Geometric Transformations

1
Geometric Transformations
2
Cartesian Coordinates
(10, 12)
10
12
Rene Descartes - 1637
3
Cartesian Coordinates
(13, 15)
Rene Descartes - 1637
4
Cartesian Coordinates
(13, 15)
(3, 3)
Rene Descartes - 1637
5
Co-ordinate Systems

• Objects in computer graphics are given numerical
  descriptions that characterize their shape and
  dimensions.
• These are in relation to some coordinate system,
  most often Cartesian co-ordinates x,y and z

6
Transforming Pictures

• Sometimes objects exhibit certain symmetries, so
  only a part of it needs to be described, and the
  rest constructed by reflecting, rotating and
  translating the original part

7
Example 1

• Object parts defined in a local co-ordinate
  system
• Larger objects are then assembled by
  duplicating and transforming each of the
  constituent parts

etc...
8
Transforming Pictures

• Sometimes objects exhibit certain symmetries, so
  only a part of it needs to be described, and the
  rest constructed by reflecting, rotating and
  translating the original part
• A designer may want to view and object from
  different vantage points, by rotating the object,
  or by moving a synthetic camera viewpoint.

9
Example 2
10
Transforming Pictures

• Sometimes objects exhibit certain symmetries, so
  only a part of it needs to be described, and the
  rest constructed by reflecting, rotating and
  translating the original part
• A designer may want to view and object from
  different vantage points, by rotating the object,
  or by moving a synthetic camera viewpoint.
• In animation, one or more objects must move
  relative to one another, so that their local
  co-ordinate systems must be shifted and rotated
  as the animation proceeds.

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Example 3
5 steps of a rotating cube animation

• At each frame of the animation, the object is
  transformed, in this case by a rotation.
• It could also be transformed by changing its size
  (scaling), or its shape (deforming), or its
  location (translation).
• Further animation effects can be achieved by not
  changing the object, but the way it is viewed
  zooming and panning the viewing window

12
Transformations

• A transformation on an object is an operation
  that changes how the object is finally drawn to
  screen
• There are two ways of understanding a
  transformation
• An Object Transformation alters the coordinates
  of each point according to some rule, leaving the
  underlying coordinate system unchanged
• A Coordinate Transformation produces a different
  coordinate system, and then represents all
  original points in this new system
• Both ways have advantages, and are closely
  related to one another

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Example OBJECT TRANSFORMATION
.4, 2
1,1
Example COORDINATE TRANSFORMATION
(1,1)
(1,1)
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2D Object Transformations

• A 2D object transformation alters each point P
  into a new point Q using a specific formula or
  algorithm.
• It therefore alters the co-ordinates of P (Px,Py)
  into new values which specify point Q (Qx,Qy)
• This can be expressed using some function T, that
  maps co-ordinate pairs into new co-ordinate
  pairs
• T (Px,Py) (Qx,Qy)
• sometimes simply denoted as T(P) Q
• e.g P is the point (2, 2), T is a transform that
  scales by a factor of 2. Then T(P) T(2, 2)
  (4,4)

P (Px, Py)
??? DO SOMETHING ???
Q (Qx, Qy)
P (2, 2)
Scale by 2
Q (4, 4)
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Affine Transformations

• Whole collections of points may be transformed by
  the same transformation T, e.g. lines or circles
• The image of a line, L, under T, is the set of
  all images of the individual points of L.
• For most mappings of interest, this image is
  still a connected curve of some shape
• For some mappings, the image of a line may no
  longer be a line
• Affine Transformations, however, do preserve
  lines, and are the most commonly-used
  transformations in computer graphics

P1 (0, 2)
P2 (2, 2)
P3 (2, 0)
P0 (0, 0)
Q1 (0, 4)
Q2 (4, 4)
Q0 (0, 0)
Q3 (4, 0)
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Elementary Transformations

• Affine transformations are usually combinations
  of four elementary transformations
• 1 Translation
• 2 Scaling
• 3 Rotation
• 4 Shearing

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Translation

• A translation moves an object into a different
  position in a scene
• This is achieved by adding an offset/translation
  vector
• So in Vector notation

t
Translate by t
Original points
Transformed points
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Scaling

• A scaling changes the size of an object with two
  scale factors, Sx and Sy

Sx, Sy
Scale by Sx, Sy
Original points
Transformed points
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Rotation

• Using the trigonometric relations, a point
  rotated by an angle q about the origin is given
  by the following equations
• So

q
Rotate by q
Original points
Transformed points
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Rotation - derivation
1
2
3
4
1
Substituting from 3 and 4
Similarly from 2
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Shearing

• A shearing shears an object in a particular
  direction, (in 2D, its either in the x or in the
  y direction
• So, a shear in the x direction would be as
  follows
• The quantity h specifies what fraction of the
  y-coordinate should be added to the x-coordinate,
  and may be positive or negative
• More generally a simultaneous shear in both the
  x and y directions would be

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Shearing
g 0, h0.5
Shear by (g,h)
Original points
Transformed points
g0.5, h0.5
Shear by (g,h)
Original points
Transformed points
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Transformations Summary

• 1 Translation
• 2 Scaling
• 3 Rotation
• 4 Shearing

T
T
T
T
(Qx,Qy) (PxhPy, Py)
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Matrix Representation

• All affine transformations in 2D can be
  generically described in terms of two generic
  equations as follows
• where a, b, c, d, tx and ty are all constants,
  and ad bc
• This can be represented with matrices as follows
• i.e.

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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
1x1
1x3
3x1
2x2
2x2
2x2
3x3
3x1
3x1
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Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
38
Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
43
Matrix Multiplication
e.g.
44
Matrix Multiplication
e.g.
45
Matrix Multiplication
e.g.
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Matrix Multiplication
e.g.
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Transformations Summ. (2)
Translation
Scaling
Rotation
Shearing
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Problem

• An affine transformation is composed of a linear
  combination followed by a translation
• Unfortunately, the translation portion is not a
  matrix multiplication but must instead be added
  as an extra term, or vector this is
  inconvenient
• What we need is a trick, so that translations
  can be represented in matrix multiplication form
• We will see (later) that this then means that
  they can be easily composed with other
  transformations, by simply multiplying the
  matrices together

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Homogeneous Coordinates

• The trick we use is to add an additional
  component 1 to both P and Q, and also a third row
  and column to M, consisting of zeros and a 1
• i.e.

Rotate by q
Translation by (tx, ty)
Scale by Sx, Sy
Shear by g, h
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Useful Properties of Affine Transformations
1 Preservation of lines 2 Preservation of
parallelism 3 Preservation of proportional
distances
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Why are they useful?

• They preserve lines, so the image of a straight
  line is another straight line.
• This vastly simplifies drawing transformed line
  segments.
• We need only compute the image of the two
  endpoints of the original line and then draw a
  straight line between them
• Preservation of co-linearity guarantees that
  polygons will transform into polygons
• Preservation of parallelism guarantees that
  parallelograms will transform into parallelograms
• Preservation of proportional distances means that
  mid-points of lines remain mid-points

52
Multiple Transformations

• It is rare that we want to perform just one
  elementary transformation.
• Usually an application requires that we build a
  complex transformation out of several elementary
  ones
• e.g. translate an object, rotate it, and scale
  it, all in one move
• These individual transformations combine into one
  overall transformation
• This is called the composition of
  transformations.
• The composition of two or more affine
  transformations is also an affine transformation

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Rotation and Scaling

• The simple versions of rotation and scaling have
  been based around the origin.
• This means that when we rotate or scale, the
  object will also move, with respect to the origin

Not only is the object rotated, but it also moves
around the origin
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Pivotal points

• Often we wish to rotate or scale with respect to
  some pivotal point, not the origin
• Most significantly, we often wish to rotate or
  scale an object about its centre, or midpoint
• In this way, the objects location does not
  change
• To do this, we relate the rotation or scaling
  about the pivotal point V, to an elementary
  rotation or scaling about the origin
• We first translate all points so that V coincides
  with the origin
• We then rotate or about the origin
• then all points are translated back, so that V is
  restored to its original location

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(x1,y1)
(0,0)
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Rotation about a pivot point
57
Coordinate Transforms
x
u
(1,1)
v
Object defined in Local Coordinate System
y
Object after transformation in Global Coordinate
System
58
Identity
59
Translation
u (0, 1, 0)
origin
v (1, 0, 0)
60
Rotation
61
Scaling
62
Composite Transformations
O
v
u
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Why composition?

• One of the main reasons for composing
  transformations is computational efficiency
• Suppose you want to apply two affine
  transformations to the N vertices of a polygonal
  object
• It requires 12 multiplications to compose the
  transformations, and 4 multiplications to apply a
  transformation to a point
• So, compose-then-apply requires 4N12
  multiplications
• (PM1Tr1)M2 Tr2
• 8N multiplications are required if each
  transformation is applied seperately
• For models with any significant number of
  vertices (which will be most), 4N12 is much less
  than 8N

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Modeling Transformations

• To make full use of the computational
  optimisation made possible by composite
  transforms, we only want to apply the
  transformations to points at the very end
• i.e. the transformation operation (multiplying
  point p by transform matrix is the very last
  thing we do in the modelling phase)

Specify Transformations (composite if necessary)
Specify points in local coords
Send to Pipeline
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Usually we model individual objects based on a
local cordinate system
This of course shouldnt mean all objects need to
share the same transformations

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transform
transform
transform
Obviously we want something more versatile

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Code

• We probably want code that looks something like
  this

scale (sxM, syM) translate (txM,
tyM) drawMan() scale (sxH, syH) translate
(txH, tyH) drawHouse() scale (sxS,
syS) translate (txS, tyS) drawSun()
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Current Transformation Matrix

• One way to do this is by using a single object to
  represent the CURRENT TRANSFORMATION MATRIX (CTM)
• All transformation functions affect only the CTM
• When a draw procedure is finally called it draws
  based on the CTM
• The CTM might be seen as representing the
  current coordinate frame for drawing
• Next 3D Transformations and OpenGL

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Transforming Images
70
Transforming Images
71
Some 3D Transformations
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Scale

• all vectors are scaled from the origin

Original
scale all axes
scale Y axis
offset from origin
distance from origin also scales
73
Scale
Or in 3D homogeneous coordinates
74
Rotation

• Rotations are anti-clockwise about the origin

rotation of 45o about the Z axis
offset from origin rotation
75
Rotation
76
Rotation

• 2D rotation of q about origin
• 3D homogeneous rotations

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Translation

• Translation only applies to points, we never
  translate vectors.
• Remember points have homogeneous co-ordinate w
  1

translate along y
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Screen Co-ordinates

• Viewport Mapping
• Clipping

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Viewing Issues

• A Rasterization Issue
• How much of the projected data should be drawn?
• Where should it appear on the display?
• e.g We could have a model of a whole room, full
  of objects such as chairs, tables, and students.
  We may want to view the whole room in one go, or
  zoom in on one single object in the room. We may
  want to display the object or scene on the full
  screen, or we may only want to display it on a
  portion of the screen

80
Screen Co-ordinates

• In our final display, i.e. the frame buffer,
  images are specified in terms of pixels (screen
  coordinates defined with integer values) where as
  our original data is in real-number coordinates
  (modelling coordinates)
• We need to apply transform to convert from
  Modelling co-ordinates to Screen Coordinates
• We can scale dimensions to change the resulting
  view
• We can even achieve a zooming in and out effect
  without changing the model by scaling dimensions
  proportionally

81
2-Dimensional Views

• A 2-dimensional view is defined by two rectangles
  we will refer to them as
• A Window given in real world co-ordinates, e.g.
  meters, feet etc defining the portion of model
  (scene) to be drawn
• A Viewport given in screen co-ordinates, pixels,
  defining the portion of the screen which will be
  used to display the contents of the window
• Note that confusingly, some textbooks sometimes
  use these terms interchangeably to describe the
  original rectangle (the window).

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The Viewport

• We need to associate the 2D viewport co-ordinate
  system with the screen co-ordinate system in
  order to determine the correct pixel associated
  with each vertex.

Real-world co-ordinates
screen co-ordinates
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Windows and Viewports
(wxmax,wymax)
(vxmax,vymax)
(vx,vy)
(wx,wy)
(vxmin,vymin)
(wxmin,wymin)
For any point (wx, wy) in the real-world window,
what is the corresponding pixel position?
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Window to Viewport Mapping
Preservation of horizontal ratios implies that

• wx - wxmin vx - vxmin
• wxmax - wxmin vxmax - vxmin

So, solving for vx
vx (wx - wx min) vxmax - vxmin
vxmin wxmax - wxmin
vy can be solved for similarly.
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Example
0.4
0.3
-0.4
-0.2

• Say we want to map this to a 640x480 viewport
• Where is the black point mapped in screen
  coordinates?

480
640
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Example

• Vxmax640
• Vxmin-0.4
• Vymax0
• Vymin480

• Wx-.2
• Wy.3
• Wxmax0
• Wxmin-0.4
• Wymax0.4
• Wymin0
• Vx (-.2-(-0.4))(640-0) 0 320
• 0-(-0.4)
• Vy (.3-0)(0-480) 480 120
• 0-(-0.4)

320
120
Repeat this for all vertices and then rasterize.