Introduction to
Computer Graphics
Farhana Bandukwala, PhD
Lecture 4: 2D Geometry and
Transformations
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Lecture 4: 2D Geometry and Transformations in Computer Graphics, Assignments of Introduction to Sociology
A part of an introduction to computer graphics lecture series by farhana bandukwala, phd. In this lecture, the focus is on 2d geometry and transformations, including coordinate systems, geometric objects and operations, homogeneous coordinates, and various types of transformations such as translation, scale, and rotation. The lecture also covers the concept of rigid body transformation and affine transformation.
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Pre 2010
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Introduction to
Computer GraphicsFarhana Bandukwala, PhD
Lecture 4: 2D Geometry and
Transformations
Outline
• Assignment 1 clarifications• Coordinate System• Geometric objects and
operations
(Angel,chapter 4)
• Homogeneous coordinates• Transformations
Coordinate System
Image maps toscreen coordinatesystem
(glRasterPos())
Differs from worldcoordinate system
2D transformations &clipping to switchbetween them
y
x y’
x’
Geometric Objects
• Points: location in two dimensional space• Vectors: direction and magnitude• Scalars: real (and complex) numbers
which act on points and vectors accordingto a set of rules
Q
A
B
= A
C
= (.5)B
Transformations: translation
A
Q
Translate 1 point:P = Q + A
P
P =
x’ y’
Q =
x y
A
d
x d
y
Define column vectors:
x’ y’
x’ y’
d
x d
y
Vector notation for translation:
Translate objectconsisting ofmultiple points
Transformations: scale (shear)
B
Shear 1 vector:B
= S
.^
A
Shear object
A
B
x
y’
A
x y
Define column vectors:
x’ y’
s
x
s
y
x y
Scale equation:x’ = s Matrix notation for scale:
x
x , y’ = s
y
y
Homogeneous coordinates
Goal: translation expressed in same form as rotation & scale (matrixmultiplication)
Solution: represent points as homogeneous coordinates (three-tuples)
P=(x,y,W) is the same as Q=(x’,y’,W’) if differ by a multiple (2,3,6) &(4,6,12)
If W is non-zero (x/W,y/W,1) is same point as (x,y,W)
When W is zero -> point at infinity
x
y’ 1
d
x
d
y
x y 1
Matrix notation for translationusing homogeneous coordinates
Homogeneous coordinates (contd.)
x
y’ 1
s
x
s
y
x y 1
Matrix notation for scale
x’ y’ 1
cos
T
-sin
T
sin
T
cos
T
x y 1
Matrix notation for rotation
Affine Transformation
Product of arbitrary rotation,translation,scaling
Preserves parallel lines
Does NOT preserve angles and lengths
square
Rotation by 45 degrees
Non-uniform scaling