UBI 516 Advanced Computer Graphics Three Dimensional Viewing Aydın Öztürk [email protected]...

UBI 516UBI 516 Advanced Computer GraphicsAdvanced Computer Graphics

Three Dimensional ViewingThree Dimensional Viewing

Aydın Öztü[email protected]

http://www.ube.ege.edu.tr/~ozturk

OverviewOverview

• Viewing a 3D scene

• Projections– Parallel and perspective

OverviewOverview

• Depth cueing and hidden surfaces

• Identifying visible lines and surfaces

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 ,

Another method for generating the rotation-transformation matrix is to calculate uvn vectors and obtain the composite rotation matrix directly. Given the vectors and , these unit vectors are calculated as

N V

),,(

),,(

),,(

321

321

321

vvv

uuu

nnn

unv

NVNV

u

NN

n



This method also automatically adjusts the direction for so that is perpendicular to . The rotation matrix for the viewing transformation is then

Vnv

10 0 0

0

0

0

321

321

321

nnn

vvv

uuu

R


The matrix for translating the viewing origin to the world origin is

1 0 0

100

0 10

0 01

0

0

0

z

y

x

R


The composite matrix for the viewing transformation is then

zyx

zyx

zyx

VCWC

nznynxt

vzvyvxt

uzuyuxt

tnnn

tvvv

tuuu

00003

00002

00001

3321

2321

1321

,

where

10 0 0

Pn

Pv

Pu

M

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

The orthographic transformation

11000

0000

0010

0001

1

'

'

'

z

y

x

z

y

x

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.


• 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.


• 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.


φ

α

(xp, yp)

(x, y, z)

(x, y)

yv

xv

zv

L

)sin(

)cos(

tan

sin

cos

1

1

1

Lzyy

Lzxx

zLz

L

Lyy

Lxx

p

p

p

p

)sin(

)cos(

tan

sin

cos

1

1

1

Lzyy

Lzxx

zLz

L

Lyy

Lxx

p

p

p

p

The oblique transformation

11000

0000

0sin10

0cos01

1

1

1

z

y

x

L

L

z

y

x

p

p

p



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

',','

',

'


1100

0100

0010

0001

d

eperspectivM


hyyhxx

dzh

z

y

x

dh

z

y

x

hphp

h

h

h

/,/

/

10100

0100

0010

0001


Projection MatrixProjection Matrix

• We talked about geometric transforms, focusing on modeling transforms– Ex: translation, rotation, scale, gluLookAt()

– These are encapsulated in the OpenGL modelview matrix

• Can also express projection as a matrix– These are encapsulated in the OpenGL

projection matrix

• 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

• To clip a line against the view volume, we need to test the relative position of the line using the view volume’s boundary plane equation.

• An end point (x,y,z) of a line segment is outside a boundary plane if

where A, B, C and D are the plane parameters for that boundary.

0 DCzByAx

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.


Viev volume

Steps For Normalized View Volumes Steps For Normalized View Volumes

• A scene is constructed by transforming object descriptions from modeling coordinates to wc.

• The world descriptions are converted to viewing coordinates.

• The viewing coordinates are transformed to projection coordinates which effectively converts the view volume into a rectangular parallelepiped.

• The parallelepiped is mapped into the unit cube called normalized projection coordinate system.

• A 3D viewport within the unit cube is constructed.• Normalized projection coordinates are converted to

device coordinates for display.

Normalized View Volumes Normalized View Volumes

x

y

z

(Xwmin, ywmin, zfront)

(Xwmax, ywmax, zback)

(Xvmin, yvmin, zwmin)

(Xvmax, yvmax, zvmax)

Unit Cube

Parallelepiped

View Volume

10002

200

20

20

200

2

)2

,2

,2

(

)2

,2

,2

(

minmax

minmax

minmax

minmax

minmax

minmax

minmaxminmaxminmax

minmaxminmaxminmax

zz

zz

yy

yy

xx

xx

zzyyxx

zzyyxx

TSP

S

T

OrthoOrthogonalgonal Projection Normalization Projection Normalization

Oblique ProjectionOblique Projection Normalization Normalization

• Angles of projection for x axis for y axis

• Shearing matrix H(, )

0

cot

cot

tan

p

p

p

p

z

zyy

zxx

xx

z

1000

0100

0cot10

0cot01

),(

H

Oblique ProjectionOblique Projection Normalization Normalization

1000

0100

0cot10

0cot01

1000

0000

0010

0001

),(orth

HPP

Finished ? No, this is a sheared view volume, so we have to apply orthogonal transformation :

P=Porth STH

PerspPerspectiveective Projection Normalization Projection Normalization

Perspective Normalization is Trickier

Perspective Projection NormalizationPerspective Projection Normalization

Consider N =

After multiplying:

p’ = Np

0100

00

0010

0001


After dividing by w’, p’ -> p’’


Quick Check• If x = z

– x’’ = -1• If x = -z

– x’’ = 1

• If x = z– x’’ = -1

• If x = -z– x’’ = 1


What about z?

• if z = zmax

• if z = zmin

• Solve for and such that zmin -> -1 and zmax ->1• Resulting z’’ is nonlinear, but preserves ordering of

points– If z1 < z2 … z’’1 < z’’2


We did it. Using matrix, N• Perspective viewing frustum transformed to cube• Orthographic rendering of cube produces same

image as perspective rendering of original frustum

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

Projections in OpenGLProjections in OpenGL

Perspective in OpenGLPerspective in OpenGL

Specification of a frustum

– near, far: positive number !!

zmax = – far

zmin = – near

glMatrixMode(GL_PROJECTION);glLoadIdentity( );glFrustum(xmin, xmax, ymin, ymax, near, far);

Perspective in OpenGLPerspective in OpenGL

Specification using the field of view

– fov: angle between top and

bottom planes– fovy: the angle of view in the

up (y) direction– aspect ratio: width / height

glMatrixMode(GL_PROJECTION);glLoadIdentity( );gluPerspective(fovy, aspect, near, far);

Parallel Parallel Viewing Viewing in OpenGLin OpenGL

Orthographic viewing function

– OpenGL provides only this parallel-viewing function

– near < far !!

no restriction on the sign

zmax = – far

zmin = – near

glMatrixMode(GL_PROJECTION);glLoadIdentity( );glOrtho(xmin, xmax, ymin, ymax, near, far);

Optional Clipping PlanesOptional Clipping Planes

– glClipPlane(id, PlaneParameters);glEnable(id);

// id = GL_CLIP_PLANE0,

GL_CLIP_PLANE1, ...// PlaneParameters = A,B,C and D of the

plane

. . .

glDisable(id);

Date post:	29-Dec-2015
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UBI 516 Advanced Computer Graphics Three Dimensional Viewing Aydın Öztürk [email protected]...

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