3D Rendering Praktikum: Shader Gallery

Jens Krüger & Polina Kondratieva – Computer Graphics and Visualization GroupJens Krüger & Polina Kondratieva – Computer Graphics and Visualization Group

computer graphics & computer graphics & visualizationvisualization

3D Rendering Praktikum:3D Rendering Praktikum:Shader GalleryShader Gallery

The Rendering PipelineThe Rendering Pipeline


Kai Bürger & Polina Kondratieva – Computer Graphics and Kai Bürger & Polina Kondratieva – Computer Graphics and Visualization GroupVisualization Group

… … what an „end user“ sees.what an „end user“ sees.

The Rendering Pipeline …The Rendering Pipeline …



The Rendering Pipeline …The Rendering Pipeline …

… … in picturesin pictures



Rendering pipelineRendering pipeline

Geometrysubsystem

Rastersubsystem

Objects in3D world

Colorimage

2D primitives



Pixel Stage

Vertex Stage

User / Driver

OverviewOverview

Transform & LightingTransform & Lighting Rasterizer

TexturingTexturingBlending/Ops

Texture 3Texture 2Texture 1Texture 0



Rendering pipelineRendering pipeline

Geometrysubsystem

Rastersubsystem

Objects in3D world

Colorimage

2D primitives

2D

Objectcoordinates

Worldcoordinates

Eyecoordinates

Modellingtransform

Viewingtransform

Normalizingtransform

Normalized(Clip-)coord.

clipping

affine

affine

Model-View-Transformation



TransformationsTransformations

- A transformation is like a function of a point

- Modeling:

- Define objects in „convenient“ coordinate systems

- Multiply-instantiated geometry

- Hierarchically defined linked figures

- Viewing:

- Window systems

- Virtual camera

- Perspective transformations

)(' PTP



Point representationPoint representation

- Represent as row or column vector

- Affects the kind of matrices we can multiply with

y

xP

yxP



Transformation representationTransformation representation

- Represent 2D transformations by a 2x2 matrix

- If the point is a column vector

- If the point is a row vector

dc

ba

y

x

dc

ba

y

x

Mpp

'

'

'

db

cayxyx

pMp T

''

'



Linear transformationsLinear transformations

- Scaling

- Reflection

y

x

s

syxP0

0

sx = sy uniform scaling

S

10

01yxP



Linear transformationsLinear transformations

- Shearing

- Rotation around origin by

1

01

ayxP

a1

cossin

sincosyxP

R(90°)



Affine transformationsAffine transformations- Affine transformations: parallelism is preserved, length & angles not

- All linear transformations can be written as matrix multiplication

- What about translation ?

- We want to write

y

x

t

t

y

xP

T t

11

010

001

1 yx

yx

tytx

tt

yxP

Homogeneous coordinates!Homogeneous coordinates!

Tzyx pppP )0(vector

Tzyx pppP )1(point



General Camera SetupGeneral Camera Setup

- Look at:- Position

- Orientation

- Frustum:- Camera parameters

- Viewport:- 2D coordinate system

Look atLook atLook atLook at

FrustumFrustumFrustumFrustum

ViewportViewportViewportViewport



Camera look atCamera look at

- Move camera C to originTranslate by –C

- Build orthonormal frame„right“ R=DxU„zenith“ U=RxD

- Adjust orientationRotation [R,U,D][X,Y,-Z]

Right

View direction

Up



Frustum (Perspective Projection)Frustum (Perspective Projection)

nf

nf

bB

tT

nb

nt

)tan(

)tan(

2

2

0100

00

00

00

2)(

2

2

nffn

nfnfbtbt

btn

lrlr

lrn

TT

tt

BB

bb

-f-f

-n-n

-z-z

yy



Last TransformationsLast Transformations

- After all the projections we have:

- Perspective division:

- Viewport transformation:

TwzyxP )(

Tddddd wzwywxwzyxP )1()(

2/)(]2/)[(

)1()2/(

)2/(

nearDfarDznearDfarD

Oyvp

Oxvp

z

y

x

P

d

ydy

xdx

v

v

v

v



Pixel Stage

Vertex Stage

User / Driver

OverviewOverview

Transform & LightingTransform & Lighting Rasterizer

TexturingTexturingBlending/Ops

Texture 3Texture 2Texture 1Texture 0



Bresenham (Line Drawing)Bresenham (Line Drawing)

pp xxmyy

myy

mxxmyxxmyy

xxmyy

kk

ppppk

ppk

1

1 1



Getting to Bresenham …Getting to Bresenham …/*********************************

* Input:

* xP: x-value of the startpoint

* yP: y-value of the startpoint

* xQ: x-value of the endpoint

* yQ: y-value of the endpoint

*********************************/

function drawLine(int xP, int yP, int xQ, int yQ) {

x = xP;

y = yP;

d = 0;

m = (yQ - yP) / (xQ - xP)

while(true) {

// put the pixel on the screen

putPixel(x, y);

if(x == xQ)

break;

x++;

d += m;

if(d > 0.5) {

y++;

d--;

}

}

}

/*********************************

* Input:

* xP: x-value of the startpoint

* yP: y-value of the startpoint

* xQ: x-value of the endpoint

* yQ: y-value of the endpoint

*********************************/


x = xP;

y = yP;

d = 0;

m = (yQ - yP) / (xQ - xP)

while(true) {

// put the pixel on the screen

putPixel(x, y);

if(x == xQ)

break;

x++;

d += m;

if(d > 0.5) {

y++;

d--;

}

}

}

Problem:• still floating point arithmetic

Obersvation:• m is rational• d is rational



Bresenham …Bresenham …


x = xP;

y = yP;

D = 0;

H = xQ - xP;

c = 2 * H;

M = 2 * (yQ - yP);

while(true) {

putPixel(x, y);

if(x == xQ) break;

x++;

D += M;

if(D > H) {

y++;

D -= c;

}

}

}


x = xP;

y = yP;

D = 0;

H = xQ - xP;

c = 2 * H;

M = 2 * (yQ - yP);

while(true) {

putPixel(x, y);

if(x == xQ) break;

x++;

D += M;

if(D > H) {

y++;

D -= c;

}

}

}

Introduce the following integer variables:

•

•

•

•

pq xxc 2:

pq yymcM 2:

dcD :

pq xxcH 5,0:

3D Rendering Praktikum: Shader Gallery

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