Coordinate Systems and Transformations

Graphic Libraries

Hardware dependent programHardware dependent program

Hardware independent programHardware 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 Systemsviewpoint 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 TransformationModel Transformation

Viewing TransformationViewing Transformation

ProjectionTransformationProjectionTransformation

Translation

P2 (x2, y2)

x

y

P1(x1, y1)o xd

yd P2

P1

y

x

z

d

P1 =x1

y1

z1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

P2 =x1

y1

z1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

+xd

yd

zd

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

=x1 + xd

y1 + yd

z1 + zd

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

= P1 + d

3D2D

Scaling y

x

z

P2

P1

3D

P1 =x1

y1

z1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ P2 =

sx1

sy1

sz1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ P2 = sP1

y

V(x, y)

V’(x’, y’)

θ xo

2D

P2 (x2, y2)

P1(x1, y1)

Reflection (about XOY Plane)

y

x

z

P2

P1

P1 =x1

y1

z1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ P2 =

x1

y1

−z1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ P2 =

1 0 00 1 00 0 −1

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

P1

o

Rotation about Z Axis - CCW by Ө

( )( )

1

1 1

1

2

2 2

2

1 1

1 1

1

r cos r sin

z

r cos +r sin +

r cos cos sin sinr cos sin sin cos

z

cos sin sin cos

z

xyz

xyz

rr

x yx y

z

ϕϕ

ϕ θϕ θ

ϕ θ ϕ θϕ θ ϕ θ

θ θθ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤−+⎢ ⎥= ⎢ ⎥

⎢ ⎥⎣ ⎦−⎡ ⎤

⎢ ⎥= +⎢ ⎥⎢ ⎥⎣ ⎦

P

P

x

y

θφ

o

P1(x1, y1)

φ z

y

x

θr

P2

P1

1x2x

1y2y

CCW

Φz

P2 (x2, y2)

o

3-D Transformation

x

y

θ

φ

o

2 1 1

2 1 1

2 1

cos sinsin cos

x x yy x yz z

θ θθ θ

= −= +=

2 1

2 1

2 1

cos sin 0sin cos 0

0 0 1

x xy yz z

θ θθ θ

−⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥=⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

P1(x1, y1, z2)

P2 (x2, y2, z2)

Rotation about Z Axis - CCW by Ө

Counter Clockwise, CCW

y

z

θ

φ

o

Rotation about X Axis -CCW by Ө

y

z

x o

2

2 1 1

2 1 1

cos sinsin cos

x xy y zz y z

θ θθ θ

== −= +

2 1

2 1

2 1

1 0 00 cos sin0 sin cos

x xy yz z

θ θθ θ

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥= −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

P1(x1, y1, z1)

P2 (x2, y2, z2)

Rotation about Y Axis -CCW by Ө

Z

X Y

φ

θP2

P1

( )

( )

1

1 1 1

1

2

2 2 1 1

2

1 1

1

1 1

2

2

cos

sin

cos cos cos sin sin

sin sin cos cos sin

cos sin

sin cos

x ry yz r

x r r ry y yz r r r

x zy

x z

x

ϕ

ϕ

ϕ θ ϕ θ ϕ θ

ϕ θ ϕ θ ϕ θ

θ θ

θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= = =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

+⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥+⎣

+

⎦

−

−

=

−

−

P

P

P [ ]

1

2 1 1

2 1

cos 0 sin0 1 0

sin 0 cosyR

xy yz z

θ θ

θ θ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦−

P

Homogeneous Representation

The representation is introduced to express all geometric transformations in the from of matrix multiplication for the convenience of manipulation.

Dummy (n+1)th coordinate to facilitate multiplication

2 1

2 1

2 1

1 0 00 1 0

0 0 11 10 0 0 1

x

y

z

dx xdy y

z zd

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

Translation

Homogeneous Representations

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000000000000

][z

y

x

ss

s

H

Scaling

Rotation

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

==

1000010000cossin00sincos

12

θθθθ

zz RandR PP

Homogeneous Representations

[ ] [ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

±±

±

==

1000010000100001

12 MandM PP

[ ]

1 0 0 0 cos 0 sin 00 cos sin 0 0 1 0 0

; 0 sin cos 0 sin 0 cos 00 0 0 1 0 0 0 1

x yR R

θ θθ θθ θ θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎡ ⎤= =⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎢

⎦ ⎣ ⎦

−⎥ ⎥

⎣

Reflection

Composition of Transformation (Concatenation of Transformation)

via Pre-Multiplication

2 1 1 1P [ ][ ] [ ]Pn nH H H−=

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.

[T]D-O

[R]y-30 [T]O-D

Step 1

Step 2 Step 3

o

Procedure:• First we translate ([T]D-O) the object at the refernece point

D to the origin O.• Then we rotate ( ) about the Y-axis• Finally, we translate ([T]O-D) the point D from the origin

back to its original position.

1 0 0 30 1 0 5

[ ]0 0 1 50 0 0 1

D OT −

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

( )30

cos( 30) 0 sin 30 00 1 0 0[ ]

sin( 30) 0 cos( 30) 00 0 0 1

YR −

− −⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥− − −⎢ ⎥⎢ ⎥⎣ ⎦

1 0 0 30 1 0 5

[ ]0 0 1 50 0 0 1

O DT −

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

30[ ]YR −

1

3 7 7 3 3 35 5 5 5 6 63 3 5 5 5 31 1 1 1 1 1

P

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

2

4.00 7.46 6.46 3.00 3.00 4.005.00 5.00 5.00 5.00 6.00 6.003.27 5.27 7.00 5.00 5.00 3.271.00 1.00 1.00 1.00 1.00 1.00

P

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

The definition of the point matrix in the homogeneous representation.

302 1[ ] [ ] [ ]O D Z D OP T R T P−

− −=

[A B C D E F]

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 x1ox,oy,oz=x2,y2,z2 components of a unit vector along y1ax,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

Dn[ ]=

1 0 0 00 1 0 00 0 0 00 0 0 1

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

,

uV01

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

= Dn[ ]

uVn1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

′ P =

uV01

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

= T[ ]P = T[ ]

xyz1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

Drop the n (zv) coordinate

In summary, to project a view of an object on the UV plane, one needs to transform each point on the object by:

Note: The inverse transforms are not needed! We don't want to go back to x - y - z coordinates.

,[ ][[ ] [] ] ]][ [vz y o oxn R R DRDT =

0 0

FrontTop

Right

X

Y

Z

Orthographic Projection

• 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

FrontTop

Right

X

Y

Z

Front

Yv,Y

Xv, X

PPv

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000000000100001

Geometric Transformations for

Generating Orthographic Projection

(Front View)

Drop Z

FrontTop

Right

X

Y

Z

Top

Xv, X

Yv

Z

( )

1 0 0 01 0 0 0 1 0 0 00 cos(90 ) sin(90 ) 00 1 0 0 0 0 1 00 sin 90 cos(90 ) 00 0 0 0 0 0 0

0 0 0 1 0 0 0 10 0 0 10vP P P

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥° − ° −⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥ ⎢ ⎥° °⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

Geometric Transformations for Generating

Orthographic Projection (Top View)

90[ ]xRDrop Z

FrontTop

Right

X

Y

Z

Right

Xv

Yv, Y

Z

( )

cos( 90 ) 0 sin( 90 ) 01 0 0 0 0 0 1 00 1 0 00 1 0 0 0 1 0 0

sin 90 0 cos( 90 ) 00 0 0 0 0 0 00 0 0 1 0 0 0 10 0 0 1

0vP P P

− ° − °⎡ ⎤ −⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥ ⎢ ⎥− − ° − °⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

Geometric Transformations for Generating

Orthographic Projection (Right View)

90[ ]yR −Drop Z

Rotations Needed for Generating Isometric Projection

1 0 0 0 cos 0 sin 00 cos sin 0 0 1 0 0

[ ] [ ]0 sin cos 0 sin 0 cos 00 0 0 1 0 0 0 1

v x yP R R P Pφ θ

θ θφ φφ φ θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Top

Right

Yv

Zv

Front

X

Z

Y

Xv

FrontTop

Right

X

Y

Z

Isometric Projection: Equally foreshorten the three main axes

°±=°±= 26.35,45 φθ

°±=°±= 26.35,45 yx rr• Rx --> Ry

• Rz --> Ry(Rx)

°±=°±= 74.54,45 )( xyz rr• Rx(Ry) --> Rz

ANGLEANYrr zxy =°±= ,45)(

Other Possible Rotation Paths