Coordinate Systems and
Transformations
Graphic Libraries
Hardware dependent program
Hardware independent program
GL Examples:
Core, GKS, PEX, OpenGL
Coordinate Systems
• Device Coordinate System:
identifies locations on the display
=>Virtual Device Coordinate System:
Usually the origin is at the low left corner and u,v
range from 0 to 1.
Coordinate Systems
• Model Coordinate System(MCS):
identifies the shapes of object and it is attached to the
object. Therefore the MCS moves with the object in the
WCS
• World Coordinate System (WCS):
identifies locations of objects in the world in the
application.
• Viewing Coordinate System (VCS):
Defined by the viewpoint and viewsite
Coordinate Systems
viewpoint and viewsite
Coordinate Systems
Projection on Screen
• Parallel Projection
• Perspective Projection
Parallel Projection
• Preserve actual dimensions and shapes of objects
• Preserve parallelism
• Angles preserved only on faces parallel to the
projection plane
• Orthographic projection is one type of parallel
projection
Perspective Projection
• Doesn’t preserve parallelism
• Doesn’t preserve actual dimensions and angles
of objects, therefore shapes deformed
• Popular in art (classic painting); architectural
design and civil engineering.
• Not commonly used in mechanical engineering
Perspective Projection
Coordinate scaling:
Transformations between
Coordinate Systems
Model Transformation
Viewing Transformation
ProjectionTransformation
Translation
2D
3D
y
y
P2 (x2, y2)
d
yd
P2
P1(x1, y1)
o
xd
⎡ x1 ⎤
⎢ ⎥
P1 = ⎢ y1 ⎥
⎢⎣ z1 ⎥⎦
x
P1
z
⎡ x1 ⎤ ⎡ x d ⎤ ⎡ x1 + xd ⎤
P 2 = ⎢⎢ y1 ⎥⎥ + ⎢⎢ y d ⎥⎥ = ⎢⎢ y1 + yd ⎥⎥ = P1 + d
⎢⎣ z1 ⎥⎦ ⎢⎣ z d ⎥⎦ ⎢⎣ z1 + zd ⎥⎦
x
Scaling
2D
y
3D
y
P2
P2 (x
2, y2)y’)
V’(x’,
P1
y)
P1(xV(x,
1, y1)
θ
x
x
o
z
⎡x1 ⎤
⎢ ⎥
P1 = ⎢y1 ⎥
⎢⎣z1 ⎥⎦
⎡sx1 ⎤
⎢ ⎥
P 2 = ⎢sy1 ⎥
⎢⎣sz1 ⎥⎦
P 2 = sP1
y
P2
Reflection (about XOY
Plane)
o
x
P1
z
⎡ x1 ⎤
⎢ ⎥
P1 = ⎢ y1 ⎥
⎢⎣ z1 ⎥⎦
⎡ x1 ⎤
⎢ ⎥
P 2 = ⎢ y1 ⎥
⎢⎣ −z1 ⎥⎦
⎡1 0 0 ⎤
P 2 = ⎢⎢ 0 1 0 ⎥⎥ P1
⎢⎣ 0 0 −1⎥⎦
Rotation about Z Axis - CCW by Ө
⎡ x1 ⎤ ⎡ r cos ϕ ⎤
P1 = ⎢⎢ y1 ⎥⎥ = ⎢⎢ r sin ϕ ⎥⎥
⎢⎣ z1 ⎥⎦ ⎢⎣ z ⎥⎦
⎡ x2 ⎤ ⎡ r cos (ϕ +θ ) ⎤
⎢
⎥
P2 = ⎢⎢ y2 ⎥⎥ = ⎢ r sin (ϕ +θ ) ⎥
⎥⎦
⎢⎣ z2 ⎥⎦ ⎢⎣
z
⎡ r cos ϕ cos θ − r sin ϕ sin θ ⎤
= ⎢⎢ r cos ϕ sin θ + r sin ϕ cos θ ⎥⎥
⎢⎣
⎥⎦
z
⎡ x1 cos θ − y1 sin θ ⎤
= ⎢⎢ x1 sin θ + y1 cos θ ⎥⎥
⎢⎣
⎥⎦
z1
φ
z
CCW
y2
y
y1
P2
θ
z
Φ
o
r
P1
x2
x1
x
P2 (x2, y2)
y
θ
o
φ
P1(x1, y1)
x
3-D Transformation
Rotation about Z Axis - CCW by Ө
x2 = x1 cos θ − y1 sin θ
y2 = x1 sin θ + y1 cos θ
z2 = z1
P2 (x2, y2, z2)
y
P1(x1, y1, z2)
θ
φ
o
x
Counter Clockwise, CCW
⎡ x2 ⎤ ⎡cos θ − sin θ 0 ⎤ ⎡ x1 ⎤
⎢ y ⎥ = ⎢sin θ
⎥⎢y ⎥
cos
θ
0
⎢ 2⎥ ⎢
⎥ ⎢ 1⎥
⎢⎣ z2 ⎥⎦ ⎢⎣ 0
0
1 ⎥⎦ ⎢⎣ z1 ⎥⎦
Rotation about X Axis CCW by Ө
z
x2 = x
y
y2 = y1 cos θ − z1 sin θ
o
x
z2 = y1 sin θ + z1 cos θ
P2 (x2, y2, z2)
z
P1(x1, y1, z1)
θ
φ
o
y
0
0 ⎤ ⎡ x1 ⎤
⎡ x2 ⎤ ⎡1
⎢ y ⎥ = ⎢0 cos θ − sin θ ⎥ ⎢ y ⎥
⎢ 2⎥ ⎢
⎥ ⎢ 1⎥
⎢⎣ z2 ⎥⎦ ⎢⎣0 sin θ
cos θ ⎥⎦ ⎢⎣ z1 ⎥⎦
Rotation about Y Axis CCW by Ө
⎡ x1 ⎤ ⎡ r cos ϕ ⎤
P1 = ⎢⎢ y1 ⎥⎥ = ⎢⎢ y1 ⎥⎥
⎢⎣ z1 ⎥⎦ ⎢⎣ r sin ϕ ⎥⎦
⎡ x2 ⎤ ⎡ r cos (ϕ − θ ) ⎤ ⎡ r cos ϕ cos θ + r sin ϕ sin θ ⎤
⎢
⎥ ⎢
⎥
=
P2 = ⎢⎢ y2 ⎥⎥ = ⎢
y1
y
1
⎥ ⎢
⎥
⎢⎣ z2 ⎥⎦ ⎢⎣ r sin (ϕ − θ ) ⎥⎦ ⎢⎣ r sin ϕ cos θ − r cos ϕ sin θ ⎥⎦
⎡ x1 cos θ + z1 sin θ ⎤
⎥
= ⎢⎢
y1
⎥
⎢⎣ − x1 sin θ + z1 cos θ ⎦⎥
⎡ x2 ⎤ ⎡ cos θ 0 sin θ ⎤ ⎡ x1 ⎤
P2 = ⎢⎢ y2 ⎥⎥ = ⎢⎢ 0
1
0 ⎥⎥ ⎢⎢ y1 ⎥⎥ = R[ y ] P1
⎢⎣ z2 ⎥⎦ ⎢⎣ − sin θ 0 cos θ ⎥⎦ ⎢⎣ z1 ⎥⎦
Z
P1
P2
θ
φ
X
Y
Homogeneous Representation
The representation is introduced to express all
geometric transformations in the from of matrix
multiplication for the convenience of manipulation.
⎡ x2 ⎤ ⎡1 0
⎢ y ⎥ ⎢0 1
Translation ⎢ 2 ⎥ = ⎢
⎢ z2 ⎥ ⎢0 0
⎢ ⎥ ⎢
⎣1 ⎦ ⎢⎣ 0 0
0 d x ⎤ ⎡ x1 ⎤
0 d y ⎥⎥ ⎢ y1 ⎥
⎢ ⎥
1 d z ⎥ ⎢ z1 ⎥
⎥⎢ ⎥
0 1 ⎥⎦ ⎣1 ⎦
Dummy (n+1)th coordinate to facilitate multiplication
Homogeneous Representations
Scaling
Rotation
P2 = [Rz ]P1
⎡sx
⎢0
[H ] = ⎢
⎢0
⎢
⎣0
and
0 0 0⎤
s y 0 0⎥
⎥
0 sz 0 ⎥
⎥
0 0 1⎦
⎡cos θ
⎢ sin θ
[Rz ] = ⎢
⎢ 0
⎢
⎣ 0
− sin θ
cos θ
0
0
0 0⎤
0 0 ⎥⎥
1 0⎥
⎥
0 1⎦
Homogeneous Representations
0
⎡1
⎢ 0 cos θ
[ Rx ] = ⎢⎢
0 sin θ
⎢
0
⎣0
0
− sin θ
cos θ
0
Reflection
P2 = [M ]P1
and
0⎤
0 ⎥⎥
;
0⎥
⎥
1⎦
⎡ cos θ
⎢ 0
⎡⎣ R y ⎤⎦ = ⎢
⎢ − sin θ
⎢
⎣ 0
⎡± 1 0 0
⎢ 0 ±1 0
[M ] = ⎢
⎢ 0 0 ±1
⎢
⎣0 0 0
0 sin θ
1
0
0 cos θ
0
0
0⎤
0⎥⎥
0⎥
⎥
1⎦
0⎤
0 ⎥⎥
0⎥
⎥
1⎦
Composition of Transformation
(Concatenation of Transformation)
via Pre-Multiplication
P2 = [ H n ][ H n −1 ]"[ H1 ]P1
An Example
Consider a 3D object. The coordinates of
the vertices are given as follows:
A=[3, 5, 3]
B=[7, 5, 3]
C=[7, 5, 5]
D=[3, 5, 5]
E=[3, 6, 5]
F=[3, 6, 3]
Rotate the 3D object by 30 degree in
clockwise (CW) direction at point D about
the Y-axis.
Step 1
[T]D-O
o
Step 2
[R]y-30
Step 3
[T]O-D
Procedure:
• First we translate ([T]D-O) the object at the refernece point
D to the origin O.
• Then we rotate ([ R]Y−30) about the Y-axis
• Finally, we translate ([T]O-D) the point D from the origin
back to its original position.
[T ] D − O
⎡1 0 0 − 3 ⎤
⎢0 1 0 − 5⎥
⎥
= ⎢
⎢0 0 1 − 5⎥
⎢
⎥
0
0
0
1
⎣
⎦
[T ]O
[ R]Y−30
− D
⎡cos(−30) 0 sin ( −30 ) 0 ⎤
⎢
⎥
0
1
0
0
⎥
=⎢
⎢ − sin(−30) 0 cos(−30) 0 ⎥
⎢
⎥
0
0
1 ⎥⎦
⎢⎣ 0
⎡1
⎢0
= ⎢
⎢0
⎢
⎣0
0 0 3⎤
1 0 5 ⎥⎥
0 1 5⎥
⎥
0 0 1⎦
[A B C D E F]
⎡3 7 7 3 3 3 ⎤
⎢5 5 5 5 6 6 ⎥
⎥
P1 = ⎢
⎢3 3 5 5 5 3 ⎥
⎢
⎥
⎣1 1 1 1 1 1 ⎦
The definition of the point
matrix in the homogeneous
representation.
−30
Z
P2 = [T ]O − D [ R] [T ]D −O P1
⎡ 4.00 7.46 6.46 3.00 3.00 4.00 ⎤
⎢5.00 5.00 5.00 5.00 6.00 6.00 ⎥
⎥
P2 = ⎢
⎢3.27 5.27 7.00 5.00 5.00 3.27 ⎥
⎢
⎥
⎣1.00 1.00 1.00 1.00 1.00 1.00 ⎦
Mapping and Parallel Projections
• Mapping involves calculating the coordinates of a point
known with rispect to a CS to a new CS.
About Transformations
• Transformations are applied for both changing WCS
in respect to MCS, but also for projections
Mapping and Parallel Projections
Where:
Px,py,pz=x2,y2,z2 components required to translate (X2,Y2,Z2) to (X1,Y1,Z1),
the following are evaluated as the translation as been performed:
nx,ny,nz=x2,y2,z2 components of a unit vector along x1
ox,oy,oz=x2,y2,z2 components of a unit vector along y1
ax,ay,az=x2,y2,z2 components of a unit vector along y1
Once mapping has been performed the Z coordinate is dropped
Mapping and Perspective
Projections
Once mapping has been performed the zv coordinate are not dropped
yet and the xv, yv coordinates are scaled following the similarity rule of
triangles (L,S needed). Then zv is dropped.
Generating Parallel Projection
Drop the n (zv) coordinate
⎡1
⎢0
[Dn ]= ⎢⎢0
⎢0
⎣
0 0 0⎤
1 0 0⎥
⎥,
0 0 0⎥
0 0 1 ⎥⎦
⎛ u⎞
⎛u⎞
⎜ V⎟
⎜ V⎟
⎜ ⎟ = [Dn ]⎜ ⎟
⎜ 0⎟
⎜n⎟
⎜ ⎟
⎜ ⎟
⎝ 1⎠
⎝1⎠
In summary, to project a view of an object on the UV plane, one needs to
transform each point on the object by:
[T ] = [ Dn ][ Rz ][ Ry ][ Rx ][ Dov ,o ]
⎛ u⎞
⎜ V⎟
P ′ = ⎜ ⎟ = [T ]P = [T]
⎜ 0⎟
⎜ ⎟
⎝ 1⎠
⎛ x⎞
⎜ y⎟
⎜ ⎟
⎜ z⎟
⎜ ⎟
⎝ 1⎠
Note: The inverse transforms are not needed! We don't want to
go back to x - y - z coordinates.
Orthographic Projection
Y
Top
Front
Right
X
Z
• Projection planes (Viewing planes) are
perpendicular to the principal axes of the
MCS of the model
• The projection direction (viewing direction)
coincides with one of the MCS axes
Y
Top
Front
Right
X
Geometric
Transformations for
Generating
Orthographic Projection
(Front View)
⎡1 0 0 0 ⎤
⎢0 1 0 0 ⎥
⎥P
Pv = ⎢
⎢0 0 0 0 ⎥
⎢
⎥
⎣0 0 0 1 ⎦
Z
Yv,Y
Front
Xv, X
Drop Z
Geometric Transformations
for Generating
Orthographic Projection
(Top View)
Y
Top
Front
Yv
Right
Top
X
Xv, X
Z
Z
0
0
⎡1 0 0 0 ⎤ ⎡1
⎢ 0 1 0 0 ⎥ ⎢ 0 cos(90 ° ) − sin(90 ° )
⎥⎢
Pv = ⎢
⎢ 0 0 0 0 ⎥ ⎢ 0 sin ( 90 ° ) cos(90 ° )
⎢
⎥⎢
0
0
⎣ 0 0 0 1 ⎦ ⎢⎣ 0
Drop Z
[ R]90
x
0⎤
⎡1
⎢0
0 ⎥⎥
P=⎢
⎢0
0⎥
⎥
⎢
1 ⎥⎦
⎣0
0
0
0
0
0 0⎤
− 1 0 ⎥⎥
P
0 0⎥
⎥
0 1⎦
Y
Top
Front
Yv, Y
Geometric Transformations
for Generating
Orthographic Projection
(Right View)
Right
Right
X
Z
Xv
sin( −90°)
− 1 0⎤
0 0 ⎥⎥
P
0 0⎥
⎥
0 1⎦
Z
⎡1 0 0 0 ⎤ ⎡ cos( −90°)
⎢0 1 0 0 ⎥ ⎢
0
⎢
⎢
⎥
Pv =
⎢ 0 0 0 0 ⎥ ⎢ − sin ( −90° )
⎢
⎥⎢
0
⎣ 0 0 0 1 ⎦ ⎢⎣
Drop Z
0
1
0
0
0⎤
⎡0 0
⎢0 1
0
0 ⎥⎥
P=⎢
⎢0 0
cos( −90°) 0 ⎥
⎥
⎢
0
1 ⎥⎦
⎣0 0
[ R ]−y90
Rotations Needed for Generating Isometric Projection
Y
Yv
Top
Front
Y
Right
Top
Front
X
Right
Xv
Z
Z
0
⎡1 0
⎢0 cos φ − sin φ
Pv = [ R]φx [ R]θy P = ⎢
⎢0 sin φ cos φ
⎢
0
⎣0 0
Zv
0 ⎤ ⎡cos θ
⎥
⎢
0⎥ ⎢ 0
0 ⎥ ⎢ − sin θ
⎥⎢
1⎦⎣ 0
X
0 sin θ 0 ⎤
⎥
1 0 0 ⎥
P
0 cos θ 0 ⎥
⎥
0 0 1 ⎦
Isometric Projection: Equally foreshorten the
three main axes
θ = ±45°, φ = ±35.26°
Other Possible Rotation Paths
• Rx --> Ry
rx = ±45°, ry = ±35.26°
• Rz --> Ry(Rx)
rz = ±45°, ry ( x ) = ±54.74°
• Rx(Ry) --> Rz
ry ( x ) = ±45°, rz = ANY ANGLE