From Vertices to Fragments Software College, Shandong University Instructor: Zhou Yuanfeng E-mail: [email protected]

From Vertices to Fragments

Software College, Shandong University

Instructor: Zhou Yuanfeng E-mail: [email protected]

Review

•Shading in OpenGL;•Lights & Material;•From vertex to fragment:

•Cohen-Sutherland2

ProjectionFragmentsClippingShading

Black box Surface hiddenTextureTransparency

3

Objectives

•Geometric Processing:

Cyrus-Beck clipping algorithm

Liang-Barsky clipping algorithm• Introduce clipping algorithm for polygons•Rasterization: DDA & Bresenham

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

6

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)

•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:

7

Cyrus-Beck Algorithm

P1

P2

A

N

R

para tspara te

112 )()( PtPPtP 0≤t≤1

0))(( AtPN )(tP,then is inside of R;

0))(( AtPN )(tP

0))(( AtPN )(tP

,then is on R or extension;

,then is outside of R.

cos)(

))((

AtPN

AtPN

0cos 90

0cos 90

0cos 90

How to get ts and te


• Intersection:

•NL ● (P(t) – A) = 0

•Substitute line equation for P(t)

P(t) = P0 + t(P1 - P0)

•Solve for t

t = NL ● (P0 – A) / -NL

● (P1 - P0)

A

NL

P(t)

Inside

OutsideP0

P1


•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


•For each edge:

10

L1

L2L3

L4

L5

P0

P1

t1

t2

t5

t3t4

para tspara te

0 1 0( ) ( ) 0, 0 1i i i i iP A P P t t N N

1 0

1 0

max{0,max{ | ( ) 0}}

min{1,min{ | ( ) 0}}s i i

e i i

t t P P

t t P P

N

NCompute PE with largest tCompute PL with smallest tClip to these two points


•When ; then • if

• if

11

1 0( ) 0i P P N

0( ) 0i iP A N

Then P0P1 is invisible;

0( ) 0i iP A N

Then go to next edge;

1 0( )i P P N

Programming:

12

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

} }

1 0min{1,min{ | ( ) 0}}e i it t P P N

1 0max{0,max{ | ( ) 0}}s i it t P P N

Output:if ( ts > te ) return nil;else return P(ts) and P(te) as the true clip intersections;

Input:If (P0 = P1 ) Line is degenerate so clip

as a point;

Liang-Barsky Algorithm (1984)

13

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, January 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)

tB=-(y0 - ybottom) / (y1 - y0)

tT=(y0 - ytop) / -(y1 - y0)


PE

PLP1

PL

PE

P0

(-1, 0)(1, 0)(0, -1)

(0, 1)

15

)(

)(

12

1

PP

APt

N

N

)(

)(

12

1

xx

XLx

12

1 )(

xx

XRx

)(

)(

12

1

yy

YBy

12

1 )(

yy

YTy

EdgeInner

normalA P1-A

Leftx=XL

（ 1 ，0 ）

（ XL ，y ）

（ x1-XL ， y1-

y ）

Rightx=XR

（ -1 ，0 ）

（ XR ，y ）（ x1-XR ， y1-y ）

Bottomy=YB

（ 0 ，1 ）

（ x ， YB ）（ x1-x ， y1-YB ）

Topy=YT

（ 0 ， -1 ）

（ x ， YT ）（ x1-x ， y1-YT ）


• 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


Let ∆x=x2 － x1 ，∆ y=y2 － y1:

,

,

,

,

4

3

2

1

yr

yr

xr

xr

,

,

,

,

14

13

12

11

yys

yys

xxs

xxs

T

B

R

L

TBRLkrst kkk , , , , /

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

18

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

19

Pipeline Clipping of Line Segments

•Clipping against each side of window is independent of other sides

Can use four independent clippers in a pipeline

20

Clipping and Normalization

•General clipping in 3D requires intersection of line segments against arbitrary plane

•Example: oblique view

21

Plane-Line Intersections

0 1

2 1

( )

( )

n p pt

n p p

0 1 2 1(( ( ( ))) n 0P P t P P

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?

23

Normalized Form

before normalization after normalization

Normalization is part of viewing (pre clipping)but after normalization, we clip against sides ofright parallelepiped

Typical intersection calculation now requires onlya floating point subtraction, e.g. is x > xmax

top view

Clipping Polygons

•Clipping polygons is more complex than clipping the individual lines

Input: polygon

Output: polygon, or nothing

25

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

26

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

27

Pipeline Clipping of Polygons

• Three dimensions: add front and back clippers

• Strategy used in SGI Geometry Engine

• Small increase in latency

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




After doing all planes, the polygon is fully clipped

































•Will this work for non-rectangular clip regions?•What would 3-D clipping involve?


• 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 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


•Edge from s to p takes one of four cases:(Gray line can be a line or a plane)

inside outside

s

p

p output

inside outside

s

p

no output

inside outside

sp

i output

inside outside

sp

i outputp output


•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


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 inside P

(p - q) • n = 0: p on P

(p - q) • n > 0: p outside P

P

np

q

P

np

q

P

np

q

44

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

45

Bounding Boxes

Can usually determine accept/reject based only on bounding box

reject

accept

requires detailed clipping

Ellipsoid collision detection

46

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.

47

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

x

ym

Scan Conversion of Line Segments

48

One pixel

49

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);}

50

Problem

•DDA = for each x plot pixel at closest y Problems for steep lines

51

Using Symmetry

•Use for 1 m 0•For m > 1, swap role of x and y

For each y, plot closest x

52

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.

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

54

Candidate Pixels

1 m 0

last pixel

candidates

Note that line could havepassed through anypart of this pixel

55

Decision Variable

-

d = x(a-b)△

d is an integerd < 0 use upper pixeld > 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

ε(xi+1)= yi+1–yi,r–0.5

=BC-AC=BA=B-A

= yi+1–(yi,r+ yi,r+1)/2

AB

C

Incremental Form

56

if ε(xi+1) ≥ 0, yi+1,r= yi,r+1, pick pixel D, the next pixel is

( xi+1, yi,r+1)

yi,r

yi,r+1

AB

xixi+1

D

C

d1

d2

yi,r

yi,r+1

A

xi xi+1

D

Cd1

d2

if ε(xi+1) < 0, yi+1,r= yi,r, pick pixel C, the next pixel is

( xi+1, yi,r)

Improvement

anew = alast – m anew = alast – (m-1)

bnew = blast + m bnew = blast + (m-1)

57

- -

- -

d = x(a-b)△

Improvement

58

•More efficient if we look at dk, the value of the decision variable at x = k

dk+1= dk – 2 y, if d△ k > 0dk+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.

59

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

60

Hidden Surface Removal

•Object-space approach: use pairwise testing between polygons (objects)

•Worst case complexity O(n2) for n polygons

partially obscuring can draw independently

61

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

62

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

63

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

64

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

65

Hard Cases

(3) Overlap in all directionsbut can one is fully on one side of the other

cyclic overlap

penetration

(4)

66

Back-Face Removal (Culling)

•face is visible iff 90 -90equivalently cos 0or v • n 0

•plane of face has form ax + by +cz +d =0but after normalization n = ( 0 0 1 0)T

•need only test the sign of c

•In OpenGL we can simply enable cullingbut may not work correctly if we have nonconvex objects

67

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

68

Efficiency

• If we work scan line by scan line as we move across a scan line, the depth changes satisfy ax+by+cz=0

Along scan line

y = 0z = - x

c

a

In screen space x = 1

69

Scan-Line Algorithm

•Can combine shading and hsr through scan line algorithm

scan line i: no need for depth information, can only be in noor one polygon

scan line j: need depth information only when inmore than one polygon

70

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

71

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

72

Simple Example

consider 6 parallel polygons

top view

The plane of A separates B and C from D, E and F

73

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

74

Polygon Scan Conversion

•Scan Conversion = Fill•How to tell inside from outside

Convex easy

Nonsimple difficult

Odd even test• Count edge crossings

Winding numberodd-even fill

75

Winding Number

•Count clockwise encirclements of point

•Alternate definition of inside: inside if winding number 0

winding number = 2

winding number = 1

76

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

77

Using Interpolation

span

C1

C3

C2

C5

C4scan line

C1 C2 C3 specified by glColor or by vertex shadingC4 determined by interpolating between C1 and C2

C5 determined by interpolating between C2 and C3

interpolate between C4 and C5 along span

78

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);} }

79

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

80

Data Structure

81

Aliasing

• Ideal rasterized line should be 1 pixel wide

•Choosing best y for each x (or visa versa) produces aliased raster lines

82

Antialiasing by Area Averaging

• Color multiple pixels for each x depending on coverage by ideal line

original antialiased

magnified

83

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

Documents

From Vertices to Fragments Software College, Shandong University Instructor: Zhou Yuanfeng E-mail: [email protected]