2D Geometric Transformations Question : How do we represent a - PDF document

2D Geometric Transformations (Chapter 5 in FVD) 1 2D Geometric Transformations • Question : How do we represent a geometric object in the plane? • Answer : For now, assume that objects consist of points and lines. A point is represented by its Cartesian coordinates: ( x,y ). • Question : How do we transform a geometric object in the plane? • Answer : Let ( A,B ) be a straight line segment and T a general 2D transformation: T transforms ( A,B ) into another straight line segment ( A’,B’ ), where A’=TA and B’=TB . 2

Translation • Translate (a,b): (x,y) (x+a,y+b) Translate(2,4) 3 Scale • Scale (a,b): (x,y) (ax,by) Scale (2,3) Scale (2,3) 4

• How can we scale an object without moving its origin (lower left corner)? Translate(-1,-1) Scale(2,3) Translate(1,1) 5 Reflection Scale(-1,1) Scale(1,-1) 6

Rotation • Rotate( θ): (x,y) (x cos( θ )+y sin( θ ), -x sin( θ )+y cos( θ )) Rotate(90) Rotate(90) 7 • How can we rotate an object without moving its origin (lower left corner)? Translate(-1,-1) Translate(1,1) Rotate(90) 8

Shear • Shear (a,b): (x,y) (x+ay,y+bx) Shear(1,0) Shear(0,2) 9 Composition of Transformations • Rigid transformation: – Translation + Rotation (distance preserving). • Similarity transformation: – Translation + Rotation + uniform Scale (angle preserving). • Affine transformation: – Translation + Rotation + Scale + Shear (parallelism preserving). • All above transformations are groups where Rigid ⊂ Similarity ⊂ Affine. 10

Matrix Notation • Let’s treat a point ( x,y ) as a 2x1 matrix (a column vector):   x     y • What happens when this vector is multiplied by a 2x2 matrix? +       a b x ax by =       +       c d y cx dy 11 2D Transformations • 2D object is represented by points and lines that join them. • Transformations can be applied only to the the points defining the lines. • A point ( x,y ) is represented by a 2x1 column vector, and we can represent 2D transformations using 2x2 matrices:       x ' a b x =        '   d    12

Scale • Scale(a,b): ( x,y ) ( ax,by )       a 0 x ax =             0 b y by • If a or b are negative, we get reflection. 13 Reflection • Reflection through the y axis: −  1 0     0 1 • Reflection through the x axis:   1 0   − 1   0 • Reflection through y=x :   0 1     1 0 • Reflection through y=-x : −   0 1   −   1 0 14

Shear, Rotation • Shear(a,b): (x,y) (x+ay,y+bx) +       1 a x x ay =       +       b 1 y y bx • Rotate( θ ): (x,y) ( x cos θ + y sin θ , - x sin θ + y cos θ ) θ θ θ + θ       cos sin x x cos y sin =       − − + θ θ θ θ       sin cos y x sin y cos 15 Composition of Transformations • A sequence of transformations can be collapsed into a single matrix:     x x [ ][ ][ ] [ ] = A B C D         y y • Note: order of transformations is important! ( otherwise - commutative groups ) translate rotate rotate translate 16

Translation +     x x a → • Translation(a,b):     +     y y b • Problem : Cannot represent translation using 2x2 matrices. • Solution : Homogeneous Coordinates 17 Homogeneous Coordinates • Homogeneous Coordinates is a mapping from R n to R n+1 : → = x y X Y W tx ty t ( , ) ( , , ) ( , , ) • Note: ( tx,ty,t ) all correspond to the same non-homogeneous point ( x,y ). E.g. (2,3,1) ≡ (6,9,3). • Inverse mapping:   X Y →   ( X , Y , W ) ,   W W 18

Translation • Translate(a,b):      +  1 0 a x x a       = + 0 1 b y y b              0 0 1   1   1  • Affine transformation now have the following form:   a b e   c d f      0 0 1  19 Geometric Interpretation W Y (X,Y,W) 1 y x (X,Y,1) X • A 2D point is mapped to a line (ray) in 3D. The non-homogeneous points are obtained by projecting the rays onto the plane Z=1. 20

• Example: Rotation about an arbitrary point: (x 0 ,y 0 ) θ • Actions: – Translate the coordinates so that the origin is at (x 0 ,y 0 ). – Rotate by θ . – Translate back. θ − θ −         1 0 x cos sin 0 1 0 x x 0 0         θ θ − = 0 1 y sin cos 0 0 1 y y         0 0                 0 0 1 0 0 1 0 0 1 1 θ − θ − θ + θ     cos sin x ( 1 cos ) y sin x 0 0     = θ θ − θ − θ sin cos y ( 1 cos ) x sin y     0 0         0 0 1 1 21 • Another example: Reflection about an Arbitrary Line: p 2 p 1 L=p 1 +t (p 2 -p 1 )=t p 2 +(1-t) p 1 • Actions: – Translate the coordinates so that P 1 is at the origin. – Rotate so that L aligns with the x- axis. – Reflect about the x-axis. – Rotate back. – Translate back. 22

Viewing in 2D (Chapter 6 in FVD) World Coordinate Device Coordinate Window Viewport Object in World g n i p p a m D 2 : D 3 2D:2D mapping Device Coordinates World Coordinates • Objects are given in world coordinates . • The world is viewed through a world- coordinate window . • The WC window is mapped onto a device coordinate viewport . 23 Viewing in 2D (cont.) Maximum range of screen Window coordinates Viewport World Coordinates Screen Coordinates (Device Coordinates) Window to Viewport Transformation 24

Window to Viewport Transformation (x max ,y max ) (u max ,v max ) (x min ,y min ) (u min ,v min ) Window in Window Window Translated to World translated scaled to viewport position in Coordinates to origin viewport size. screen coordinates. u max -u min v max -v min M wv = T(u min ,v min ) S( , ) T(-x min ,-y min ) x max -x min y max -y min u max -u min 0 0 1 0 u min 1 0 -x min x max -x min v max -v min = 0 1 v min 0 0 0 1 -y min y max -y min 0 0 1 0 0 1 0 0 1 u max -u min u max -u min 0 x max -x min (-x min ) + u min x max -x min v max -v min v max -v min = 0 y max -y min (-y min ) + v min y max -y min 0 0 1 25