3.1 Projection.pdf

8/10/2019 3.1 Projection.pdf

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Pr i n Vi w


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Content

Coordinate systems

Orthographic projection


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Graphical Coordinator Systems

A coordinate system is needed to input, store anddisplay model geometry and graphics.

Fourdifferent types of coordinate systems are used

modeling and for different tasks.


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Model (or World, Database) Coordinator System

e re erence space o e mo e w respec o w c a

of the geometrical data is stored.

It is a Cartesian system which forms the default coordinate

system used by a software system.

z

y

Y

x

X

Z


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Viewing Coordinate System

View Plane

Textbook setup - forYv

ara e ro ec on,

the viewer is atinfinity.

Zv Xv

Viewing Direction

X

YZu o e auSetup

Viewer


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A 3-D Cartesian coordinate system (right hand of left hand) in

which a projection of the modeled object is formed. VSC wil l be

discussed in detail under Perspective or Parallel Projections.

Viewin Coor. S stem

Model Coor Sys


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P


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Viewer

Viewer


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Parallel Projection

Preserve actual dimensions and shapes of objects

Angles preserved only on faces parallel to the

projection plane

Orthographic projection is one type of parallel

projection


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Perspective Projection

Doesnt preserve parallelism

of objects, therefore shapes deformed

Po ular in art classic aintin architecturaldesign and civil engineering.

Not commonly used in mechanical engineering


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Projection View

Set Up the Viewing Coordinate System (VCS)

i) Define the view reference point P = Px, Py , Pz( )T

v

n = Nx, Ny , Nz( )T

N x2 + N y

2 + Nz2 =1and

iii) Define the "up" direction

v

V = Vx , Vy , Vz v

n,v

V v

n = 0

v

u =v

V v

nr

vv

This also defines an orthogonal vector,

, ,

Define the View Window in


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Generating Parallel ProjectionProblem: for a given computer model, we know its x-y-zcoordinates in MCS; and we need to find its u-v-n

s- s .

Getting the u-v-n coordinates of the objects by transforming the objects

and u-v-n coordinate system together to fully align u-v-n with x-y-z axes,

s s

Viewin Coor. S stem

Model Coor Sys- Translate Ov to O.

- Align the n axis with the Z axis.- Fully aligning u-v-n with x-y-z


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Generatin Parallel Pro ection 1First transform coordinates of objects into the u-v-n coordinates (VCS),

1 0 0 0vx

0 1 0 0v

.

i.e. Overlapping u - v n withx - y - z

i) Translate O to O.

ii) Align the axis with the Zaxis.

D[ ] =0 0 1 0 vz

0 0 0 1

v

n

The procedure is identical to the

transformations used to prepare for

the rotation about an axis.

A = Nx , B = N y, C = Nz

L = Nx2 + N2 + Nz

2

V = Ny2 + Nz

2


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Rotating about X: [R]x; and Rotating about Y: [R]y1 2

Fully aligning u-v-n with x-y-z

Then, rotate

3 about theZaxis to align

withX

andv

withY


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Generating Parallel Projection (3)

At this point, is given by whereV Vx, V y, 0( )T

Rotate 3 about theZaxis to align with Xand vwith Y

Vx

Vy

= Ry[ ] Rx[ ] Dov ,o[ ]

V x

V y

1

z

1

We need to rotate by an angle 3 about the Zaxis

L = Vx2 + Vy

2, sin3 =

VxL

, cos3 =Vy

L

Rz[ ]=

x

Vx L Vy L 0 0

0 0 1 0

Result:

0 0 0 1


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Generating Parallel Projection (4)

Dn =

1 0 0 0

0 1 0 0

,

u

V

= Dn

u

V

Drop the n coordinate

0 0 0 1

1

1

, ,

transform each point on the object by:

,[ ][[ ] [] ] ]][ [ vz y o oxn R R DRDT =

u

V

x

y

=0

1

= = z

1

Note: The inverse transforms are not needed! We don't want to

go back to x - y - z coordinates.


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Y

r ograp c ro ec on

Front

Top

Right

X

Z

Projection planes (Viewing planes) are

perpendicular to the principal axes of the

MCS of the model

The projection direction (viewing direction)

coincides with one of the MCS axes


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Geometric

Front

TopRight

Y Transformations for

Generating

X

(Front View)

Z

0001

Yv,YPPv

=

0000

ront

Dro Z,


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Geometric Transformations

for Generating

Front

TopRight

Y

Yv

(Top View)

X TopXv, X

Z Z

1 0 0 01 0 0 0 1 0 0 0

0 cos 90 sin 90 0

( )0 sin 90 cos(90 ) 00 0 0 0 0 0 0

0 0 0 1 0 0 0 10 0 0 1

0vP P P= =

90[ ]

x

RDrop Z


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Geometric Transformations

for Generating

Front

TopRight

Y

Yv,Y

r ograp c ro ec on(Right View)

X

Right

Z

cos( 90 ) 0 sin( 90 ) 01 0 0 0 0 0 1 0

0 1 0 00 1 0 0 0 1 0 0

= =

( )sin 90 0 cos( 90 ) 00 0 0 0 0 0 00 0 0 1 0 0 0 10 0 0 1

0v

90[ ]y

R

Drop Z


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Rotations Needed for Generating Isometric Projection

YvYTop

Y

Top

RightFront Xv

Front

X

Zv XZ

1 0 0 0 cos 0 sin 0

[ ] [ ] 0 sin cos 0 sin 0 cos 0v x yP R R P P

= =


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three main axes

== 26.35,45


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Rx --> R

== 26.35,45 yx rr

Rz --> Ry(Rx)

== ., xyzRx(Ry) --> Rz

ANGLEANYrr zxy == ,45)(

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3.1 Projection.pdf