projection viewsprojection views · • projection planes (viewing planes) are perpendicular to the...
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Projection ViewsProjection Views
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Content
• Coordinate systems
• Orthographic projection
• (Engineering Drawings)• (Engineering Drawings)
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Graphical Coordinator Systems
A coordinate system is needed to input, store and display model geometry and graphics.
Four different types of coordinate systems are used in a CAD system at different stages of geometricin a CAD system at different stages of geometric modeling and for different tasks.
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Model (or World, Database) Coordinator System
Th f f th d l ith t t hi h ll• The reference space of the model with respect to which allof 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 SystemView Plane
View WindowTextbook setup - for P ll l P j ti
YvView WindowParallel Projection,
the viewer is at infinity. Zv
Xv
Viewing Direction
ViewerViewer
ZA t CAD D f lt
X
YZAutoCAD Default
Setup
Viewer
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Viewing Coordinate System (VCS)Viewing Coordinate System (VCS)
A 3-D Cartesian coordinate system (right hand of left hand) in which a projection of the modeled object is formed. VSC will be discussed in detail under Perspective or Parallel Projections.
Viewing Coor. SystemViewing Coor. System
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
• Preserve parallelismPreserve parallelism
• Angles preserved only on faces parallel to the projection planep j p
• Orthographic projection is one type of parallel projection
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Perspective Projection
• Doesn’t preserve parallelism
• Doesn’t preserve actual dimensions and anglesDoesn t preserve actual dimensions and angles of objects, therefore shapes deformed
• Popular in art (classic painting); architectural p ( p g);design and civil engineering.
• Not commonly used in mechanical engineering
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Geometric Transformations for GeneratingGeometric Transformations for Generating Projection View
ii) Define the line of the sight vector
Set Up the Viewing Coordinate System (VCS)i) Define the view reference point P = Px, Py , Pz( )T
v n (normalized)ii) Define the line of the sight vector n
v n = Nx, Ny , Nz( )T Nx
2 + Ny2 + Nz
2 =1
( )T
(normalized)
and iii) Define the "up" direction
v V = Vx , Vy , Vz( )T
⊥v n ,
v V ⋅
v n = 0
v u =
v V ×
v n r
u v V
v n ( )
This also defines an orthogonal vector, forms the viewing coordinates
u , V , n ( ) forms the viewing coordinates
Define the View Window incoordinatesu v n− − coordinatesu v n− −
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Generating Parallel Projectiong jProblem: for a given computer model, we know its x-y-z
coordinates in MCS; and we need to find its u-v-ncoordinates in VCS and X Y in WCScoordinates in VCS and Xs-Ys in WCS.
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, then drop the n (the depth) component to get X and Ythen drop the n (the depth) component to get Xs and Ys
Viewing Coor. SystemViewing Coor. System
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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Generating Parallel Projection (1)Generating Parallel Projection (1)
First transform coordinates of objects into the u-v-n coordinates (VCS), then drop the n component. (n is the depth)
[ ]
1 0 0 −0vx
0 1 0 −0vy
⎡ ⎢ ⎢ ⎢
⎤ ⎥ ⎥ ⎥
then drop the n component. (n is the depth)i.e. Overlapping u - v – n with x - y - z
i) Translate Ov to O.
ii) Align the axis with the Z axis.
D[ ] =y
0 0 1 −0 vz
0 0 0 1⎣
⎢ ⎢ ⎢
⎦
⎥ ⎥ ⎥
v n
) v
The procedure is identical to the transformations used to prepare for the rotation about an axis.
A = Nx , B = N y, C = Nz
L = N x2 + Ny
2 + Nz2
y
V = N y2 + Nz
2
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Generating Parallel Projection (2)Generating Parallel Projection (2)
Rotating about X: [R]x; and Rotating about Y: [R]y1θ 2θ1 2
Fully aligning u-v-n with x-y-z
Then, rotate Ө3 about the Z axis to align ū with X and v with Y
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Generating Parallel Projection (3)
At this point, is given by where V ′ V x, ′ V y, 0( )TRotate Ө3 about the Z axis to align ū with X and v with Y
′ V x
′ V y
0
⎛ ⎜ ⎜ ⎜ ⎜
⎞ ⎟ ⎟ ⎟ ⎟
= Ry[ ] Rx[ ] Dov ,o[ ]
V x
V y
V
⎛ ⎜ ⎜ ⎜ ⎜
⎞ ⎟ ⎟ ⎟ ⎟ 0
1⎝
⎜ ⎜ ⎜
⎠
⎟ ⎟ ⎟
Vz
1⎝
⎜ ⎜
⎠
⎟ ⎟
We need to rotate by an angle Ө3 about the Z axis
L = ′ V x2 + ′ V y
2 , sinθ3 =′ V x
L, cosθ3 =
′ V yL
′ V L − ′ V L 0 0⎡ ⎤
Rz[ ]=
V r L V x L 0 0
′ V x L ′ V y 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
0 0 0 0
⎡ ⎢ ⎢ ⎢
⎤ ⎥ ⎥ ⎥ ,
u
V
0
⎛ ⎜ ⎜ ⎜
⎞ ⎟ ⎟ ⎟
= Dn[ ]
u
V
n
⎛ ⎜ ⎜ ⎜
⎞ ⎟ ⎟ ⎟
Drop the n coordinate
0 0n[ ]0 0 0 0
0 0 0 1⎣ ⎢ ⎢ ⎦
⎥ ⎥
0
1⎝ ⎜ ⎜
⎠ ⎟ ⎟
n[ ]n
1⎝ ⎜ ⎜
⎠ ⎟ ⎟
In summary to project a view of an object on the UV plane one needs to
0 0
In summary, to project a view of an object on the UV plane, one needs to transform each point on the object by:
,[ ][[ ] [] ] ]][ [vz y o oxn R R DRDT =
′ P
u
V
⎛ ⎜ ⎜
⎞ ⎟ ⎟ T[ ]P T[ ]
x
y
⎛ ⎜ ⎜
⎞ ⎟ ⎟ ′ P =
0
1⎝
⎜ ⎜ ⎜
⎠
⎟ ⎟ ⎟
= T[ ]P = T[ ] 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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O th hi P j tiY
Orthographic Projection
FrontTop
Right
X
ZZ
• 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
FrontTop
Right
Y Transformations for Generating
Orthographic ProjectionX
Orthographic Projection (Front View)
Z
⎥⎥⎤
⎢⎢⎡
00100001
F
Yv,Y PPv
⎥⎥⎥
⎦⎢⎢⎢
⎣
=
100000000010
Front
Xv X
⎥⎦
⎢⎣ 1000
Drop ZXv, X Drop Z
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Geometric Transformations for Generating
Orthographic Projection
FrontTop
Right
YYv
Orthographic Projection (Top View)
X Top
Xv, X
Z Z
1 0 0 01 0 0 0 1 0 0 00 cos(90 ) sin(90 ) 00 1 0 0 0 0 1 0
⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥° − ° −⎢ ⎥⎢ ⎥ ⎢ ⎥
( )0 cos(90 ) s (90 ) 00 1 0 0 0 0 1 00 sin 90 cos(90 ) 00 0 0 0 0 0 0
0 0 0 1 0 0 0 10 0 0 10vP P P⎢ ⎥⎢ ⎥ ⎢ ⎥= =
⎢ ⎥⎢ ⎥ ⎢ ⎥° °⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
90[ ]xRDrop Z
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Geometric Transformations for Generating
O th hi P j ti
FrontTop
Right
YYv, Y
Orthographic Projection (Right View)
XRight
XvZZ
XvZ
cos( 90 ) 0 sin( 90 ) 01 0 0 0 0 0 1 00 1 0 00 1 0 0 0 1 0 0
P P P
− ° − °⎡ ⎤ −⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥⎢ ⎥ ⎢ ⎥( )sin 90 0 cos( 90 ) 00 0 0 0 0 0 0
0 0 0 1 0 0 0 10 0 0 10vP P P
⎢ ⎥⎢ ⎥ ⎢ ⎥− − ° − °⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦
90[ ]yR −Drop Z
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Rotations Needed for Generating Isometric Projection
YvYTop
Right
Y
Top
RightFront
Z
Xv
Front Right
X
Zv X
Z
Z
1 0 0 0 cos 0 sin 00 cos sin 0 0 1 0 0φ θ
θ θφ φ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥0 cos sin 0 0 1 0 0
[ ] [ ]0 sin cos 0 sin 0 cos 00 0 0 1 0 0 0 1
v x yP R R P Pφ θ φ φφ φ θ θ
⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦0 0 0 1 0 0 0 1⎣ ⎦ ⎣ ⎦
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Isometric Projection: Equally foreshorten theIsometric Projection: Equally foreshorten the three main axes
°±=°±= 26.35,45 φθ
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Other Possible Rotation Paths
• Rx --> Ry
Other Possible Rotation Paths
°±=°±= 26.35,45 yx rry
• Rz --> Ry(Rx)
°±=°±= 745445 )(rr ±± 74.54,45 )( xyz rr• Rx(Ry) --> Rz
ANGLEANYrr zxy =°±= ,45)(