viewing3d pipeline

Graphics

Graphics Lab @ Korea University

cgvr.korea.ac.kr

3D Viewing

고려대학교 컴퓨터 그래픽스 연구실

CGVR


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3d Rendering Pipeline

Model TransformationModel Transformation

LightingLighting

Viewing TransformationViewing Transformation

Projection TransformationProjection Transformation

ClippingClipping

Viewport TransformationViewport Transformation

Scan ConversionScan Conversion

3D Primitives

Image

This is a pipelined sequence of operations to draw a 3D primitive into a

2D image for direct illumination

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In Pipeline

Transform into3d world coordinate system


LightingLighting



ClippingClipping



Image

3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image

Illustrate according to lighting and reflectance

3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image


Transform into 3D viewing coordinate system

3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image




3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image




Clip primitives outside window’s view

3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image





Transform into viewport

3D Primitives

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In Pipeline



LightingLighting



ClippingClipping



Image






Draw pixels(includes texturing, hidden surface etc.)

3D Primitives

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Transformation



LightingLighting



ClippingClipping



Image






Draw pixels(includes texturing, hidden surface etc.)

3D Primitives

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Transformation

3D Object Coordinate





p(x’, y’)

P(x, y, z)

3D World Coordinate

3D Viewing Coordinate

2D Projection Coordinate

2D Device Coordinate



3D World Coordinate

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Viewing Transformation






p(x’, y’)

P(x, y, z)

3D World Coordinate Viewing

Transformation 3D Viewing Coordinate



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Mapping from world to Viewing coordinates Origin moves to eye position Up vector maps to Y axis Right vector maps to X axis

X

Y

ZCamera

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Transformation from WC to VC

Transformation sequences

1. Translate the view reference point to the origin of the

WC system

2. Apply rotations to align the xv, yv, and zv axes with the world axes

General sequence of translate-rotate transformation

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Transformation from WC to VC (cont’)

Translation view reference point(x0, y0, z0)

Rotation rotate around the world xw axis to bring zv into the

xwzw plane rotate around the world yw axis to align the zw and zv

axis final rotation is about the zw axis to align the yw and yv

axis

1000

100

010

001

0

0

0

z

y

x

T

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Transformation from WC to VC (cont’)

Rotation by uvn system Calculate unit uvn vectors

N : view-plane normal vector V : view-up vector U : perpendicular to both N and V

Form the composite rotation matrix

321

321

321

, ,

, ,

, ,

vvv

uuu

nnn

unv

NV

NVu

N

Nn

1000

0

0

0

321

321

321

nnn

vvv

uuu

R TRM VCWC ,

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Camera Models

The most common model is pin-hole camera All captured light rays arrive along paths toward focal

point without lens distortion (everything is in focus) Sensor response proportional to radiance

Other models consider… Depth of field Motion blur Lens distortion

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Viewing Parameters

Position Eye position(px, py, pz)

Orientation View direction(dx, dy, dz) Up direction(ux, uy, uz)

Aperture Field of view(xfov, yfov)

Film plane “look at” point View plane normal

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Viewing Coordinate

Canonical coordinate system Convention is right-handed (looking down – z axis) Convention for projection, clipping, etc.

X

Y

Viewing up vector maps to Y axis

Viewing right vector maps to X axis

Viewing back vector maps to Z axis (potting out of page)

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Transformation matrix maps camera basis vectors to canonical vectors in viewing coordinate system

Right

UpBack

Eye(0, 0, 1)

(0, 1, 0)

(1, 0, 0)

Matrix

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p(x’, y’)

P(x, y, z)

3D World Coordinate

Projection

Transformation




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Projection

General definition Transform points in n-space to m-space(m<n)

In computer graphics Map viewing coordinates to 2D screen coordinates

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Taxonomy of Projections

Planar geometric projection

Parallel Perspective

Orthographic Oblique

Top

Front

Side

Axonometric Cabinet

Cavalier

Other

One-point

Two-point

Three-point

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Parallel & Perspective

Parallel Projection

Perspective Projection

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Top

Front

Side

Axonometric Cabinet

Cavalier

Other

One-point

Two-point

Three-point

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Parallel Projection

Center of projection is at infinity Direction of projection (DOP) same for all points

View Plane

DOP

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Top

Front

Side

Axonometric Cabinet

Cavalier

Other

One-point

Two-point

Three-point

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Parallel Projection View Volume

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Orthographic & Oblique

Orthographic parallel projection the projection is perpendicular to the view plane

Oblique parallel projection The projectors are inclined with respect to the view

plane

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Orthographic Projections

DOP perpendicular to view plane

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Orthographic Projections

DOP perpendicular to view plane

Front

Top Side

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Orthographic Coordinates

yyxx pp ,

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Oblique Projections

DOP not perpendicular to view plane

Cavalier(DOP at 45 )

Cabinet(DOP at 63.4 )

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Oblique Projections

DOP not perpendicular to view plane Cavalier projection

Cabinet projection 4.63,2tan

45,1tan

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Parallel Projection Matrix

General parallel projection transformation

)sin(

)cos(tan

,tan

sin,cos

1

1

1

Lzyy

Lzxx

zLz

LL

z

LyyLxx

p

p

pp

Where L1 is the inverse of tan α , which is also the value of L when z=1

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General parallel projection transformation

11000

0000

0sin10

0cos01

1

1

z

y

x

L

L

w

z

y

x

p

p

p

p

CGVR


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1000

0000

0sin10

0cos01

1

1

L

L

parallelM

CGVR


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Top

Front

Side

Axonometric Cabinet

Cavalier

Other

One-point

Two-point

Three-point

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Map points onto “view plane” along “projectors” emanating from “center of projection”(cop)

View PlaneCenter of Projection

Proj

ecto

rs

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How many vanishing point?

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Three-point

perspective

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Three-point

perspective

Two-point perspectiv

e

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Three-point

perspective

Two-point perspectiv

e

One-point perspectiv

e

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Perspective Projection View Volume

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Compute 2D coordinates from 3D coordinates with similar triangles

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Compute 2D coordinates from 3D coordinates with similar triangles

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Perspective Projection Matrix

4x4 matrix representation?

1

p

p

p

p

w

Dz

zyDy

zxDx

1????

????

????

????

z

y

x

w

z

y

x

p

p

p

p

Dzw

zz

yy

xx

p

'

'

'

CGVR


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4x4 matrix representation?

10100

0100

0010

0001

z

y

x

Dw

z

y

x

p

p

p

p

1

p

p

p

p

w

Dz

zyDy

zxDx

Dzw

zz

yy

xx

p

'

'

'

CGVR


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11

00

0100

0010

0001

1000

0000

0010

0001

11

00

0000

0010

0001

DD

PERM

Perspective projection

Perspective transformation

Orthographicprojection

1001

0000

0010

0001

x

PER

D

M

101

0

0000

0010

0001

y

PER

D

M

Center of Projection on the x axis Center of Projection on the y axis

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10

0000

0010

0001

or

10

0000

0010

0001

tssr

2-point perspectives

1

0000

0010

0001

tsr

3-point perspectives

CGVR


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Top

Front

Side

Axonometric Cabinet

Cavalier

Other

One-point

Two-point

Three-point

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Perspective vs. Parallel

Perspective projection+ Size varies inversely with distance – looks realistic

– Distance and angles are not(in general) preserved

– Parallel line do not (in general) remain parallel

Parallel projection + Good for exact measurements

+ Parallel lines remain parallel

– Angles are not (in general) preserved

– Less realistic looking

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Classical Viewing

Education

viewing3d pipeline