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A.Aruna/Assistant professor/IT/SNSCE 1

Transformation and object modeling is extended from 2D by including z coordinate

Object can translate by specifying 3D Vector

Can be moved in each of the three coordinates

Rotation – select any spatial orientation of the rotation axis

Express in 3*3 Matrix form04/21/23A.Aruna/Assistant professor/IT/SNSCE

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In 3D Homogeneous representationa point is translated from position P

to position P’

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11000

100

010

001

1

'

'

'

z

y

x

t

t

t

z

y

x

z

y

x

PTP '

3

An object is translated in 3D dimensional by transforming each of the defining points of the objects.


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zyx tttT ,, zyx ,,

',',' zyx


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Designate the axis of rotation and amount of angular rotation


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A.Aruna/Assistant professor/IT/SNSCE

Positive rotation angles produce counterclockwise rotations about a coordinate axis

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11000

0100

00cossin

00sincos

1

'

'

'

z

y

x

z

y

x

sincos' yxx

cossin' yxy zz '

PRP z )('

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xzyx 04/21/23

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11000

0cossin0

0sincos0

0001

1

'

'

'

z

y

x

z

y

x

sincos' zyz cossin' zyy

xx '

PRP x )('

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11000

0cos0sin

0010

0sin0cos

1

'

'

'

z

y

x

z

y

x

sincos' xzz

cossin' xzx yy '

PRP y )('

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An object is to be rotated about an axis that is parallel to one of the coordinate axes

1. Translate the object so that the rotation axis coincides with the parallel coordinate axis

2. Perform the specified rotation about that axis3. Translate the object so that the rotation axis

is moved back to its original position

PTRTP x )(' 1 TRTR x )()( 1

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An object is to be rotated about an axis that is not parallel to one of the coordinate axes

1. Translate the object so that the rotation axis passes through the coordinate origin.

2. Rotate the object so that the axis of rotation coincide with one of the coordinate axes.

3. Perform the specified rotation about that coordinate axis.

4. Apply inverse rotations to bring the rotation axis back to its original orientation.

5. Apply the inverse Translation to bring the rotation axis back to its original position.

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Draw any shape, then moving translation matrix.

Good Luck


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P is scaled to P' by S:

Called theScaling matrix

S =

1000

000

000

000

z

y

x

s

s

s


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Scaling with respect to the coordinate origin

11000

000

000

000

1

'

'

'

z

y

x

s

s

s

z

y

x

z

y

x

PSP '


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Scaling with respect to a selected fixed position (xf, yf, zf)

1. Translate the fixed point to origin

2. Scale the object relative to the coordinate origin

3. Translate the fixed point back to its original position

1000

)1(00

)1(00

)1(00

),,(),,(),,(fzz

fyy

fxx

fffzyxfff zss

yss

xss

zyxTsssSzyxT


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About an axis: equivalent to 180˚rotation about that axis


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3D Reflections

1000

0100

0010

0001

zRF


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3D Shearing• Modify object shapes• Useful for perspective projections:

– E.g. draw a cube (3D) on a screen (2D)– Alter the values for x and y by an

amount proportional to the distance from zref


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3D Shearing

1000

0100

10

01

refzyzy

refzxzx

zshear

zshsh

zshsh

M


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Shears

1000

0100

010

001

b

a

SH z


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PUZZLE


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4 = i


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1 = h


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3D Viewing

The steps for computer generation of a view of a three dimensional scene are somewhat analogous to the processes involved in taking a photograph.

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Camera Analogy

1. Viewing position2. Camera orientation3. Size of clipping window

Position

Orientation

Window (aperture)

of the camera

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Viewing Pipeline The general processing steps for modeling and

converting a world coordinate description of a scene to device coordinates

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Viewing Pipeline

1. Construct the shape of individual objects in a scene within modeling coordinate, and place the objects into appropriate positions within the scene (world coordinate).

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Viewing Pipeline

2. World coordinate positions are converted to viewing coordinates.

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Viewing Pipeline

3. Convert the viewing coordinate description of the scene to coordinate positions on the projection plane.

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Viewing Pipeline4. Positions on the projection plane, will then mapped to the Normalized coordinate and output device.

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Viewing Coordinates

Camera Analogy

Viewing coordinates system described 3D objects with respect to a viewer.

A Viewing (Projector) plane is set up perpendicular to zv and aligned with (xv,yv).

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Specifying the Viewing Coordinate System (View Reference Point)

We first pick a world coordinate position called view reference point (origin of our viewing coordinate system).

P0 is a point where a camera is located.

The view reference point is often chosen to be close to or on the surface of some object, or at the center of a group of objects.

wx

wy

wz 0

P

Position04/21/23

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Specifying the Viewing Coordinate System (Zv Axis)

Next, we select the positive direction for the viewing zv axis,

by specifying the view plane normal vector, N. The direction of N, is from the look at point (L) to the view

reference point.

Look

Vector

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Specifying the Viewing Coordinate System (yv Axis)

Finally, we choose the up directionup direction for the view by

specifying a vector VV, called the view up vectorview up vector. This vector is used to establish the positive direction for the

yv axis.

V V is projected into a plane that is perpendicular to the normal vector.

Up Vector

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Look and Up Vectors Look Vector the direction the camera is pointing three degrees of freedom; can be any vector in 3-

space Up Vector

determines how the camera is rotated around the Look vector

for example, whether you’re holding the camera horizontally or vertically (or in between)

projection of Up vector must be in the plane perpendicular to the look vector (this allows Up vector to be specified at an arbitrary angle to its Look vector)

Up vectorLook vector

Projection of up vector

Position04/21/23

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Specifying the Viewing Coordinate System (xv Axis)

Using vectors N and V, the graphics package computer

can compute a third vector U, perpendicular to both N and

V, to define the direction for the xv axis.

P0

P0

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The View Plane Graphics package allow users to choose the position of the view plane along the zv axis by specifying the view plane distance from the viewing origin.

The view plane is always parallel to the xvyv plane.

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Obtain a Series of View To obtain a series of view of a scene, we can keep the view reference point fixed and change the direction of N.

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Simulate Camera Motion To simulate camera motion through a scene, we can keep N fixed and move the view reference point around.

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Transformation from World to Viewing

Coordinates

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Viewing Pipeline Before object description can be projected to the view plane, they must be transferred to viewing coordinates.

World coordinate positions are converted to viewing coordinates.

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Transformation from World to Viewing Coordinates

Transformation sequence from world to viewing coordinates:

TRM zyzVCWC

RR,

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Transformation from World to Viewing Coordinates

Another Method for generating the rotation-

transformation matrix is to calculate unit uvn vectors and form the composite rotation matrix directly:

),,(

),,(

),,(

321

321

321

vvv

uuu

nnn

unvNV

NVu

N

Nn

1000

0

0

0

321

321

321

nnn

vvv

uuu

R

TRM VCWC , 04/21/23

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Projection

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Viewing Pipeline Convert the viewing coordinate description of the scene to coordinate positions on the projection plane.

Viewing 3D objects on a 2D display requires a mapping from 3D to 2D.

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Projection Projection can be defined as a mapping of point P(x,y,z) onto its image in the projection plane.

The mapping is determined by a projectorprojector that passes through P and intersects the view plane ( ).

),,( zyxP

P

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Projection Projectors are lines from center (reference) of projection through each point in the object.

The result of projecting an object is dependent on the spatial relationship among the projectors and the view plane.

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Projection

Parallel ProjectionParallel Projection : Coordinate position are transformed to the view plane along parallel lines.

Perspective Perspective ProjectionProjection: : Object positions are transformed to the view plane along lines that converge to the projection reference (center) point.

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Parallel Projection Coordinate position are transformed to the view plane along parallel lines.

Center of projection at infinity results with a parallel projection.

A parallel projection preserves relative proportion of objects, but dose not give us a realistic representation of the appearance of object.

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Perspective Projection Object positions are transformed to the view plane along lines that converge to the projection reference (center) point.

Produces realistic views but does not preserve relative proportion of objects.

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Perspective Projection Projections of distant objects are smaller than the projections of objects of the same size are closer to the projection plane.

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Parallel and Perspective Projection

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Parallel Projection

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Parallel Projection Projection vector: Defines the direction for the projection

lines (projectors). Orthographic ProjectionOrthographic Projection: Projectors (projection vectors)

are perpendicular to the projection plane. Oblique ProjectionOblique Projection: Projectors (projection vectors) are

not not perpendicular to the projection plane.

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Orthographic Parallel Projection

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Orthographic Parallel Projection Orthographic projection used to produce

the front, side, and top views of an object.

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Orthographic Parallel Projection FrontFront, sideside, and rearrear orthographic projections of an object

are called elevationselevations.

TopTop orthographic projection is called a planplan view.

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Multi View Orthographic04/21/23

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Orthographic Parallel Projection Axonometric orthographicAxonometric orthographic projections

display more than one face of an object.

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Orthographic Parallel Projection Isometric ProjectionIsometric Projection: Projection plane intersects each coordinate axis in which the object is defined (principal axes) at the same distant from the origin.

Projection vector makes equal angles with all of the three principal axes.

Isometric projection is obtained by aligning the projection vector with the cube diagonal.

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Orthographic Parallel Projection Dimetric ProjectionDimetric Projection:

Projection vector makes equal angles with exactly two of the principal axes.

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Orthographic Parallel Projection Trimetric ProjectionTrimetric Projection: Projection

vector makes unequal angles with the three principal axes.

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Transformation

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Convert the viewing coordinate description of the

scene to coordinate positions on the Orthographic parallel projection plane.

Orthographic Parallel Projection Transformation

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Since the view plane is placed at position zvp along the zv axis. Then any point (x,y,z) in viewing coordinates is transformed to projection coordinates as:

Orthographic Parallel Projection Transformation

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0000

0010

0001

ParallelicOrthographM

yyxx pp ,

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Oblique Parallel Projection

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Oblique Parallel Projection Projection are not perpendicular to the viewing plane.

Angles and lengths are preserved for faces parallel the plane of projection.

Preserves 3D nature of an object.

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Transformation

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scene to coordinate positions on the Oblique parallel projection plane.

Oblique Parallel Projection Transformation

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Oblique Parallel Projection Point (x,y,z) is projected to position (xp,yp) on the view plane.

Projector (oblique) from (x,y,z) to (xp,yp) makes an angle

with the line (LL) on the projection plane that joins (xp,yp) and (x,y).

Line LL is at an angle with the horizontal direction in the projection plane.

α

φ

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φsin

φcos

Lyy

Lxx

p

p

L

zαtan

1 αtan

zL

zL

)φsin(

)φcos(

1

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p

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L

L

ParallelM

L

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L

01 L

Orthographic Projection:Orthographic Projection:

90

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0001

ParallelicOrthographM

yyxx pp ,

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Oblique Parallel Projection Angles, distances, and parallel lines in the plane are projected accurately.

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Cavalier ProjectionCavalier Projection:

Preserves lengths of lines perpendicular to the viewing plane. 3D nature can be captured but shape seems distorted. Can display a combination of front, and side, and top views.

4530 and45

1tan

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Cabinet ProjectionCabinet Projection:

Lines perpendicular to the viewing plane project at ½ ½ of their length.

A more realistic view than the cavalier projection. Can display a combination of front, and side, and top views.

4.63

2tan

4530 and

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Cavalier & Cabinet Projection

Cavalier Cabinet

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Perspective Projection

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Perspective Projection In a perspective projection, the center of projection is

at a finite distance from the viewing plane. Produces realistic views but does not preserve

relative proportion of objects The size of a projection object is inversely

proportional to its distance from the viewing plane.

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Perspective Projection Parallel lines that are not parallel to the viewing

plane, converge to a vanishing pointvanishing point. A vanishing point is the projection of a point at

infinity.

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Vanishing Points Each set of projected parallel lines will have a separate vanishing points.

There are infinity many general vanishing points.

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Perspective Projection The vanishing point for any set of lines that are parallel to one of the principal axes of an object is referred to as a principal vanishing point.

We control the number of principal vanishing points (one, two, or three) with the orientation of the projection plane.

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Perspective Projection The number of principal vanishing points in a projection is determined by the number of principal axes intersecting the view plane.

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One Point PerspectiveOne Point Perspective (z-axis vanishing point)04/21/23

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Two Point Perspective Two Point Perspective (z, and x-axis vanishing points)04/21/23

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Two Point Perspective Two Point Perspective 04/21/23

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ThreeThree Point PerspectivePoint Perspective(z, x, and y-axis vanishing points)04/21/23

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Perspective Projection Transformation

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scene to coordinate positions on the perspective projection plane.


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Suppose the projection reference point at position zprp along the zv axis, and the view plane at zvp.


uzzzz

yuyy

xuxx

prp)(

10 u

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uzzzz

yuyy

xuxx

prp)(

vpzz

zz

zzu

prp

vp

On the view plane:

prp

p

prp

vpprp

p

prp

p

prp

vpprp

p

zz

dy

zz

zzyy

zz

dx

zz

zzxx

vpprpp zzd

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Perspective Projection Transformationvpzz On the view plane:

prp

p

prp

vpprp

p

prp

p

prp

vpprp

p

zz

dy

zz

zzyy

zz

dx

zz

zzxx

1100

00

0010

0001

z

y

x

dzd

dzzdz

h

z

y

x

pprpp

pprpvppvph

h

h

p

prp

d

zzh

hyyhxx hphp ,

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Perspective Projection TransformationSpecial Cases:Special Cases: 0vpz

1

1

1

1

prpprp

prpp

prpprp

prpp

zzy

zz

zyy

zzx

zz

zxx

prp

p

prp

vpprp

p

prp

p

prp

vpprp

p

zz

dy

zz

zzyy

zz

dx

zz

zzxx

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prp

p

prp

vpprp

p

prp

p

prp

vpprp

p

zz

dy

zz

zzyy

zz

dx

zz

zzxx

Perspective Projection TransformationSpecial Cases:Special Cases: The projection reference point is at the viewing coordinate origin:0prpz

Zprp=0

vp

vpp

vp

vpp

zzy

z

zyy

zzx

z

zxx

1

1

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Summery

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Summary

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12/24/2015 a.aruna/assistant professor/it/snsce 1

Documents

axis of rotation

parallel coordinate

specified rotation

rotation axisexpress

angular rotation

coordinate axestranslate

object relative

position p