Geometric Transformations For Computer Graphics: Presentation By R.sakthivel Murugan, Ap/mech, Kcet

Geometric Transformations for Computer Graphics

Presentation by R.Sakthivel Murugan, AP/Mech, KCET.

Coordinate Systems • • • • • • •

World Coordinate System Object Coordinate System Hierarchical Coordinate Systems Viewpoint Coordinate System Model Window Coordinate System Screen Coordinate System Viewport Coordinate System

World Coordinate System • Also known as the "universe" or sometimes "model" coordinate system. • This is the base reference system for the overall model, ( generally in 3D ), to which all other model coordinates relate.

Object Coordinate System • When each object is created in a modelling program, the modeler must pick some point to be the origin of that particular object, and the orientation of the object to a set of model axes.

Hierarchical Coordinate Systems • Objects in a scene are arranged in a hierarchy, so that the "position" of one object in the hierarchy is relative to its parent in the hierarchy scheme, rather than to the world coordinate system.

View point Coordinate Systems • This coordinate system is based upon the viewpoint of the observer, and changes as they change their view.

Model Window Coordinate System • Coordinate system refers to the subset of the overall model world that is to be displayed on the screen

Screen Coordinate System • 2D coordinate system refers to the physical coordinates of the pixels on the computer screen, based on current screen resolution. ( E.g. 1024x768 )

Viewport Coordinate System • Coordinate system refers to a subset of the screen space where the model window is to be displayed.

2D Transformations What is transformations? – The geometrical changes of an object from a current state to modified state. Why the transformations is needed? – To manipulate the initially created object and to display the modified object without having to redraw it.

Types of 2D Transformations • Translation • Rotations • Scaling • Reflections

2D Translation (x’,y’)

(tx,ty)

(x,y)

• A translation moves all points in an object along the same straight-line path to new positions. • The path is represented by a vector, called the translation or shift vector.

2D Translation P  ( x, y ) T  (t x , t y )

?

P'  ( x' , y ' ) x'  x  t x ty =4

y'  y  t y By Matrix Form,

(2, 2)

tx = 6

 x '   x  t x   y '   y   t       y P'  P  T T  TransformationMatrix

2D Rotation P’

 P

• A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. • First, we’ll assume the pivot is at the origin.

2D Rotation Review Trigonometry => cos  = x/r , sin = y/r

P’(x’, y’)

x = r. cos , y = r.sin  => cos (+ ) = x’/r •x’ = r. cos (+ ) •x’ = r.coscos -r.sinsin •x’ = x.cos  – y.sin  =>sin (+ ) = y’/r y’ = r. sin (+ ) •y’ = r.cossin + r.sincos •y’ = x.sin  + y.cos 



r y’  x’

P(x,y) r



x

y

2D Rotation x'  r cos  cos   r sin  sin  y '  r cos  sin   r sin  cos  x'  x cos   y sin  y '  x sin   y cos  By Matrix Form,

 x' cos   sin    x   y '   sin  cos    y       P'  R  P R  RotationMatrix

Scaling P’

P

• Scaling changes the size of an object and involves two scale factors, Sx and Sy for the xand y- coordinates respectively. • Scales are about the origin.

Scaling P  ( x, y ) S  (s x , s y )

Sx Sy

x'  s x x y'  s y y

y

x

f

, yf



x

 x'  s x  y '   0    P'  S  P

0  x s y   y 

S-> Scale factor

2D Reflections y

y

y

x

x

x Reflection about X axis

Reflection about Y axis

Reflection about X & Y axis

P  ( x, y ) P '  ( x' , y ' )  [ X ,Y ]

P  ( x, y ) P'  ( x' , y ' )  [ X , Y ]

P  ( x, y ) P '  ( x' , y ' )  [ X ,Y ]

1 0   x  P'     0  1  y  P '  Pm  P

  1 0  x  P'      0 1  y  P '  Rm  P

 1 0   x  P'      0  1  y  P'  Z m  P

Specifying 2D Transformations • Translation – T(tx, ty) – Translation distances

• Scale – S(sx,sy) – Scale factors

• Rotation – R() – Rotation angle

Homogeneous Coordinates w x

 x

• Let’s move our problem into 3D. • Let point (x, y) in 2D be represented by point (x, y, 1) in the new space. • Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a). • A point in 2D can be represented in many ways in the new space. • (2, 4) ---------- (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc.

Homogeneous Coordinates Why do we use 1 for the last coordinate? • The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name – homogeneous coordinates.

2D Translation

2D Rotation

2D Scaling

 x  1 0 t x   x   y  0 1 t    y  , P  T t , t  P  x y y       1  0 0 1   1   x  cos   y   sin      1   0

 x   S x  y   0     1   0

 sin  cos  0

0 Sy 0

0  x  0   y  , P  R    P 1   1 

0  x  0   y  , P  S  S x , S y   P 1   1 

Composite transformations • More complex geometric & coordinate transformation can be built from the basic transformation by using the process of composition of function. • We can represent any sequence of transformations as a single matrix. – No special cases when transforming a point • matrix • vector.

– Composite transformations • matrix • matrix.

Inverse transformations:

1 0 t x   cos  T1  0 1 t y  , R 1    sin  0 0 1   0

sin  cos  0

0 0 1 S x 0 , S 1   0 1 S y  0 1  0

0 0 1 

P  M2  M1  P    M2  M1   P  M  P

Composite transformations:



 



P  T  t2 x , t2 y  T  t1x , t1 y   P  T  t2 x , t2 y   T  t1x , t1 y   P

Composite translations:

1 0 t2 x  1 0 t1x  1 0 t1x  t2 x  0 1 t   0 1 t   0 1 t  t  2y   1y  1y 2y    0 0 1  0 0 1  0 0 1  T  t2 x , t2 y   T  t1x , t1 y   T  t1x  t2 x , t1 y  t2 y 

P  R  2  R 1   P  R  2   R 1   P R  2   R 1   R 1   2 

Composite Rotations:

P  R 1   2   P

 S2 x  0   0 Composite Scaling:

0 S2 y 0

0   S1x 0    0 1   0

0 S1 y 0

0   S1x  S2 x 0    0 1   0

0 S1 y  S2 y 0

0 0  1 

S  S2 x , S2 y   S  S1x , S1 y   S  S1x  S2 x , S1 y  S2 y 

3D Translation Very similar to 2D. Using 4x4 matrices rather than 3x3.

x  x  t x

Translation

y

y  y  t y z  z  tz

 x, y, z  z

 x, y, z

x

 x   1  y  0    z  0     1  0

0 1 0 0

0 tx   x  0 t y   y   1 tz   z     0 1  1

3D Scaling y

y

x  x  S x y  y  S y z  x  S z

z

x

z

Enlarging object also moves it from origin

 x   S x  y  0 P       z   0    1  0

0 Sy 0 0

0 0 Sz 0

0  x  0  y    SP 0  z     1  1 

x

3D Rotations Rotation about X-axis ( Y-Z Plane)

Rotation about Y-axis ( Z-X Plane)

Rotation about Z-axis ( X-Y Plane)

0  x' 1  y ' 0 Cos P'       z '  0 Sin    0  1  0

0  Sin Cos 0

 x'  Sin  y '  0 P'       z '  Cos    1  0

0 Cos 1 0 0  Sin

 x' Cos  y '  Sin P'       z'  0    1  0

 Sin Cos 0

0

0

0

0  x  0  y    0 z    1  1  0  x  0  y    0 z    1  1 

0 0  x  0 0  y   1 0  z     0 1  1 

Thank You