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
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
© 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
Inverse transformations
• Inverse translation matrix
• Two-Dimensional translation matrix
• Two-Dimensional translation matrix
100
t10
t01
T y
x1
100
0cossin
0sincos
R 1
100
0s
10
00s
1
Sx
x
1
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
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
© 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 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
© 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 in three-dimensional space (2)
• 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
© 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 in three-dimensional space (3)
• 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
© 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 in three-dimensional space (4)
• Three-dimensional rotation definition– Assume looking in the negative direction along the axis– Positive angle rotation produce counterclockwise
rotationsabout a coordinate axis
© 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 in three-dimensional space (5)
• 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
© 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 in three-dimensional space (6)
• 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
© 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
• Inverse of a rotation matrix
RR 1
sinsin,coscos
1000
0100
00cossin
00sincos
1000
0100
00cossin
00sincos
1
zz RR
TRR 1 : orthogonal matrix
Geometric transformations in three-dimensional space (7)
© 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
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
© 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
Matrix Concatenation Properties
• Associative properties
• Transformation is not commutative (CopyCD!)– Order of transformation may affect transformation
position
321321321 M)MM()MM(MMMM
© 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
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
© 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
Two-dimensional composite transformation (2)
• 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
© 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
Two-dimensional composite transformation (3)
• 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
© 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
Two-dimensional composite transformation (4)
• Example of pivot-point rotation
© 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
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)
© 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
Two-dimensional composite transformation (5)
• 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
© 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
Two-dimensional composite transformation (6)
• Example of fixed-point scaling
© 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
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 ,,,, ,,
© 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
Two-dimensional composite transformation (7)
• 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
© 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
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
© 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
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
?
© 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
General Rotation (2/2)
zyx RRRR
1000
0100
00cossin
00sincos
1000
0cos0sin
0010
0sin0cos
1000
0cossin0
0sincos0
0001
© 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
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
© 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
Instance Transformation (2/2)
TRSM
1000
000
000
000
1000
0100
00cossin
00sincos
1000
100
010
001
z
y
x
z
y
x