advanced computer graphics three dimensional viewing

Advanced Computer GraphicsAdvanced Computer Graphics

Three Dimensional ViewingThree Dimensional Viewing

OverviewOverview

• Viewing a 3D scene

• Projections– Parallel and perspective

OverviewOverview

• Surface rendering

OverviewOverview

• Exploded and cutaway views

OverviewOverview

• 3D and stereoscopic viewing

3D Viewing Pipeline3D Viewing Pipeline

ModelingTransformation

ViewingTransformation

ProjectionnTransformation

NormalizationTransformation

and Clipping

ViewportTransformation

MC

WC

VC PC

NC

DC

Viewing CoordinatesViewing Coordinates

• Generating a view of an object in 3D is similar to photographing the object.

• Whatever appears in the viewfinder is projected onto the flat film surface.

• Depending on the position, orientation and aperture size of the camera corresponding views of the scene is obtained.

Specifying The View CoordinatesSpecifying The View Coordinates

• For a particular view of a scene first we establish viewing-coordinate system.

• A view-plane (or projection plane) is set up perpendicular to the viewing z-axis.

• World coordinates are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.

xw

zw

yw

xv

zv

yv

P0=(x0 , y0 , z0)


• To establish the viewing reference frame, we first pick a world coordinate position called the view reference point.

• This point is the origin of our viewing coordinate system. If we choose a point on an object we can think of this point as the position where we aim a camera to take a picture of the object.


• Next, we select the positive direction for the viewing z-axis, and the orientation of the view plane, by specifying the view-plane normal vector, N.

• We choose a world coordinate position P and this point establishes the direction for N.

• OpenGL establishes the direction for N using the point P as a look at point relative to the viewing coordinate origin.

xw

zw

yw

xv

zv

P0

P

N

xv

yv


• Finally, we choose the up direction for the view by specifying view-up vector V.

• This vector is used to establish the positive direction for the yv axis.

• The vector V is perpendicular to N.• Using N and V, we can compute a

third vector U, perpendicular to both N and V, to define the direction for the xv axis.

xw

zw

yw

xv

zv

yv

P0

P

N

V


To obtain a series of views of a scene , we can keep the the view reference point fixed and change the direcion of N. This corresponds to generating views as we move around the viewing coordinate origin.

P0

V

N

N

Transformation From World To Transformation From World To Viewing CoordinatesViewing Coordinates

Conversion of object descriptions from world to viewing coordinates is equivalent to transformation that superimpoes the viewing reference frame onto the world frame using the translation and rotation.

xw

yw

zw

xv

yv

zv


First, we translate the view reference point to the origin of the world coordinate system

xw

yw

zw

xv

yv

zv


Second, we apply rotations to align the xv,, yv and zv axes with the world xw, yw and zw axes, respectively.

xw

yw

zw

xv

yv

zv

xv

yv

zv


If the view reference point is specified at word position (x0, y0, z0), this point is translated to the world origin with the translation matrix T.

1000

100

010

001

0

0

0

z

y

x

T


• The rotation sequence requires 3 coordinate-axis transformation depending on the direction of N.

• First we rotate around xw-axis to bring zv into the xw

-zw plane.

1000

00

00

0001

CosSin

SinCosxR


Then, we rotate around the world yw axis to align the zw and zv axes.

1000

00

0010

00

CosSin

SinCos

yR


The final rotation is about the world zw axis to align the yw and yv axes.

1000

0100

00

00

CosSin

SinCos

zR


The complete transformation from world to viewing coordinate transformation matrix is obtaine as the matrix product

TRRRM xyzvcwc ,

Transformation From World To Viewing Transformation From World To Viewing Coordinates: Coordinates: An Example For 2d SystemAn Example For 2d System

y

x

x′y′

Θ=300

0 2 4 6

0

2

4

6

2

2

P=(5,5)

P0=(4,3)

y

x

x′y′

Θ=300

2 4 6

0

2

4

6

2 2

P

P0

Translation:

100

310

401

T


Rotation

0

2

4

6

y

xx′

y′

2 4 6

P

P0

100

0866.05.0

05.0866.0

R


New coordinates

100

310

401

100

0866.05.0

05.0866.0

.vcwcM

1

232.1

866.1

1

5

5

100

598.0866.0500.0

964.4500.0866.0

1

'

'

y

x


Alternative Method

xTRx

R

v

n

'

100

0866.0500.0

0500.0866.0

)866.0500.0(

)500.0866.0( y

x

x′y′

Θ=300

0

1

2

3

1

1

P

P01 2 3

nv


ProjectionsProjections

• Once WC description of the objects in a scene are converted to VC we can project the 3D objects onto 2D view-plane.

• Two types of projections:

-Parallel Projection

-Perspective Projection

Classical ViewingsClassical Viewings

• Hand drawings : Determined by a specific relationship between the object and the viewer.

Parallel ProjectionsParallel Projections

Coordinate Positions are transformed to the view plane along parallel lines.

P1

P2 P′2

P′1

View

Plane

Parallel ProjectionsParallel Projections

• Orthographic parallel projection The projection is perpendicular to the view plane.

• Oblique parallel projecion The parallel projection is not perpendicular to the view plane.

Orthographic Parallel ProjectionOrthographic Parallel Projection

Oblique Parallel ProjectionOblique Parallel Projection

– The projectors are still ortogonal to the projection plane– But the projection plane can have any orientation with

respect to the object.– It is used extensively in architectural and mechanical design.

Oblique Parallel ProjectionOblique Parallel Projection

• Preserve parallel lines but not angles– Isometric view : Projection plane is placed

symmetrically with respect to the three principal faces that meet at a corner of object.

– Dimetric view : Symmetric with two faces.– Trimetric view : General case.

Perspective ProjectionsPerspective Projections

• First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance

• Objects closer to viewer look larger• Parallel lines appear to converge to single

point


In perspective projection object positions are transformed to the view plane along lines that converge to a point called the projection reference point (or center of projection)


• In the real world, objects exhibit perspective foreshortening: distant objects appear smaller

• The basic situation:

When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world:

How tall shouldthis bunny be?


The geometry of the situation is that of similar triangles. View from above:

d

P (x, y, z) X

Z(0,0,0)x′ = ?


View plane

(xp, yp)

Desired result for a point [x, y, z, 1]T projected onto the view plane:

dzdz

y

z

ydy

dz

x

z

xdx

z

y

d

y

z

x

d

x

',','

',

'


• When a camera used to take a picture, the type of lens used determines how much of the scene is caught on the film.

• In 3D viewing, a rectangular view window in the view plane is used to the same effect. Edges of the view window are parallel to the xv-yv axes and window boundary positions are specified in viewing coordinates.

View VolumesView Volumes

View VolumesView Volumes

zv

window

Front Plane

Back Plane

View volume

Parallel ProjectionParallel Projection Perspective Projection

Front Plane

Back Plane

View volume (frustum)

window

Projection Reference Point

ClippingClipping

• An algorithm for 3D clipping identifies and saves all surface segments within the view volume for display.

• All parts of object that are outside the view volume are discarded.

Clipping LinesClipping Lines

Just like the case in two dimensions, clipping removes objects that will not be visible from the scene

The point of this is to remove computational effort

3-D clipping is achieved in two basic steps

– Discard objects that can’t be viewed

• i.e. objects that are behind the camera, outside the field of view, or too far away

– Clip objects that intersect with any clipping plane

Discard ObjectsDiscard Objects

Discarding objects that cannot possibly be seen involves comparing an objects bounding box/sphere against the dimensions of the view volume

– Can be done before or after projection

Clipping Polygon SurfaceClipping Polygon Surface

• To clip a polygon surface, we can clip the individual polygon edges.

• First we test the coordinate extends against each boundary of the view volume to determine whether the object is completely inside or completely outside of that boundary.

• If the object has intersection with the boundary then we apply intersection calculations.


• The projection operation can take place before the view- volume clipping or after clipping.

• All objects within the view volume map to the interior of the specified projection window.

• The last step is to transform the window contents to a 2D view port.

NormalisationNormalisation

The transformed volume is then normalised around position (0, 0, 0) and the z axis is reversed

Dividing Up The WorldDividing Up The World

Similar to the case in two dimensions, we divide the world into regions

This time we use a 6-bit region code to give us 27 different region codes

The bits in these regions codes are as follows:

bit 6

Far

bit 5

Near

bit 4

Top

bit 3

Bottom

bit 2

Right

bit 1

Left

Region CodesRegion Codes

3D Line Clipping Example (cont…)3D Line Clipping Example (cont…)

When then simply continue as per the two dimensional line clipping algorithm


Viev volume

3D Polygon Clipping (cont…)3D Polygon Clipping (cont…)

In this case we first try to eliminate the entire object using its bounding volume

Next we perform clipping on the individual polygons using the Sutherland-Hodgman algorithm we studied previously

OpenGL Projection CommandsOpenGL Projection Commands

OpenGL OpenGL Look-At FunctionLook-At Function

• OpenGL utility function

– VRP: eyePoint (eyex, eyey, eyez)

– VPN: – ( atPoint – eyePoint ) (atx, aty, atz) – (eyex, eyey, eyez)

– VUP: upPoint – eyePoint (upx, upy, upz)

gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz);

look-at positioning

Projections in OpenGLProjections in OpenGL

• Angle of view, field of view :– Only objects that fit within

the angle of view of the camera appear in the image

• View volume, view frustum :– Be clipped out of scene– Frustum – truncated

pyramid

advanced computer graphics three dimensional viewing

Documents

view reference point

view plane

viewing coordinate origin

view coordinatesto

viewing zaxis

world origin

view coordinatesfinally

view coordinatesfor