2D Raster Graphics

 Integer grid  Sequential (left-right, top-down) scan

Computer Graphics Line drawing

 A very important operation  used frequently, block diagrams, bar charts, engineering drawing, architecture plans, etc.  curves as concatenation of small line segments  Criteria  line should appear straight

illuminate nearest grid point

Computer Graphics Line drawing  Line should terminate correctly  Line should have a constant intensity

specify both end points instead of end point + slope +length

5 / 4 2 brightness adjustment (antialiasing) 5/4 intensity ~ # of dots/unit length

Computer Graphics Line drawing

 Line should not have “gaps”

y=f(x) if |slope|<1 x=f(y) if |slope|>1

y=f(x)

Computer Graphics Line drawing

 Line should be drawn as fast as possible  Brute-force method  DDA (digital differential analyzer) y  mx  b  1 fp * 1 fp + for(i  xo ;i  xn ;i  )

yi  m i  b

yi1  mxi1  b

 m(xi 1)  b 1 fp +

 mxi  b  m

 yi  m

Computer Graphics Bresenham’s Line Algorithm integer operations only

0  slope 1

(0,0)  (x2  x1, y2  y1) ti

si (x , y ) (0,0) i1 i1

if s  t or s  t  0  ti else s  t or s  t  0  si

Computer Graphics o 45 possible next intersection possible range current intersection range

Computer Graphics Bresenham’s Line Algorithm

ti dy si (xi1 1) dy dx s  (x 1)  y (xi1, yi1) i dx i1 i1 (0,0) dy t  ( y 1)  (x 1) i i1 dx i1 dy floating point s  t  2 (x 1)  2 y 1 i i dx i1 i1 dx(si  ti )  2(xi1dy  yi1dx)  2dy  dx Integer!! di

Computer Graphics Bresenham’s Line Algorithm

di  2(xi1dy  yi1dx)  2dy  dx

di1  2(xidy  yidx)  2dy  dx

 di1  di  2dy(xi  xi1)  2dx( yi  yi1)

 di1  di  2dy  2dx( yi  yi1)

 di1  di  2dy  2dx( yi  yi1)

Computer Graphics Bresenham’s Line Algorithm di1  di  2dy  2dx( yi  yi1) if di  0 choose ti

 yi  yi1 1

 di1  di  2(dy  dx) if di  0 choose si

 yi  yi1

 di1  di  2dy initial condition d1  2dy  dx (x0, y0 )  (0,0) •Complexity: 1 left shift + 2 integer additions

Computer Graphics Circle Drawing

 Symmetry reduces drawing to 1/8

(-y,x) (y,x) x  xc  r cos

y  yc  r sin (-x,y) (x,y) (-x,-y) (x,-y)

(-y,-x) (y,-x)

Computer Graphics Bresenham’s Circle Algorithm integer operations only

(x , y )  (0,0) (xi1, yi1) c c 45o    90o ti

si (0,0)

2 2 2 D(si )  (xi1 1)  ( yi1 1)  r | D(si )|| D(ti )|  ti 2 2 2 D(ti )  (xi1 1)  yi1  r | D(si )|| D(ti )|  si

Computer Graphics Bresenham’s Circle Algorithm

di | D(si )|| D(ti )| D(si )  D(ti ) 2 2 2 2 di  2r  2(xi1 1)  ( yi1 1)  yi1 2 2 2 2 di1  2r  2(xi1  2)  ( yi 1)  yi 2 2  di1  di  4xi1  6  2( yi  yi1 )  2( yi  yi1)

Computer Graphics Bresenham’s Circle Algorithm d1  3 2r (x0, y0 )  (0,r) if di  0 choose ti

 yi  yi1, di1  di  4xi1  6 if di  0 choose si

 yi  yi1 1, di1  di  4xi  4 yi  6

•Complexity: only integer and shift operations

Computer Graphics Other primitives

 Ellipse  symmetry reduces to 1/4  Bresenhem’s ellipse algorithm  Curve  difficult  approximation using short line segments  general curve forms (Bezier, B-spline, etc.)  Characters  rectangular grid patterns

Computer Graphics Polygon Filing

 Arbitrary # of sides  Convex or concave  Holes Y

X Computer Graphics Scan Line Algorithm

 Edge table  sort edges by scanline (using min Y) ym a x  record  x coordinate of ymin  ymax yi1  x to be added yi

(x, ymin )

Computer Graphics Scan Line Algorithm  Set y to smallest y in ET  Initialize AET to be Null  Repeat until AET and ET are empty  move from ET bucket y to AET those edges whose ymin=y  sort edges in AET by x (insertion sort)  fill in pixel values in between x pairs  remove from AET those edges whose ymax = y  increment y by 1  update x for all edges in AET x  x  x

Computer Graphics Polygon Patterned Filling

Y Y

logical and

ComputerX Graphics Polygon Patterned Filling

 Pattern can be anchored at

 a fixed point: transparent object moves on a patterned background

 a polygon corner: patterned object

Computer Graphics 2D Transformation

 For animation, manipulation, user interaction  translation, rotation, scaling

x' x Tx           y'  y Ty  x' cos  sinx        y' sin cos  y

x' Sx 0 x        y'  0 S y  y

Computer Graphics A Very Common Confusion

 What is being transformed? Points or coordinate system?  For CG, pipeline operations are always applied to features (points, lines, curves, planes)  But you can think in either way:  Points are physically moved in a fixed coordinate system (e.g., in modeling transform), or  A coordinate system is moved, while points stay stationary (e.g., in viewing transform)  Both interpretations are useful

Computer Graphics 2D Rigid Transformations

 A rigid transformation maps one coordinate system into another  Preserves distances and angles  To transform points from one coordinate frame to another, find the rigid transformation that brings the two coordinate frames in alignment  Translate so that their origins coincide  Rotate so that their axes coincide (x with x, y with y, and z with z)

Computer Graphics 2D examples y y’ y’ y

x’ x x’ x Rot

Trans y’ P y

x’

x Trans+Rot Computer Graphics 2D Translation

 Translate the coordinate system by (Tx,Ty)  What is the translated point?

y T  P  P  x P   Ty 

x (Tx,Ty)

Computer Graphics 2D rotation matrix

 Rotate θ counterclockwise P  RP  What is the transformation R?

y’ y

P  cos sin  P   P x’  sin cos   x

Computer Graphics 2D Rigid Transformations

 A rigid transformation moves an object from one location to another location  Preserves distances and angles  To transform points from one place to another, find the rigid transformation that  Translate so that the object moves  Rotate so that the object reorients

Computer Graphics 2D examples P y P P’ y P’

x x

Trans y Rot

Trans+Rot P’ Computer Graphics 2D Translation

 Translate the coordinate system by (Tx,Ty)  What is the translated point?

y (Tx,Ty) T  P  P  x P   Ty 

x (-Tx,-Ty)

Computer Graphics 2D rotation matrix

 Rotate θ counterclockwise P  RP  What is the transformation R?

y P’ cos  sin  P  P   P sin cos  x -

Computer Graphics 2D Transformation

 For animation, manipulation, user interaction  translation, rotation, scaling

x' x Tx           y'  y Ty  x' cos  sinx        y' sin cos  y

x' Sx 0 x        y'  0 S y  y

Computer Graphics 2D Transformation (cont.)

 Inconsistent representation for translation  Cannot be concatenated  Troublesome for  Hierarchical transforms  Interactive, incremental display

P  Rn (R3 (R2 (R1P  T1)  T2 )  T3 ) Tn

Computer Graphics Homogeneous Coordinates

 consistent representation for all three  can be concatenated & pre-computed

(x, y)  (wx, wy, w), w  0 (wx, wy, w)  (wx / w, wy / w)

Computer Graphics x' 1 0 Tx x  y'  0 1 T  y    y    1  0 0 1 1 x' cos  sin 0x  y'  sin cos 0 y       1   0 0 11

x' Sx 0 0x  y'   0 S 0 y    y    1   0 0 11 x' x  y'  (TRS ) y      1  1

Computer Graphics How about other transforms?

 For example, reflection

Mirrored images of each other

Computer Graphics  Try to represent the new transform as a composite of T, R, S

(a,b)

cos  sin 0 1 0 0 cos  sin 0   sin cos 0 0 1 0 sin cos 0        0 0 1 0 0 1  0 0 1 b b    tan 1( )   tan 1( ) a a

Computer Graphics Clipping Against Upright Rectangular Window

 Points

if xmin  x  xmax & ymin  y  ymax then accept otherwise reject

Computer Graphics Clipping Against Upright Rectangular Window

 Lines – trivially accepted if both end points inside – otherwise Points

x1  t(x2  x1)  x'1 t'(x'2 x'1 )

y1  t( y2  y1)  y'1 t'( y'2  y'1 ) 0  t,t' 1

(x1, y1),(x2 , y2 ) : end points of line

(x'1 , y'1 ),(x'2 , y'2 ) : end points of window boundary

Computer Graphics Cohen-Sutherland Line-Clipping Algorithm xmin xmax

1001 1000 1010 ymax  Outcodes bit1 --above: ymax-y 0001 0000 0010 bit2 --below: y-ymin ymin bit3 --right of: xmax-x bit4 --left of: x-xmin 0101 0100 0110

Computer Graphics  Trivially-accept: both end points having outcode 0000  Trivially-reject: corresponding bits in two outcodes are set, or outcode1 & outcode2 nonzero

1001 1000 1010

0001 0000 0010

0101 0100 0110 Computer Graphics  Neither: need more testing  E.g. mid-point algorithm

 divide a line segment (x1, y1),(x2, y2 ) into two line segments

(x1, y1),((x1  x2 ) / 2,( y1  y2 ) / 2)

((x1  x2 ) / 2,( y1  y2 ) / 2),(x2, y2 )  test each line independently  recursive division if necessary  guarantee to stop in O(logn) steps

Computer Graphics Cyrus-Beck (Liang-Basky) Line Clipping

 Can be more efficient when intersection tests are unavoidable  Work in the parameter (t) space to locate true intersections before calculating 2D coordinates  Work for all kinds of clipping polygons and in 3D  Two basic steps:  find intersections (t)  classify intersections

Computer Graphics Cyrus-Beck (Liang-Basky) Line Clipping  Find intersections

ax  by  c  0

N (P(t)  PE )  0

PE N (Po  t(P1  Po )  PE )  0 t N (P  P )  N (P  P )  0 P 1 o o E 1 N (P  P ) N (P(t)  PE )  0 t  o E  N (P1  Po ) N (P(t)  PE )  0 Po P (t)

N (P(t)  PE )  0

N  (a,b)

Computer Graphics  Intersections outside (0,1) range are not valid  But intersection inside (0,1) range might not be valid either

Computer Graphics Cyrus-Beck (Liang-Basky) Line Clipping

 Classify intersections

N (P1  Po )  0 PE (potentially entering)

N (P1  Po )  0 PL (potentially leaving)

Computer Graphics  Locate the largest PE point & t>0 P1  Locate the smallest PL point & t<1  PE < PL for a valid line PE

Computer Graphics Polygon Clipping (Sutherland-Hodgman)

 Given an ordered sequence of polygon vertices  And a convex clipping polygon  Output ordered clipped polygon vertices  Using divide-and-conquer, one clipping edge at a time

Computer Graphics IN OUT IN OUT

IN OUT IN OUT

Computer Graphics 1 1

4 2 5 2 4

3 3

Original Right boundary clipping

Computer Graphics 7 7

6 6

1 8 5 5

1 4 4 2 3 2 3

Bottom boundary clipping Left boundary clipping

Computer Graphics 7 6

8 4 9 3

1 2

Top boundary clipping

Computer Graphics Other Primitives

 Use of extents (extents for a whole string, words, individual characters)  Divide and Conquer

Computer Graphics