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Chapter 6

Computer Graphics and Visualization

SSE, Mukka

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Classical Viewing

Introduce the classical views

Compare and contrast image formation bycomputer with how images have been formed byarchitects, artists, and engineers

Learn the benefits and drawbacks of each type ofview


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Classical Viewing

Viewing requires three basic elements One or more objects

A viewer with a projection surface

Projectors that go from the object(s) to the projectionsurface

Classical views are based on the relationship amongthese elements The viewer picks up the object and orients it how she

would like to see it

Each object is assumed to constructed from flatprincipal faces Buildings, polyhedra, manufactured objects


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Planar GeometricProjections

Standard projections project onto a plane

Projectors are lines that either

converge at a center of projection

are parallel

Such projections preserve lines

but not necessarily angles

Nonplanar projections are needed for applicationssuch as map construction


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Classical Projections


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Perspective vs Parallel

Computer graphics treats all projections the sameand implements them with a single pipeline

Classical viewing developed different techniques fordrawing each type of projection

Fundamental distinction is between parallel andperspective viewing even though mathematicallyparallel viewing is the limit of perspective viewing


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Taxonomy of PlanarGeometric Projections

parallel perspective

axonometricmultiview

orthographic oblique

isometric dimetric trimetric

2 point1 point 3 point

planar geometric projections


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PerspectiveProjection


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Parallel Projection


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Orthographic ProjectionProjectors are orthogonal to projection surface


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Multiview Orthographic

ProjectionProjection plane parallel to principal face

Usually form front, top, side views

isometric (not multiview

orthographic view)front

sidetop

in CAD and architecture,

we often display threemultiviews plus isometric


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Advantages and

Disadvantages Preserves both distances and angles

Shapes preserved

Can be used for measurements

Building plans

Manuals

Cannot see what object really looks like becausemany surfaces hidden from view

Often we add the isometric


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Axonometric ProjectionsAllow projection plane to move relative to object

classify by how many angles of

a corner of a projected cube are

the same

none: trimetric

two: dimetric

three: isometric

q 1

q 3q 2


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Axonometric Projections

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The distinguishing feature of axonometric projection is the inclined

position of the object with respect to the plane of projection.

The axonometric axis is the intersection of the three edges of the cube

at the corner (O).

Axonometric projections are classified as isometric projection (a),

dimetric projection (b), and trimetric projection (c).


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Types of AxonometricProjections


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Advantages and

DisadvantagesLines are scaled (foreshortened) but can find scaling

factors

Lines preserved but angles are not Projection of a circle in a plane not parallel to the projectionplane is an ellipse

Can see three principal faces of a box-like objectSome optical illusions possible

Parallel lines appear to divergeDoes not look real because far objects are scaled

the same as near objectsUsed in CAD applications


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Oblique Projection

Arbitrary relationship between projectors andprojection plane


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Advantages and

DisadvantagesCan pick the angles to emphasize a particular

face

Architecture: plan oblique, elevation obliqueAngles in faces parallel to projection plane are

preserved while we can still see around side

In physical world, cannot create with simplecamera; possible with bellows camera orspecial lens (architectural)


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Perspective Projection

Projectors coverge at center of projection


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Vanishing Points

Parallel lines (not parallel to the projectionplan) on the object converge at a single point in

the projection (the vanishing point)Drawing simple perspectives by hand uses

these vanishing point(s)

vanishing point


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Three-Point Perspective

No principal face parallel to projection plane

Three vanishing points for cube


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Two-Point Perspective

On principal direction parallel to projectionplane

Two vanishing points for cube


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One-Point Perspective

One principal face parallel to projection plane

One vanishing point for cube


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Advantages and

DisadvantagesObjects further from viewer are projected smaller

than the same sized objects closer to the viewer

(diminution) Looks realistic

Equal distances along a line are not projected intoequal distances (nonuniform foreshortening)

Angles preserved only in planes parallel to theprojection plane

More difficult to construct by hand than parallelprojections (but not more difficult by computer)


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Computer Viewing

There are three aspects of the viewing process, all ofwhich are implemented in the pipeline,

Positioning the camera Setting the model-view matrix

Selecting a lens

Setting the projection matrix

Clipping

Setting the view volume


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The OpenGL Camera In OpenGL, initially the object and camera frames are

the same

Default model-view matrix is an identity

The camera is located at origin and points in the

negative z directionOpenGL also specifies a default view volume that is a

cube with sides of length 2 centered at the origin

Default projection matrix is an identity


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DefaultProjection

Default projection is orthogonalclipped out

z=0

2


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Moving the Camera Frame

If we want to visualize object with both positiveand negative z values we can either Move the camera in the positive z direction

Translate the camera frame

Move the objects in the negative z direction

Translate the world frame

Both of these views are equivalent and are determined

by the model-view matrixWant a translation (glTranslatef(0.0,0.0,-d);)

d > 0


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Moving Camera back from

Origin

default frames

frames after translation by d

d > 0


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Moving the Camera

We can move the camera to any desired position by asequence of rotations and translations

Example: side view Rotate the camera

Move it away from origin

Model-view matrix C = TR


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OpenGL code

Remember that last transformation specified is firstto be applied

glMatrixMode(GL_MODELVIEW)glLoadIdentity();glTranslatef(0.0, 0.0, -d);glRotatef(90.0, 0.0, 1.0, 0.0);


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The LookAt FunctionThe GLU library contains the function

gluLookAt to form the required modelviewmatrix through a simple interface

Note the need for setting an up directionStill need to initialize Can concatenate with modeling transformations

Example: isometric view of cube aligned with

axesglMatrixMode(GL_MODELVIEW):glLoadIdentity();gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0., 1.0. 0.0);


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


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Other Viewing APIs

The LookAt function is only one possible API forpositioning the camera

Others include View reference point, view plane normal, view up

(PHIGS, GKS-3D)

Yaw, pitch, roll

Elevation, azimuth, twist Direction angles

P j i d


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Projections andNormalization

The default projection in the eye (camera) frame isorthogonal

For points within the default view volume

Most graphics systems use view normalization

All other views are converted to the default view bytransformations that determine the projection matrix

Allows use of the same pipeline for all views

xp = xyp = y

zp = 0

H C di


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Homogeneous CoordinateRepresentation

xp = x

yp = y

zp

= 0

wp = 1

pp = Mp

M =

1000

0000

0010

0001

In practice, we can let M = I and set

the zterm to zero later

default orthographic projection


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Simple Perspective

Center of projection at the origin

Projection plane z= d, d< 0


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Perspective Equations

Consider top and side views

xp =

dz

x

/

dz

x

/

yp =

dz

y

/

zp = d


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Homogeneous Coordinate Form

M =

0/100

0100

0010

0001

d

consider q = Mp where

1

zy

x

dz

z

y

x

/

q = p =


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Perspective Division

However w 1, so we must divide by w to return fromhomogeneous coordinates

Thisperspective division yields

the desired perspective equations

We will consider the corresponding clipping volumewith the OpenGL functions

xp =

dz

x

/

yp =

dz

y

/

zp = d


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OpenGL Orthogonal Viewing

glOrtho(left,right,bottom,top,near,far)

near and far measured from camera


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OpenGL Perspective

glFrustum(left,right,bottom,top,near,far)


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Using Field of ViewWith glFrustumit is often difficult to get the desired

view

gluPerpective(fovy, aspect, near,far) often provides a better interface

aspect = w/h

front plane

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