Lecture computer graphics and virtual reality slides lesson 4 fundamental algorithms

Lesson 4

Fundamental
algorithms

Trinh Thanh Trung School of ICT, HUST

Content

1. Clipping algorithms
2. Polygon drawing
3. Anti-aliasing

1.
Clipping algorithms

Clipping

■ A fundamental task in graphics is to keep those
parts of an object that lie outside a selected view
from being drawn
■ Clipping is the removal of all objects or part of
objects in a modelled scene that are outside the
real-world window.

Clipping

■ For each point to be rendered on the screen

xmin  x  xmax
ymin  y  ymax

xmin xmax
ymax

ymin

Line clipping

■ Lines are defined by their endpoints, so it should
be possible just to examine these (in a similar way
to points) and determine whether or not to clip
without considering every pixel on the line
■ We often have windows that are either very
large, i.e. nearly the whole scene fits inside, or very
small, i.e. most of the scene lies inside the window
■ Hence, most lines may be either trivially
accepted or rejected

Cohen-Sutherland Outcode

Line clipping

Cohen-Sutherland Outcode

■ Each point on all lines are first assigned an
“outcode” defining their position relative to the
clipping rectangle

■ If p.x <= xmin: P.code or 0001
■ If p.y <= ymin: P.code or 0100

■ If p.x >= xmax: P.code or 0010
■ If p.y >= ymax: P.code or 1000

Cohen-Sutherland Outcode

■ If P1.code OR P2.code == 0000

□ Accept line

■ If P1.code AND P2.code != 0000

□ Remove line

■ In case the line crosses the screen, the end
points will be redefined by the intersections of the
line and the boundary of the display.

Cohen-Sutherland

■ The Cohen-Sutherland line-clipping algorithm is
particularly fast for “trivial” cases, i.e. lines
completely inside or outside the window.
■ Non-trivial lines, i.e. ones that cross a boundary
of the window, are clipped by computing the
coordinates of the new boundary endpoint of the
line where it crosses the edge of the window

Cyrus-beck & Liang-barsky

Line clipping

Cyrus-beck & Liang-barsky

■ The Cohen-Sutherland algorithm requires the
window to be a rectangle, with edges aligned with
the co-ordinate axes
■ It is sometimes necessary to clip to any convex
polygonal window, e.g. triangular, hexagonal, or
rotated.
■ Cyrus-beck and Liang-Barsky line clippers better
optimise the intersection calculations for clipping to
window boundary
■ Nicholl-Lee-Nicholl reducing redundant
boundary clipping by identifying edge and corner
regions

Cyrus-beck & Liang-barsky

■ x = x1 + (x2 - x1)u = x1 + uDx
■ y = y1 + (y2 - y1)u = y1 + uDy
■ xmin  x1 + Dx.u  xmax  x  [xm, xM]
■ ymin  y1 + Dy.u  ymax  y  [ym, yM]
■ Pk u  qk k = 1, 2, 3, 4

q1 = x1 − xm P1 = −Dx
 
q2 = xM − x1 P2 = Dx
q3 = y1 − ym P3 = −Dy
q4 = yM − y1 P4 = Dy

Sutherland-Hodgman

Polygon clipping

Sutherland-Hodgman clipping

■ Sutherland-Hodgman basic routine

□ Go around polygon one vertex at a time
□ Current vertex has position p
□ Previous vertex had position s, and it has been

added to the output if appropriate

Sutherland-Hodgman clipping

■ Edge from s to p takes one of four cases:

inside outside inside outside inside outside inside outside

s p p s

p s p s

p output i output no output i output
p output

Sutherland-Hodgman clipping

1. s inside plane and p inside plane

o Add p to output
o Note: s has already been added

2. s inside plane and p outside plane

o Find intersection point i
o Add i to output

3. s outside plane and p outside plane

o Add nothing

4. s outside plane and p inside plane

o Find intersection point i
o Add i to output, followed by p

2.
Polygon drawing

Polygon scan conversion

■ Polygon scan conversion is a classic general purpose
algorithm.
■ For each scan-line we determine the polygon edges
that intersect it, compute spans representing the interior
portions of the polygons along this scan-line and fill the
associated pixels.
■ This represents the heart of a scan-line rendering

algorithm used in many commercial products including
Renderman and 3D Studio MAX.

Polygon scan conversion

■ Use mid-point algorithm
to specify the boundaries
between polygons
■ Basic algorithm:

□ Specify the intersections
between scanline and
polygon boundaries

□ Sort x value incrementally
□ Fill the pixels between

each pair of x