[PDF] - ibl ) Sfzrn size (32 point, 1 point = 1/72 inch) serif ( better PDF Document

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Einführung in Visual Computing

Werner Purgathofer

Institut für Computergraphik und Algorithmen

2. Vorlesungseinheit “Computergraphik”

Triangles and Polygon Filling

Attributes of Graphics Output Primitives in 2D

points, lines characters

in 2D and 3D

triangles

Werner Purgathofer / Einf. in Visual Computing 2

triangles

ther polygons

((filled) ellipses and other curves)

color type: solid, dashed, dotted,… Points and Line Attributes

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width, line caps, corners pen and brush options … text attributes

font (e.g. Courier, Arial, Times, Roman, …) styles (regular, bold, italic, underline,…) size (32 point, 1 point = 1/72 inch)

proportionally sized vs. fixed space fonts

Character Attributes

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p p y p

string attributes

rientation

alignment (left, center, right, justify)

v e r t i c a l horizontal

Displayed primitives generated by the raster algorithms discussed in Chapter 3 have a jagged, or stairstep, appearance. Displayed primitives generated by the raster algorithms discussed in Chapter 3 have a jagged, or stairstep, appearance. Displayed primitives generated by the raster algorithms discussed in Chapter 3 have a jagged, or stairstep, appearance. Displayed primitives generated by the raster algorithms discussed in Chapter 3 have a jagged,

r

stairstep, appearance.

font (typeface)

design style for (family of) characters

Courier, Times, Arial, …

serif (better readable), if (b tt l ibl ) Sfzrn

Sf rn

Character Primitives

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sans serif (better legible)

definition model

bitmap font (simple to define and display), needs more space (font cache)

utline font (more costly, less space,

geometric transformations)

Sfzrn

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Example: Newspaper

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Character Generation Examples

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the letter B represented with an 8x8 bilevel bitmap pattern and with an outline shape defined with straight line and curve segments fill styles

hollow, solid fill, pattern fill

fill options

edge type, width, color

pattern specification

through pattern tables

Area-Fill Attributes (1) 4 0 4 0 4 0 4 0 4 0

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through pattern tables tiling (reference point)

4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 4 0 0 4 combination of fill pattern with background colors soft fill

combination of colors

pattern background and

r

xor

Area-Fill Attributes (2)

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combination of colors antialiasing at object borders semitransparent brush simulation example: linear soft-fill

xor replace

F...foreground color B...background color

P = tF + (1  t)B

Triangle and Polygon Attributes color material transparency texture f d t il

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surface details reflexion properties, … → defined illumination produces effects General Polygon Fill Algorithms triangle rasterization

ther polygons: what is inside?

scan-line fill method flood fill method

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(α, β ,γ)

barycentric coordinates clean borders between adjacent triangles

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3

General Polygon Fill Algorithms triangle rasterization

ther polygons: what is inside?

scan-line fill method flood fill method

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“interior”, “exterior” for self-intersecting polygons? General Polygon Fill Algorithms triangle rasterization

ther polygons: what is inside?

scan-line fill method flood fill method

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interior pixels along a scan line passing through a polygon area General Polygon Fill Algorithms triangle rasterization

ther polygons: what is inside?

scan-line fill method flood fill method

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starting from a seed point fill until you reach a border (α, β ,γ) Triangles: Barycentric Coordinates

P0(x0, y0) P2(x2, y2) (0, 0, 1) (0 60 0 13 0 27)

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P1(x1, y1) (1, 0 ,0) (0, 1, 0) P = αP0 + βP1 + γP2 triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1} (0.60, 0.13, 0.27)

(α, β ,γ) Barycentric Coordinates

P0(x0,y0,z0) P2(x2,y2,z2) (0, 0, 1) (0 60 0 13 0 27)

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P1(x1,y1,z1) (1, 0 ,0) (0, 1, 0) P = αP0 + βP1 + γP2 triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1} (0.60, 0.13, 0.27)

for all x for all y /* use a bounding box!*/ {compute (α, β ,γ) for (x,y) ; if (0<α<1) and (0<β<1) and (0<γ<1) Triangle Rasterization Algorithm <1 <1 <1

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{c = αc0 + βc1 + γc2 ; draw pixel (x,y) with color c }}

P = αA + βB + γC triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1}

c = αc0 + βc1 + γc2 ; with color c interpolates values at the corners (vertices) linearly inside the triangle (and along edges)

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4

line through P1, P2: l12(x,y) = a12x+b12y+c12=0 then α = l12(x,y) / l12(x0,y0) β,γ analogous Computing (α, β ,γ) for P(x,y)

P2(x2, y2) l ( ) 0 P(x, y)

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P = αP0 + βP1 + γP2 triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1} P0(x0, y0) P1(x1, y1) (α, β ,γ) l12(x,y)=0

Barycentric Coordinates Example

0.6 0 0 0.4 0 2 0.2 0 4 0.0 0 6 0.4 0.0 0.6 0.2 0.2 0.6 0.0 0.4 0.6 0.2 0.0 0.8 0.0 0.2 0.8 0.0 0.0 1.0 1.2 0 6 1.0 0 4 0.8 0 2

0.2

0 8 1.2

0.8

0.6 1.0

0.6

0.6 0.8

0.4

0.6 0.6

0.2

0.6

0.2

0.6 0.6 1.2

1.0

0.8 1.0

0.8

0.8 0.8

0.6

0.8 0.6

0.4

0.8 0.4

0.2

0.8

0.2

0.4 0.8 1.2

1.2

1.0 1.0

1.0

1.0 0.8

0.8

1.0 0.6

0.6

1.0 0.4

0.4

1.0 0.2

0.2

1.0

0.2

0.2 1.0 1.2

1.4

1.2 1.0

1.2

1.2 0.8

1.0

1.2 0.6

0.8

1.2 0.4

0.6

1.2 0.2

0.4

1.2 0.0

0.2

1.2

0.2

0.0 1.2

P = αP0 + βP1 + γP2 triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1}

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1.0 0.0 0.0 0.0 1.0 0.0 0.8 0.2 0.0 0.6 0.4 0.0 0.4 0.6 0.0 0.2 0.8 0.0 0.8 0.0 0.2 0.6 0.2 0.2 0.4 0.4 0.2 0.2 0.6 0.2 0.0 0.8 0.2 0.0 0.4 0.2 0.4 0.4 0.4 0.6 0.4 1.2 0.0

0.2

1.0 0.2

0.2

0.8 0.4

0.2

0.6 0.6

0.2

0.4 0.8

0.2

0.2 1.0

0.2

0.0 1.2

0.2
0.2

1.4

0.2

1.2

0.2

0.0

0.2

1.2 0.0 1.2

0.4

0.2 1.0

0.2

0.2

0.2

1.0 0.2

0.6

0.4

0.4

0.4

0.2

0.4 0.8 0.4

Avoiding to Draw Borders twice don‘t draw the outline of the triangle!

result would depend on rendering order

draw only pixels that are inside exact triangle

i.e. pixels with α, β ,γ = 0, 1 are not drawn

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clean borders between adjacent triangles

Pixels exactly on a Border holes if both triangles leave pixels away simplest solution: draw both pixels better: arbitrary choice based on some test

e.g. only right boundaries

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clean borders between adjacent triangles

What is Inside a Polygon?

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dd-even rule

nonzero-winding- number rule ??? area-filling algorithms

“interior”, “exterior” for self-intersecting polygons?

dd-even rule

nonzero winding number rule

Inside-Outside Tests

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nonzero-winding-number rule same result for simple polygons

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5

inside/outside switches at every edge straight line to the outside:

even # edge intersections = outside

dd # edge intersections = inside

What is Inside?: Odd-Even Rule

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point is inside if polygon surrounds it straight line to the outside:

same # edges up and down = outside different # edges up and down = inside

What is Inside?: Nonzero Winding Number

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Polygon Fill Areas polygon classifications

convex: no interior angle > 180° concave: not convex

concavity test

vector method

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vector method

all vector cross products have the same sign

 convex

rotational method

rotate polygon-edges onto x-axis, always same direction  convex

for polygon area (solid-color, patterned)

scan-line polygon fill algorithm

intersection points located and sorted consecutive pairs define interior span attention with vertex intersections

Fill-Area Primitives

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exploit coherence (incremental calculations)

flood fill algorithm

Scan-Line Fill: Sorted Edge Table

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

The sorted edge table contains all polygon edges sorted by lowest y-value

ymax xstart 1/m

sort all edges by smallest y-value edge entry: [max y-value, x-intercept, inverse slope] Scan-Line Fill: Sorted Edge Table

yB xC 1/mCB yD xC 1/mCD

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active-edge list

for each scan line contains all edges crossed by that scan line incremental update

consecutive intersection pairs (spans) filled

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Sorted Edge Table

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

The sorted edge table contains all polygon edges sorted by lowest y-value

ymax xstart 1/m

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List When processing from bottom to top, keep a list

f all active edges

A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yF xA 1/mAF yB xA 1/mAB A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

incremental update! Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yF xA 1/mAF yB xA 1/mAB A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yB xA 1/mAB A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yB xA 1/mAB A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

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Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yB xA 1/mAB A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yB xA 1/mAB yB xC 1/mCB yD xC 1/mCD A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yB xA 1/mAB yB xC 1/mCB yD xC 1/mCD A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yD xC 1/mCD A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yD xC 1/mCD A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yE xD 1/mDE A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

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Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

yE xF 1/mFE yE xD 1/mDE A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

Sorted Edge Table / Active Edge List

B C D E yB xC 1/mCB yD xC 1/mCD yE xD 1/mDE

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Active Edge List

A F yF xA 1/mAF yB xA 1/mAB yE xF 1/mFE

incremental update of intersection point

1

 

 k k

y y m x x

k k

1

 



slope of polygon boundary line: m Scan-Line Fill: Incremental Update

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m

(for 2 successive scanlines) yk +1 yk (xk, yk) (xk+1, yk+1) intersection points along scan lines that intersect polygon vertices Scan-Line Fill: Intersecting Vertices

1 1 2 2 2

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vertices.

→ either special handling (1 or 2 intersections?) → or move vertices up or down by ε

pixel filling of area

areas with no single color boundary start from interior point “flood” internal region

Flood-Fill Algorithm

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4-connected, 8-connected areas reduce stack size by eliminating several recursive calls

area must be distinguishable from boundaries Flood-Fill: Boundary and Seed Point

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example: area defined within multiple color boundaries

seed point

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Definition: Definition: Flood-Fill: Connectedness

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Definition: 4-connected means, that a connection is

nly valid in these 4

directions Definition: 8-connected means, that a connection is valid in these 8 directions Examples for 4- and 8-connected

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has an 8-connected border an 8-connected area has a 4-connected border a 4-connected area Simple Flood-Fill Algorithm void floodFill4 (int x, int y, int new, int old) { int color; /* set current color to new */ getPixel (x, y, color); if (color = old) {

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setPixel (x, y); floodFill4 (x–1, y, new, old); /* left */ floodFill4 (x, y+1, new, old); /* up */ floodFill4 (x+1, y, new, old); /* right */ floodFill4 (x, y–1, new, old) /* down */ }} 1 2 4 3 recursion Bad Behavior of Simple Flood-Fill

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sequence floodFill4 produces too high stacks (recursion!) solution:

incremental horizontal fill (left to right)

Span Flood-Fill Algorithm

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( g ) recursive vertical fill (first up then down)

1 2 recursion Good Behavior of Span Flood-Fill

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sequence

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Span Flood-Fill Example Stack: x

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Span Flood-Fill Example B Stack: x A B

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A 1 2 3 Span Flood-Fill Example Stack: A B C D F E 4 5 6 7 8 9 10 11

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A 1 2 3 D E F C D Span Flood-Fill Example Stack: A C D E G E 4 5 6 7 8 9 10 11 12 13 14

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A 1 2 3 E F G C D Span Flood-Fill Example Stack: A C D E E=H 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

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A 1 2 3 E G H C D Span Flood-Fill Example Stack: A C D E 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 (E)

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A 1 2 3 E H C D

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Span Flood-Fill Example Stack: A C D I 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 J

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A 1 2 3 I J C I 20 21 22 Span Flood-Fill Example Stack: A C I J 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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A 1 2 3 J C I 20 21 22 Span Flood-Fill Example Stack: A C I K 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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A 1 2 3 K C K 20 21 22 24 25 Span Flood-Fill Example Stack: A C K L 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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A=N 1 2 3 L M N C M 20 21 22 24 25 26 27 28 29 30 L Span Flood-Fill Example Stack: A C L M 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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(A) 1 2 3 M N C M 20 21 22 24 25 26 27 28 29 30 L 31 32 33 Span Flood-Fill Example Stack: C L M 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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1 2 3 C 20 21 22 24 25 26 27 28 29 30 L 31 32 33 34 35

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Span Flood-Fill Example Stack: C L 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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1 2 3 C 20 21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 37 Span Flood-Fill Example Stack: C 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 23

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1 2 3 20 21 22 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 finished!