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```AML710 CAD
LECTURE 3
Background Material
Revision of Vector Algebra
Revision of Matrix Algebra
Graphics Standards
Coordinate Systems
Points and Vectors
v = a − b;
a, b ∈ E , v ∈ R
3
3
v = (a + w) − (b + w); where w is an arbitrary vector
b’
a’
v
w
w
b
a
v
A vertex or point denotes location
Whereas a vector has length (magnitude) and
direction
Barycentric Conditions
a+b+c 1
g=
= 3 a + 13 b + 13 c
3
v = (a + w) − (b + w); where w is an arbitrary vector
c
g
a
b
• For vectors both addition and subtraction are defined
• Whereas for vertices (points) it results in convex
combination
• This leads to the definition of convex hull of point sets
Revision of Matrix Algebra
• Consider the following 2 x 2 matrices
A = arc = r1
r2
c1
c2
a11
a21
a12 =
a21
a22
a11
a11 + b11
A+ B =
a21 + b21
a12
a22
; B = brc =
a12 + b12
a22 + b22
b11
b12
b21 b22
Matrix Operations
• Multiplication by a scalar quantity
kA =
ka11
ka12
ka21 ka22
• Matrix multiplication
a11
C = AB =
a21
a12 b11 b12
a11b11 + a12b21
=
a22 b21 b22
a21b11 + a22b21
a11b12 + a12b22
a21b12 + a22b22
• Defined for matrices of r x c, c x r dimensions
• This operation is comparable to vector dot product
Matrix Operations
• Transpose of a matrix – Reflection along leading diagonal
A=
• Identity matrix
a11
a12
a21
a22
A′ =
a11
a21
a12
a22
1 0
I=
0 1
AA−1 = A−1 A = I
• Every matrix has a transpose. If At=A-1, such a matrix is
called orthonormal
Graphics Standard
•Common Graphics System
Application Data
Structure model
Application
Program
Graphics
System
Graphics Kernel
System
•Graphics System with
standard
Device
Driver
I/p o/p
device
Graphics Standards
• GKS an ISO and ANSI standard. Device independent host
system independent and application independent
• PHIGS – Programmers Hierarchical interactive
Graphics System
• VDM – Virtual Device Metafile. Defines the function to
represent a picture.
• VDI – Virtual Device Interface. Lies between GKS and
PHIGS
• IGES – initial graphics exchange specification. It is an
ANSI standard
• NAPLPS – North American Presentation Level Protocol
Syntax
Coordinate Systems
• Three types of coordinate systems are generally used in
• Model Coordinate System (MCS)
or Database CS/ World CS
• Working Coordinate System (WCS)
• Screen Coordinate System (SCS)
or Device CS
Model Coordinate System
• It is the reference space of the model with respect to which
all the model geometrical data is stored. It is a Cartesian
system with its X, Y, Z aligned with the characteristics
dimension of the model under consideration. The choice of
origin is arbitrary.
Y
H
Z
P
B
X
D
Working Coordinate System
• This is basically an auxiliary coordinate system used in
place of MCS. For convenience while we develop the
geometry by data input this kind of coordinate system is
useful. It is very useful when a plane (face) in MCS is not
aligned along any orthogonal planes. It is a user defined
system that facilitates the geometrical construction.
Y
X
Z
H
P
B
D
• While user inputs data in WCS the software transforms it to
MCS
Screen Coordinate System
• In contrast to MCS and WCS, Screen Coordinate System
is a two-dimensional device-independent system whose
origin is usually located at the lower left corner of the
display screen.
• The SCS is important for display, screen input and
Y
P
Z
X
• For Raster Graphics, the pixel grid serves as the range
of SCS. For a 1024x1280, the range is (0,0) to
(1024,1280)
Digital Image
• We know that the SCS is important for display, screen
• A digital image or image on the screen is represented
using pixels.
• The size of representation depends on the size of the
image, the horizontal and the vertical resolution of the
screen which are usually indicated as so many pixels
per unit length. E.g. dots per inch, pixels per inch etc.
Y
P
Z
X
```
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