From Vertices to Fragments
Software College, Shandong University
Instructor: Zhou Yuanfeng
E-mail: yuanfeng.zhou@gmail.com
Review
• Shading in OpenGL;
• Lights & Material;
• From vertex to fragment:
2
• Cohen-Sutherland
Black box
Projection
Fragments
Clipping
Shading
Surface hidden
Texture
Transparency
Objectives
• Geometric Processing:
Cyrus-Beck clipping algorithm
Liang-Barsky clipping algorithm
• Introduce clipping algorithm for polygons
• Rasterization: DDA & Bresenham
3
Cohen-Sutherland
• In case IV:
• o1 & o2 = 0
• Intersection: Clipping Lines by Solving
Simultaneous Equations
4
Solving Simultaneous Equations
• Equation of a line:
- Slope-intercept: y = mx + h difficult for vertical line
- Implicit Equation: Ax + By + C = 0
- Parametric: Line defined by two points, P0 and P1
• P(t) = P0 + (P1 - P0) t
• x(t) = x0 + (x1 - x0) t
• y(t) = x0 + (y1 - y0) t
Parametric Lines and
Intersections
For L1 :
x=x0l1 + t(x1l1 – x0l1)
y=y0l1 + t(y1l1 – y0l1)
For L2 :
x=x0l2 + t(x1l2 – x0l2)
y=y0l2 + t(y1l2 – y0l2)
The Intersection Point:
x0l1 + t1 (x1l1 – x0l1) = x0l2 + t2 (x1l2 – x0l2)
y0l1 + t1 (y1l1 – y0l1) = y0l2 + t2 (y1l2 – y0l2)
6
Cyrus-Beck Algorithm
• Cyrus-Beck algorithm (1978) for polygons
Mike Cyrus, Jay Beck. "Generalized two- and three-dimensional
clipping". Computers & Graphics, 1978: 23-28.
• Given a convex polygon R:
R
P(t )  ( P2  P1 )t  P1 0≤t≤1
N
P2
A
N A( P
t)0
,then
A) P (t ) is inside of R;
N  ( P(t ) 
) (
 N P(t )  A  cos 
N  ( P(t )  A)  0 ,then P (t ) is on R or extension;
  90
cos  0
) isoutside
,then
N  ( P(t )  A) 0 
90 P (tcos
 0 of R.
  90
cos  0
P1
para te
para ts
How to get ts and te
7
Cyrus-Beck Algorithm
A
• Intersection:
• NL ● (P(t) – A) = 0
Inside
P(t)
P0
Outside
NL
• Substitute line equation for P(t)
P(t) = P0 + t(P1 - P0)
• Solve for t
t = NL ● (P0 – A) / -NL ● (P1 - P0)
P1
Cyrus-Beck Algorithm
• Compute t for line intersection with all edges;
• Discard all (t < 0) and (t > 1);
• Classify each remaining intersection as
- Potentially Entering Point (PE)
- Potentially Leaving Point (PL) (How?)
NL●(P1 - P0) < 0 implies PL
NL●(P1 - P0) > 0 implies PE
Note that we computed this term in when computing t
Cyrus-Beck Algorithm
• For each edge:
Ni  ( P0  Ai )  Ni  ( P1  P0 )ti  0,
L3
t3
L2
t4
L1 t1
0  ti  1
P1
L4
L5
t2
para te
P0
t5
Compute PE with largest t
Compute PL with smallest t
Clip to these two points
para ts
ts  max{0, max{ti | Ni  ( P1  P0 )  0}}
te  min{1, min{ti | Ni  ( P1  P0 )  0}}
10
Cyrus-Beck Algorithm
• When Ni  ( P1  P0 )  0 ; then Ni  ( P1  P0 )
• if Ni  ( P0  Ai )  0
Then P0P1 is invisible;
• if Ni  ( P0  Ai )  0
Then go to next edge;
11
Programming:
for（k edges of clipping polygon）
{
solve Ni·(p1-p0);
solve Ni·(p0-Ai);
if ( Ni·(p1-p0) = = 0 ) //parallel with the edge
{
if ( Ni·(p0-Ai) < 0 )
break; //invisible
else
go to next edge;
}
else // Ni·(p1-p0) != 0
{
solve ti;
if ( Ni·(p1-p0) < 0 )
else
}
Input:
If (P0 = P1 )
Line is degenerate so clip
as a point;
Output:
if ( ts > te ) return nil;
else
return P(ts) and P(te) as the
true clip intersections;
te  min{1, min{ti | Ni  ( P1  P0 )  0}}
ts  max{0, max{ti | Ni  ( P1  P0 )  0}}
}
12
Liang-Barsky Algorithm (1984)
The ONLY algorithm named for Chinese
people in Computer Graphics course books
Liang, Y.D., and Barsky, B., "A New Concept and Method for
Line Clipping", ACM Transactions on Graphics, 3(1):1-22,
13
January 1984.
Liang-Barsky Algorithm (1984)
• Because of horizontal and vertical clip lines:
Many computations reduce
• Normals
• Pick constant points on edges
• solution for t:
tL=-(x0 - xleft) / (x1 - x0)
tR=(x0 - xright) / -(x1 - x0)
PE
tB=-(y0 - ybottom) / (y1 - y0) PE
tT=(y0 - ytop) / -(y1 - y0)
P0
PL
(0, -1)
(1, 0)
(0, 1)
PL
(-1, 0)
P1
Liang-Barsky Algorithm (1984)
N  ( P1  A)
N  ( P2  P1 )
Edge
Inner
normal
A
Left
x=XL
（1，0）
（XL，y）
（x1-XL， y1-y）
( x1  XL)
 ( x 2  x1 )
Right
x=XR
（-1，0）
（XR，y）
（x1-XR，y1-y）
 ( x1  XR)
x 2  x1
Bottom
y=YB
（0，1）
（x，YB）
（x1-x，y1-YB）
( y1  YB )
 ( y 2  y1 )
Top
y=YT
（0，-1）
（x，YT）
（x1-x，y1-YT）
P1-A
t
 ( y1  YT )
y 2  y1
15
Liang-Barsky Algorithm (1984)
Let ∆x=x2－x1，∆y=y2－y1:
r1  x,
s1  x1  xL ,
r2  x,
s2  xR  x1 ,
r3  y,
s3  y1  y B ,
r4  y,
s4  yT  y1 ,
tk  sk / rk , k  L, R, B, T
• When rk<0, tk is entering point; when rk>0, tk is
leaving point. If rk=0 and sk<0, then the line is
invisible; else process other edges
16
Comparison
• Cohen-Sutherland:
- Repeated clipping is expensive
- Best used when trivial acceptance and rejection is
possible for most lines
• Cyrus-Beck:
- Computation of t-intersections is cheap
- Computation of (x,y) clip points is only done once
- Algorithm doesn’t consider trivial accepts/rejects
- Best when many lines must be clipped
• Liang-Barsky: Optimized Cyrus-Beck
• Nicholl et al.: Fastest, but doesn’t do 3D
Clipping as a Black Box
• Can consider line segment clipping as a
process that takes in two vertices and
produces either no vertices or the vertices
of a clipped line segment
18
Pipeline Clipping of Line
Segments
• Clipping against each side of window is
independent of other sides
- Can use four independent clippers in a pipeline
19
Clipping and Normalization
• General clipping in 3D requires
intersection of line segments against
arbitrary plane
• Example: oblique view
20
Plane-Line Intersections
(( P 0  ( P1  t ( P 2  P1)))  n  0
n  ( p 0  p1)
t
n  ( p 2  p1)
21
Point-to-Plane Test
• Dot product is relatively expensive
- 3 multiplies
- 5 additions
- 1 comparison (to 0, in this case)
• Think about how you might optimize or
special-case this?
Normalized Form
top view
before normalization
after normalization
Normalization is part of viewing (pre clipping)
but after normalization, we clip against sides of
right parallelepiped
Typical intersection calculation now requires only
a floating point subtraction, e.g. is x > xmax
23
Clipping Polygons
• Clipping polygons is more complex than
clipping the individual lines
- Input: polygon
- Output: polygon, or nothing
Polygon Clipping
• Not as simple as line segment clipping
- Clipping a line segment yields at most one line
segment
- Clipping a polygon can yield multiple polygons
• However, clipping a convex polygon can
yield at most one other polygon
25
Tessellation and Convexity
• One strategy is to replace nonconvex
(concave) polygons with a set of
triangular polygons (a tessellation)
• Also makes fill easier (we will study later)
• Tessellation code in GLU library, the best is
Constrained Delaunay Triangulation
26
Pipeline Clipping of Polygons
• Three dimensions: add front and back clippers
• Strategy used in SGI Geometry Engine
• Small increase in latency
27
Sutherland-Hodgman
Clipping
Ivan Sutherland, Gary W. Hodgman: Reentrant
Polygon Clipping. Communications of the ACM, vol.
17, pp. 32-42, 1974
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully
clipped
Sutherland-Hodgman
Clipping
• Basic idea:
- Consider each edge of the viewport individually
- Clip the polygon against the edge equation
- After doing all planes, the polygon is fully clipped
• Will this work for non-rectangular clip regions?
• What would 3-D
clipping involve?
Sutherland-Hodgman
Clipping
• Input/output for algorithm:
- Input: list of polygon vertices in order
- Output: list of clipped poygon vertices consisting
of old vertices (maybe) and new vertices
(maybe)
• Note: this is exactly what we expect from
the clipping operation against each edge
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:
(Gray line can be a line or a plane)
inside
outside
inside
outside
inside
outside
p
s
p
p output
s
i output
p
inside
p
s
no output
i output
p output
outside
s
Sutherland-Hodgman
Clipping
• Four cases:
- s inside plane and p inside plane
• Add p to output
• Note: s has already been added
- s inside plane and p outside plane
• Find intersection point i
• Add i to output
- s outside plane and p outside plane
• Add nothing
- s outside plane and p inside plane
• Find intersection point i
• Add i to output, followed by p
Sutherland-Hodgman
Clipping
42
Point-to-Plane test
• A very general test to determine if a point
p is “inside” a plane P, defined by q and n:
(p - q) • n < 0:
(p - q) • n = 0:
(p - q) • n > 0:
q
p inside P
p on P
p outside P
q
q
n
p
n
n
p
p
P
P
P
Bounding Boxes
• Rather than doing clipping on a complex
polygon, we can use an axis-aligned bounding
box or extent
- Smallest rectangle aligned with axes that
encloses the polygon
- Simple to compute: max and min of x and y
44
Bounding Boxes
Can usually determine accept/reject based
only on bounding box
reject
accept
requires detailed
clipping
Ellipsoid collision detection
45
Rasterization
• Rasterization (scan conversion)
- Determine which pixels that are inside primitive
specified by a set of vertices
- Produces a set of fragments
- Fragments have a location (pixel location) and
other attributes such color, depth and texture
coordinates that are determined by interpolating
values at vertices
• Pixel colors determined later using color,
texture, and other vertex properties.
46
Scan Conversion of Line
Segments
• Start with line segment in window
coordinates with integer values for
endpoints
• Assume implementation has a
write_pixel function
y = mx + h
y
m
x
47
Scan Conversion of Line
Segments
One pixel
48
DDA Algorithm
• Digital Differential Analyzer (1964)
- DDA was a mechanical device for numerical
solution of differential equations
- Line y=mx+h satisfies differential equation
dy/dx = m = y/x = y2-y1/x2-x1
• Along scan line x = 1
For(x=x1; x<=x2, x++) {
y += m; //note:m is float number
write_pixel(x, round(y), line_color);
}
49
Problem
• DDA = for each x plot pixel at closest y
- Problems for steep lines
50
Using Symmetry
• Use for 1  m  0
• For m > 1, swap role of x and y
- For each y, plot closest x
51
Bresenham’s Algorithm
• DDA requires one floating point addition per step
• We can eliminate all fp through Bresenham’s
algorithm
• Consider only 1  m  0
- Other cases by symmetry
• Assume pixel centers are at half integers
(OpenGL has this definition)
Bresenham, J. E. (1 January 1965). "Algorithm for computer
control of a digital plotter". IBM Systems Journal 4(1): 25–30.
52
Bresenham’s Algorithm
• Observing:
If we start at a pixel that has been
written, there are only two candidates for
the next pixel to be written into the frame
buffer
53
Candidate Pixels
1m0
candidates
last pixel
Note that line could have
passed through any
part of this pixel
54
Decision Variable
d = △x(a-b)
d is an integer
d < 0 use upper pixel
d > 0 use lower pixel
How to compute a and b?
b-a =(yi+1–yi,r)-( yi,r+1-yi+1)
=2yi+1–yi,r–(yi,r+1)
= 2yi+1–2yi,r–1
A
B
C
-
ε(xi+1)= yi+1–yi,r–0.5
=BC-AC=BA=B-A
= yi+1–(yi,r+ yi,r+1)/2
55
Incremental Form
if ε(xi+1) ≥ 0, yi+1,r= yi,r+1, pick pixel D, the next pixel is
( xi+1, yi,r+1)
if ε(xi+1) < 0, yi+1,r= yi,r, pick pixel C, the next pixel is
( xi+1, yi,r)
yi,r+1
d1
yi,r
xi
D
D
d2
B
A
yi,r+1
C
yi,r
xi+1
d2
A
d1
xi
C
xi+1
56
Improvement
-
-
anew = alast – m
bnew = blast + m
-
anew = alast – (m-1)
bnew = blast + (m-1)
d = △x(a-b)
57
Improvement
• More efficient if we look at dk, the value of
the decision variable at x = k
dk+1= dk – 2△y,
if dk > 0
dk+1= dk – 2(△y - △x), otherwise
•For each x, we need do only an integer
addition and a test
•Single instruction on graphics chips
multiply 2 is simple.
58
BSP display
•
•
•
•
Type Tree
- Tree* front;
- Face face;
- Tree *back;
End
Algorithm DrawBSP(Tree T; point: w)
- //w 为视点
- If T is null then return; endif
- If w is in front of T.face then
• DrawBSP(T.back,w);
• Draw(T.face,w);
• DrawBSP(T.front,w);
- Else // w is behind or on T.face
• DrawBSP(T.front,w);
• Draw(T.face,w);
• DrawBSP(T. back,w);
- Endif
end
59
Hidden Surface Removal
• Object-space approach: use pairwise
testing between polygons (objects)
partially obscuring
can draw independently
• Worst case complexity O(n2) for n polygons
60
Image Space Approach
• Look at each projector (nm for an n x m
frame buffer) and find closest of k
polygons
• Complexity O(nmk)
• Ray tracing
• z-buffer
61
Painter’s Algorithm
• Render polygons a back to front order so
that polygons behind others are simply
painted over
B behind A as seen by viewer
Fill B then A
62
Depth Sort
• Requires ordering of polygons first
- O(n log n) calculation for ordering
- Not every polygon is either in front or behind all
other polygons
• Order polygons and deal with
easy cases first, harder later
Polygons sorted by
distance from COP
63
Easy Cases
• (1) A lies behind all other polygons
- Can render
• (2) Polygons overlap in z but not in either x
or y
- Can render independently
64
Hard Cases
(3) Overlap in all directions
but can one is fully on
one side of the other
(4)
cyclic overlap
penetration
65
Back-Face Removal (Culling)
•face is visible iff 90    -90
equivalently cos   0
or v • n  0

•plane of face has form ax + by +cz +d =0
but after normalization n = ( 0 0 1 0)T
•need only test the sign of c
•In OpenGL we can simply enable culling
but may not work correctly if we have nonconvex objects
66
z-Buffer Algorithm
• Use a buffer called the z or depth buffer to store
the depth of the closest object at each pixel
found so far
• As we render each polygon, compare the depth
of each pixel to depth in z buffer
• If less, place shade of pixel in color buffer and
update z buffer
67
Efficiency
• If we work scan line by scan line as we
move across a scan line, the depth
changes satisfy ax+by+cz=0
Along scan line
y = 0
z = - a x
c
In screen space x
=1
68
Scan-Line Algorithm
• Can combine shading and hsr through
scan line algorithm
scan line i: no need for depth
information, can only be in no
or one polygon
scan line j: need depth
information only when in
more than one polygon
69
Implementation
• Need a data structure to store
- Flag for each polygon (inside/outside)
- Incremental structure for scan lines that stores
which edges are encountered
- Parameters for planes
70
Visibility Testing
• In many realtime applications, such as
games, we want to eliminate as many
objects as possible within the application
- Reduce burden on pipeline
- Reduce traffic on bus
• Partition space with Binary Spatial
Partition (BSP) Tree
71
Simple Example
consider 6 parallel polygons
top view
The plane of A separates B and C from D, E and F
72
BSP Tree
• Can continue recursively
- Plane of C separates B from A
- Plane of D separates E and F
• Can put this information in a BSP tree
- Use for visibility and occlusion testing
73
Polygon Scan Conversion
• Scan Conversion = Fill
• How to tell inside from outside
- Convex easy
- Nonsimple difficult
- Odd even test
• Count edge crossings
- Winding number
odd-even fill
74
Winding Number
• Count clockwise encirclements of point
winding number = 1
winding number = 2
• Alternate definition of inside: inside if
winding number  0
75
Filling in the Frame Buffer
• Fill at end of pipeline
- Convex Polygons only
- Nonconvex polygons assumed to have been
tessellated
- Shades (colors) have been computed for
vertices (Gouraud shading)
- Combine with z-buffer algorithm
• March across scan lines interpolating shades
• Incremental work small
76
Using Interpolation
C1 C2 C3 specified by glColor or by vertex shading
C4 determined by interpolating between C1 and C2
C5 determined by interpolating between C2 and C3
interpolate between C4 and C5 along span
C1
C4
scan line
C2
C5
span
C3
77
Flood Fill
• Fill can be done recursively if we know a seed
point located inside (WHITE)
• Scan convert edges into buffer in edge/inside
color (BLACK)
flood_fill(int x, int y) {
if(read_pixel(x,y)= = WHITE) {
write_pixel(x,y,BLACK);
flood_fill(x-1, y);
flood_fill(x+1, y);
flood_fill(x, y+1);
flood_fill(x, y-1);
}
}
78
Scan Line Fill
• Can also fill by maintaining a data structure of all
intersections of polygons with scan lines
- Sort by scan line
- Fill each span
vertex order generated
by vertex list
desired order
79
Data Structure
80
Aliasing
• Ideal rasterized line should be 1 pixel wide
• Choosing best y for each x (or visa versa)
produces aliased raster lines
81
Antialiasing by Area
Averaging
• Color multiple pixels for each x depending on
coverage by ideal line
antialiased
original
magnified
82
Polygon Aliasing
• Aliasing problems can be serious for
polygons
- Jaggedness of edges
- Small polygons neglected
- Need compositing so color
of one polygon does not
totally determine color of
pixel
All three polygons should contribute to color
83