Visible Surface Determination

CS418 Computer Graphics

John C. Hart

Painter’s Algorithm

• Display polygons inback-to-front order

• Sort polygons by z-value

– Which vertex?

– O(n log n)

• Problems…

-z

Quadtree Algorithm

• Sort polygons

• Subdivide screen until each region contains one or zero edges

• Invented by John Warnock in 1969

Quadtree Algorithm

• Sort polygons

• Subdivide screen until each region contains one or zero edges

• Invented by John Warnock in 1969

Quadtree Algorithm

• Sort polygons

• Subdivide screen until each region contains one or zero edges

• Invented by John Warnock in 1969

Quadtree Algorithm

• Sort polygons

• Subdivide screen until each region contains one or zero edges

• Invented by John Warnock in 1969

Quadtree Algorithm

• Sort polygons

• Subdivide screen until each region contains one or zero edges

• Invented by John Warnock in 1969

Z-Buffer

Key Observation: Each pixel displays color of only one triangle, ignores everything behind it

• Don’t need to sort triangles, just find for each pixel the closest triangle

• Z-buffer: one fixed or floating point value per pixel

• Algorithm:

For each rasterized fragment (x,y)

If z > zbuffer(x,y) then

framebuffer(x,y) = fragment color

zbuffer(x,y) = z

-far-far-far-far-far-far-far-far-far-far-far-far

fram

ebuf

fer

zbuf

fer

Z-Buffer

Key Observation: Each pixel displays color of only one triangle, ignores everything behind it

• Don’t need to sort triangles, just find for each pixel the closest triangle

• Z-buffer: one fixed or floating point value per pixel

• Algorithm:

For each rasterized fragment (x,y)

If z > zbuffer(x,y) then

framebuffer(x,y) = fragment color

zbuffer(x,y) = z

-far-.1-.2-.3-.4-.5-.6-.7-.8-far-far-far

fram

ebuf

fer

zbuf

fer

Z-Buffer

Key Observation: Each pixel displays color of only one triangle, ignores everything behind it

• Don’t need to sort triangles, just find for each pixel the closest triangle

• Z-buffer: one fixed or floating point value per pixel

• Algorithm:

For each rasterized fragment (x,y)

If z > zbuffer(x,y) then

framebuffer(x,y) = fragment color

zbuffer(x,y) = z

-far-.1-.2-.3-.4-.3-.1-.7-.8-far-far-far

fram

ebuf

fer

zbuf

fer

Z-Buffer

• Get fragment z-values by interpolating z-values at vertices during rasterization

• Perspective projection destroysz-values, setting them all to –d

• Need a perspective distortion that preserves at least the ordering ofz-values

-far-.1-.2-.3-.4-.3-.1-.7-.8-far-far-far

fram

ebuf

fer

zbuf

fer

Normalized View Volume

x

y

z

x

y

z

W2V

Model

View

Persp

ModelCoords

WorldCoords

ViewingCoords

ClipCoords

ScreenCoords

(left,top,-near)

(right,bottom,-near)

(0,0,-far)

glFrustum(left,right,bottom,top,near,far)

(-1,1,1)

z1 -1

x

1

-1

y

-1

1

Perspective Projection

screen

-z

y

zview

yview

d

yclip

clip view

view

viewclip

view /

y y

d z

yy

z d

view

viewviewview

viewview view

viewviewview

view

/1

1

/1

/1/ 0 1

1

x

z dxx

yy y

z dzz

dz dd

Perspective Distortion

screen

-z

y

zview

yview

1

yclip

viewclip

view

yy

z

view

viewviewview

viewviewview

viewviewview

viewview

1

1

1 0 1

1

x

zxx

yyy

zzz

zz

Distorted z-Values

viewclip

view

yy

z

view

viewviewview

viewviewview

viewviewview

viewview

1

1

1 0 1

1

x

zxx

yyy

zzz

zz

-z

- – /z

-

1/-z curve

z1 z2

- – /z2

- – /z1

if z1 > z2 then - – /z1 > - – /z2

Normalized Perspective Distortion

2 near right left

right left right left

2 near top bottom

top bottom top bottom

far near 2 far near

far near far near1 0

x

y

z

x

y

z

(left,top,-near)

(right,bottom,-near)

(0,0,-far)

(-1,1,1)

z1 -1

x

1

-1

y

-1

1

Hierarchical Z-Buffer

• Invented by Ned Green in 1994

• Creates a MIP-map of the z-buffer

– z-value equal to farthest z-value of its four children

• Before rasterizing a triangle…

– Check z-value of its nearest vertex against z-value of the smallest quadtree cell containing the triangle

– If z-test fails, then the entire triangle is hidden and need not be rasterized

• Works best when displayingfront-to-back

-.2-.3-.4-.5-.6-.7-.8-far

fram

ebuf

fer

hierarchical zbuffer

-.3

-.5

-.7

-far

-.5

-far

-far