GEOMETRIC OBJECTS AND
TRANSFORMATIONS
Introduction


In computer graphics many applications need to alter
or manipulate a picture, for example, by changing its
size, position or orientation.
This can be done by applying a geometric
transformation to the coordinate points defining the
picture.
Geometric Objects
Points:
 fundamental geometric object is a point.
 In a three-dimensional geometric system,
a point is a location in space.
The only property that a point possesses is that
point’s location; a mathematical point has neither a
size nor a shape.

Vectors

In computer graphics, we often connect points with
directed line segments, A directed line segment has
both magnitude—its length—and direction—its
orientation—and thus is a vector.
Coordinates free geometry


Points exist in space regardless of any reference or
coordinate system. Thus, we do not
need a coordinate system to specify a point or a
vector.
Coordinates free geometry con..
Definition: Geometry

The branch of mathematics concerned with the
properties of and relationships between points, lines,
planes, and figures.
2D Transformations

What is transformations?
 The
geometrical changes of an object from a current
state to modified state.
 A rule for moving every point in a plane figure to a
new location.
Affine transformation

In geometry, an affine transformation is a
transformation which preserves straight lines (all
points lying on a line initially still lie on a line after
transformation) and ratios of distances between
points lying on a straight line. It does not necessarily
preserve angles or lengths, but does have the
property that sets of parallel lines will remain
parallel to each other after an affine transformation.
Affine transformation con..


Examples of affine transformations include
translation, geometric contraction, expansion,
reflection, rotation
An affine transformation is equivalent to a linear
transformation followed by a translation
Types of Transformation



Translation
Rotation
Scaling
Translation



A translation moves all points in an
object along the same straight-line
path to new positions.
The path is represented by a vector,
called the translation or shift vector.
?
We can write the components:
p'x = px + tx
ty=4
p'y = py + ty

or in matrix form:
P' = P + T
tx
x’
x
y’ = y + ty
(2, 2)
tx= 6
Translation con…


In a translation a figure slides up or down, or left or
right. No change in shape or size. The location
changes.
In graphing translation, all x and y coordinates of a
translated figure change by adding or subtracting.
Rotation


A rotation repositions
all points in an object
along a circular path in
the plane centered at
the pivot point.
First, we’ll assume the
pivot is at the origin.
P’

P
Rotation
• Review Trigonometry
=> cos  = x/r , sin = y/r
• x = r. cos , y = r.sin 
P’(x’, y’)
=> cos (+ ) = x’/r
•x’ = r. cos (+ )
•x’ = r.coscos -r.sinsin
•x’ = x.cos  – y.sin 
=>sin (+ ) = y’/r
y’ = r. sin (+ )
•y’ = r.cossin + r.sincos
•y’ = x.sin  + y.cos 

r
y’

x’
P(x,y)
r

y
x
Identity of Trigonometry
Rotation
• We can write the components:
p'x = px cos  – py sin 
p'y = px sin  + py cos 
P’(x’, y’)
• or in matrix form:
P' = R • P
• Rotation matrix
cos 
R
 sin 
 sin  
cos  

y’

x’
P(x,y)
r

x
y
Rotation
Example

Find the transformed point, P’, caused by rotating P=
(5, 1) about the origin through an angle of 90.
cos
 sin 

 sin    x   x  cos  y  sin  
   

cos   y   x  sin   y  cos 
5  cos 90  1 sin 90


5  sin 90  1 cos 90
5  0  11


5 1  1 0
 1
 
5

Scaling
• Scaling changes the size of an
object and involves two scale
factors, Sx and Sy for the xand y- coordinates
respectively.
• Scales are about the origin.
• We can write the components:
p'x = sx • px
p'y = sy • py
or in matrix form:
P' = S • P
Scale matrix as:
sx
S
0
0
s y 
P’
P
Scaling
• If the scale factors are in between 0
and 1  the points will be moved
closer to the origin  the object will
be smaller.
• Example :
•P(2, 5), Sx = 0.5, Sy = 0.5
•Find P’ ?
P(2, 5)
P’
Scaling
• If the scale factors are in between 0
and 1  the points will be moved
closer to the origin  the object will
be smaller.
• Example :
•P(2, 5), Sx = 0.5, Sy = 0.5
•Find P’ ?
•If the scale factors are larger than 1
 the points will be moved away from
the origin  the object will be larger.
• Example :
•P(2, 5), Sx = 2, Sy = 2
•Find P’ ?
P’
P(2, 5)
P’
Scaling
• If the scale factors are the same, Sx
= Sy  uniform scaling
• Only change in size (as previous
example)
•If Sx  Sy  differential scaling.
•Change in size and shape
•Example : square  rectangle
•P(1, 3), Sx = 2, Sy = 5 , P’ ?
P’
P(1, 2)
Matrix Math

Why do we use matrix?
 More
convenient organization of data.
 More efficient processing
 Enable the combination of various concatenations

Matrix addition and subtraction
a
b

c
d
=
a c
b d
Matrix Math

Type of matrix
a
b
Row-vector
a
b
Column-vector
Matrix Math


We’ll use the column-vector representation for a
point.
Which implies that we use pre-multiplication of the
transformation – it appears before the point to be
transformed in the equation.
 A B   x   Ax  By 
C D   y   Cx  Dy 

   


What if we needed to switch to the other
convention?
Matrix Representation of Transformations



Point in column-vector:
x
y
1
Our point now has three coordinates. So our matrix is
needs to be 3x3.
Translation
 x 1 0 t x   x 
 y  0 1 t    y 
y  
  
 1  0 0 1   1 
Matrix Representation

Rotation
 x  cos
 y   sin 
  
 1   0

 sin 
cos
0
0  x 
0   y 
1  1 
Scaling
 x   s x
 y   0
  
 z   0
0
sy
0
0   x



0    y
Sz   z 
Composite Transformation

We can represent any sequence of transformations
as a single matrix.
No special cases when transforming a point – matrix •
vector.
 Composite transformations – matrix • matrix.


Composite transformations:
Rotate about an arbitrary point – translate, rotate,
translate
 Scale about an arbitrary point – translate, scale,
translate

Order of operations
Example:
1.
2.
Translate
Rotate
1.
2.
Rotate
Translate
Homogenous Coordinates
y
y
w
x

x





Let’s move our problem into 3D.
Let point (x, y) in 2D be represented by point (x, y, 1) in the new
space.
Scaling our new point by any value a puts us somewhere along a
particular line: (ax, ay, a).
A point in 2D can be represented in many ways in the new space.
(2, 4) ---------- (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc.
Homogenous Coordinates
• We can always map back to the original 2D point by dividing by
the last coordinate
• (15, 6, 3) --- (5, 2).
• (60, 40, 10) - ?.
• Why do we use 1 for the last coordinate?
• The fact that all the points along each line can be mapped back to
the same point in 2D gives this coordinate system its name –
homogeneous coordinates.
Frames In OpenGl
OpenGL is based on a pipeline model, the first part
of which is a sequence of operations on vertices,
many of which are geometric.
 We can characterize such operations by a sequence
of transformations or as a sequence of
changes of frames for the objects specified by an
application program.

OpenGl Pipeline
1.Object (or model) coordinates
2. World coordinates
3. Eye (or camera) coordinates
4. Clip coordinates
5. Normalized device coordinates
6. Window (or screen) coordinates
Data Structure for object representation