Introduction to
Geometric Transformations
幾何轉換與座標轉換
KUAS ME, C. F. Chang
1
Contents of this section:

Classification of Transformations

Mathematic Representation of Geometric Transformations

Definitions Related to Geometric Transformations

Remarks on Homogeneous Coordinates

The Graphics Pipeline

OpenGL Functions for Setting up Transformations
KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
2
1
Classification of
Transformations

Transformations are herein classified into two types:



Geometric transformation
(fix coordinates and manipulate objects, e.g., moving object)
Coordinate transformation
(fix objects and manipulate coordinates, e.g., moving camera
The two transformations can have the same result
Moving object
Moving camera
KUAS ME, C. F. Chang
3
Mathematic Representation of
Geometric Transformations
a. Translation, rotation, scaling, and shearing are all examples of
geometric transformations.
b. In the following, we will see that all the transformations of an
object can be achieved by post multiplying a transformation
matrix M by the object’
s vertices P.
P’= M P
m11 m12 m13 m14 
x'  
x



y '  

m m m m
y
  21 22 23 24 

m31 m32 m33 m34 
z '  
z 

 

m41 m42 m43 m44 
w' 
w


KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
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2
Mathematic Representation of
Geometric Transformations (cont)
P’= M P
∵(P’
)T =( M P )T=PT MT
∴ Some graphics textbooks use a convention of premultiplying
matrices by the row vectors of the object:
x '
y ' z ' w '

x y
z
m11 m21 m31 m41 


m m m m 
 12 22 32 42 
w

m13 m23 m33 m43 


m14 m24 m34 m44 

KUAS ME, C. F. Chang
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Definitions Related to
Geometric Transformations

Affine transformation: A transformation, which preserves the
parallelism of lines, but not necessarily angles or lengths.

Coordinate systems: right-handed and left-handed coordinate
systems
KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
6
3
Definitions Related to
Geometric Transformations (cont)
 Homogeneous


coordinates :
In homogeneous coordinates, each cartesian point (x,y,z)
is represented in a redundant manner by 4 coordinates,
(wx, wy, wz, w).
Setting w=0 can be thought of as a point at infinity or,
more simply, a direction.
KUAS ME, C. F. Chang
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Remarks on
Homogeneous Coordinates

Homogeneous coordinate systems use n+1 coordinates [X1,
X2, …, Xn+1] to represent points [x1, x2, …, xn] in n-space and
satisfy
xi = Xi / Xn+1, i = 1, 2, …, n.
[Ex1]: A point [x,y] is represented in homogeneous coordinates as [X, Y, W].
∵ x = X / W, y = Y / W

X=Wx,Y=Wy
∴ [X, Y, W] = [W x, W y, W] = W [x, y, 1]
Thus,
Thus
a. Two sets of homogeneous coordinates with different values of W
represent the same point,
b. x = X and y = Y if W=1.
KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
8
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Remarks on
Homogeneous Coordinates
[Ex2]:
For a straight line, we have
x = X / W, y = Y / W
a1 x + a2 y + a3=0
a1
(X/W) + a2 (Y/W) + a3=0
a1
X + a2 Y + a3 W = 0
(This is a homogeneous equation with each term having the same degree)


a1
a2
X 

a3 
Y

0

W


KUAS ME, C. F. Chang
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Remarks on
Homogeneous Coordinates
[Ex3]: For curves of two degrees, we have
a x2 + b x y + c y2 + d x +e y + f = 0
x = X / W, y = Y / W
a
(X/W)2 + b (X/W) (Y/W) + c (Y/W)2 + d (X/W) +e (Y/W) + f = 0
a
X2 + b X Y + c Y2 + d XW +e YW + f W 2 = 0
(Degree=2 for each term, homogeneous equation)

X
Y
aX bY dW 



Z
 cY eW 0


fW



X
Y
a b d

X 



Z
0
c
e
Y



0


0 0 f
W




KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
10
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The Graphics Pipeline


Most rendering algorithms can be separated into two or
more distinct stages.
The graphics pipeline is a useful abstraction of the
rendering process as well as a block diagram for dedicated
graphics hardware
(created by using Pov-Ray retracing)
KUAS ME, C. F. Chang
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Camera Analogy





Setting up tripod and pointing the
camera at the scene
= viewing transformation
Arranging the models to be
photographed into the desired
composition
= modeling transformation
Choosing a camera lens or
adjusting the zoom
= projection transformation
Determining how large you want
the final photograph to be - for
example, enlarge or reduce the
physical photograph
= viewport transformation
Each step uses transformations (i.e., a
series of change in coordinate systems) and
yields a transformation matrix
KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
12
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OpenGL Functions for
Setting up Transformations
object
v
e
r
t
e
x
eye
Modelview
Matrix
Projection
Matrix
Model view
Projection
Model:
glTranslatef()
glRotatef()
glScalef()
View:
gluLookAt()
Normalized
device
clip
Perspective
Division
Perspective:
glFrustum()
gluPerspective()
Orthogonal:
glOrtho()
gluOrtho2D()
window
Viewport
Transform
glViewport()
KUAS ME, C. F. Chang
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Stages of Vertex Transformation
v
ve
vc
vw
ve M ModelViewv
vc M Pr oject ve M Pr oject M ModelViewv
vw MViewPort vc MViewPort M Pr oject M ModelViewv
vw Mv , where M MViewPort M Pr oject M ModelView
KUAS ME, C. F. Chang
國立高雄應用科技大學機械系 電腦繪圖學 C. F. Chang
14
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