Viewing

CS418 Computer Graphics

John C. Hart

Graphics Pipeline

Homogeneous Divide

Homogeneous Divide

ModelCoords

ModelCoords

ModelXform

ModelXform

WorldCoords

WorldCoords

ViewingXform

ViewingXform

StillClip

Coords.

StillClip

Coords.ClippingClipping

WindowCoordinates

WindowCoordinates

Windowto

Viewport

Windowto

Viewport

ViewportCoordinates

ViewportCoordinates

ClipCoords.

ClipCoords.

ViewingCoords

ViewingCoords

PerspectiveDistortion

Graphics Pipeline

Homogeneous Divide

Homogeneous Divide

ModelCoords

ModelCoords Model

Xform

ModelXform World

Coords

WorldCoords Viewing

Xform

ViewingXform

StillClip

Coords.

StillClip

Coords. ClippingClipping

WindowCoordinates

WindowCoordinates

Windowto

Viewport

Windowto

ViewportViewport

Coordinates

ViewportCoordinates

ClipCoords.

ClipCoords.

ViewingCoords

ViewingCoords

PerspectiveDistortion

W2V Persp View Model0

1 1

s m

s m

m

x x

y y

z

Graphics Pipeline

Homogeneous Divide

Homogeneous Divide

ModelCoords

ModelCoords Model

Xform

ModelXform World

Coords

WorldCoords Viewing

Xform

ViewingXform

StillClip

Coords.

StillClip

Coords. ClippingClipping

WindowCoordinates

WindowCoordinates

Windowto

Viewport

Windowto

ViewportViewport

Coordinates

ViewportCoordinates

ClipCoords.

ClipCoords.

ViewingCoords

ViewingCoords

PerspectiveDistortion

0

1

s

s

x

y

1

m

m

m

x

y

z

M

Transformation Order

0

1

s

s

x

y

1

m

m

m

x

y

z

M

x

y

z

glutSolidTeapot(1); glRotate3f(-90, 0,0,1);glTranslate3f(0,1,0);glutSolidTeapot(1);

glTranslate3f(0,1,0);glRotate3f(-90, 0,0,1);glutSolidTeapot(1);

0

1

s

s

x

y

1

m

m

m

x

y

z

M R T 0

1

s

s

x

y

1

m

m

m

x

y

z

M T R

Viewing Transformation

Homogeneous Divide

Homogeneous Divide

ModelCoords

ModelCoords Model

Xform

ModelXform World

Coords

WorldCoords Viewing

Xform

ViewingXform

StillClip

Coords.

StillClip

Coords. ClippingClipping

WindowCoordinates

WindowCoordinates

Windowto

Viewport

Windowto

ViewportViewport

Coordinates

ViewportCoordinates

ClipCoords.

ClipCoords.

ViewingCoords

ViewingCoords

PerspectiveDistortion

W2V Persp View Model0

1 1

s m

s m

m

x x

y y

z

Viewing Transformation

W2V

Model

View

Persp

ModelCoords

WorldCoords

ViewingCoords

ClipCoords

ScreenCoords

x

y

z

x

ylookatpoint

eyepoint

x

y

z

WorldCoordinates

ViewingCoordinates

x

y

z

eye pointlookat point eye point(at origin)

lookat point(along –z axis)

x

y

z

WorldCoordinates

ViewingCoordinates

x

yeye pointlookat point lookat point

(along –z axis)

x

y

z

WorldCoordinates

ViewingCoordinates

x

yeye pointlookat point

y'x'

Why notthese axes?

x

y

z

WorldCoordinates

ViewingCoordinates

y

z

eye pointlookat point

up vectorup vector

up

1. Specify up direction in world coordinates

2. Transform up vector into viewing coordinates

3. Rotate about z-axis until up vector in x-y plane.

x

x

y

z

WorldCoordinates

ViewingCoordinates

y

z

eye pointlookat point

up vectorup vector

up

1. Specify up direction in world coordinates

2. Transform up vector into viewing coordinates

3. Rotate about z-axis until up vector in x-y plane.

x

Lookat Transformation

xy

z

lookatpoint

eyepoint

upvector

Lookat Transformation

1. Translate the eye point to the origin

xy

z

1

1

1

1

x

y

z

Lookat Transformation

1. Translate the eye point to the origin

2. Rotate the view vector into the negative z-axis

xy

z

1

1

1

1

x

yR

z

x

y

z

Easier Way

• Orthogonalize lookat vector system

x

y

z

upv

Easier Way

• Orthogonalize lookat vector system– Let r = v up/||vup||

x

y

z

up

r

v

Easier Way

• Orthogonalize lookat vector system– Let r = v up/||vup||– Let u = r v

x

y

z

upuv

r

Easier Way

• Orthogonalize lookat vector system– Let r = v up/||vup||– Let u = r v– Create rotation matrix from

<r,u,-v> to <x.y,z>

R r = x, R u = y, R v = -z

x

y

z

upuv

r

R=¿

Construct Lookat

• Translate eye point to origin• Rotate view into -z axis

– Let v = (lookat - eye)/||lookat - eye||– Let r = v up/||vup||– Let u = r v

x

y

z

xy

z

lookatpoint

eyepoint

upvector

Viewing Transformation

W2V

Model

View

Persp

ModelCoords

WorldCoords

ViewingCoords

ClipCoords

ScreenCoords

x

y

z

x

y

z

lookatpoint

eyepoint