1. Geometric Transformations
Changing an object’s position (translation),
orientation (rotation) or size (scaling)
Others transformations: reflection and shearing
operations
2. Basic 2D Geometric Transformations
2D Translation
x’ = x + tx , y’ = y + ty
P’=P+T
Translation moves the object without deformation
P
P’
T
y
x
t
t
T
y
x
P
y
x
P ,
'
'
',
3. Basic 2D Geometric Transformations
2D Translation
To move a line segment, apply the transformation
equation to each of the two line endpoints and redraw the
line between new endpoints
To move a polygon, apply the transformation equation to
coordinates of each vertex and regenerate the polygon
using the new set of vertex coordinates
5. Basic 2D Geometric Transformations
2D Rotation
Rotation axis
Rotation angle
rotation point or pivot point (xr,yr)
yr
xr
θ
6. Basic 2D Geometric Transformations
2D Rotation
At first, suppose the pivot point is at the origin
x’=r cos(θ+Φ) = r cos θ cos Φ - r sin θ sin Φ
y’=r sin(θ+Φ) = r cos θ sin Φ + r sin θ cos Φ
x = r cos Φ, y = r sin Φ
x’=x cos θ - y sin θ
y’=x sin θ + y cos θ
Φ
(x,y)r
r θ
(x’,y’)
8. Basic 2D Geometric Transformations
2D Rotation
Rotation for a point about any specified position (xr,yr)
x’=xr+(x - xr) cos θ – (y - yr) sin θ
y’=yr+(x - xr) sin θ + (y - yr) cos θ
Rotations also move objects without deformation
A line is rotated by applying the rotation formula to each
of the endpoints and redrawing the line between the new
end points
A polygon is rotated by applying the rotation formula to
each of the vertices and redrawing the polygon using new
vertex coordinates
10. Basic 2D Geometric Transformations
2D Scaling
Scaling is used to alter the size of an object
Simple 2D scaling is performed by multiplying object
positions (x, y) by scaling factors sx and sy
x’ = x · sx
y’ = y · sx
or P’ = S·P
y
x
s
s
y
x
y
x
0
0
'
'
11. Basic 2D Geometric Transformations
2D Scaling
Any positive value can be used as scaling factor
Values less than 1 reduce the size of the object
Values greater than 1 enlarge the object
If scaling factor is 1 then the object stays unchanged
If sx = sy , we call it uniform scaling
If scaling factor <1, then the object moves closer to the origin
and If scaling factor >1, then the object moves farther from the
origin
x’ x
12. Basic 2D Geometric Transformations
2D Scaling
We can control the location of the scaled object by
choosing a position called the fixed point (xf,yf)
x’ – xf = (x – xf) sx y’ – yf = (y – yf) sy
x’=x · sx + xf (1 – sx)
y’=y · sy + yf (1 – sy)
Polygons are scaled by applying the above formula to each
vertex, then regenerating the polygon using the
transformed vertices
14. 5.2 Matrix Representations and
Homogeneous Coordinates
Many graphics applications involve
sequences of geometric transformations.
Hence we consider how the matrix
representations can be reformulated so that
such transformation sequence can be
efficiently processed.
Each of three basic two-dimensional
transformations (translation, rotation and
scaling) can be expressed in the general
matrix form
15. P and P ’ = column vectors,
coordinate position
M1 = 2 by 2 array containing
multiplicative factors, for
translation M1 is the identity
matrix
M2 = two-element column matrix
containing translational terms, for
rotation or scaling M2 contains
the translational terms associated
with the pivot point or scaling
fixed point
21
'
MPMP
17. To produce a sequence of transformations
such as scaling followed by rotation then
translation, we could calculate the
transformed coordinates one step at a time.
A more efficient approach is to combine the
transformations so that the final coordinate
positions are obtained directly from the
initial coordinates, without calculating
intermediate coordinate values.
18. Homogeneous Coordinates
Multiplicative and translational terms for a
two-dimensional geometric transformations
can be combined into a single matrix if we
expand the representations to 3 by 3
matrices.
Then we can use the third column of a
transformation matrix for the translation
terms, and all transformation equations can
be expressed as matrix multiplications.
19. But to do so, we also need to expand the
matrix representation for a two-dimensional
coordinate position to a three-element column
matrix.
A standard technique for accomplishing this
is to expand each two-dimensional
coordinate-position representation (x,y) to a
three-element representation (xh,yh,h), called
homogeneous coordinates, where the
homogeneous parameter h is a nonzero
value such that
,
h
y
y
h
x
x hh
20. A convenient choice is simply to set h=1.
Each two-dimensional position is then
represented with homogeneous coordinate
(x,y,1).
The term “homogeneous coordinates” is
used in mathematics to refer to the effect of
this representation on Cartesian equations.
23. 2D Composite Transformations
We can setup a sequence of transformations as a
composite transformation matrix by calculating
the product of the individual transformations
P’=M2·M1·P
P’=M·P
24. 2D Composite Transformations
Composite 2D Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
34. Reflection
Image is generated relative to an axis of reflection by
rotating the object 180° about the reflection axis
Reflection about the line y=0 (the x axis)
100
010
001
35. Other Two Dimensional
Transformations
Reflection
Reflection about the line x=0 (the y axis)
Reflection about the origin
100
010
001
100
010
001
36. Other Two Dimensional
Transformations
Shear
Transformation that distorts the shape of an object
such that the transformed shape appears as the
object was composed of internal layers that had been
caused to slide over each other
y
x
(0,1) (1,1)
(1,0)(0,0)
y
x
(2,1) (3,1)
(1,0)(0,0)
shx=2
37. Other Two Dimensional
Transformations
Shear
An x-direction shear relative to the x axis
An y-direction shear relative to the y axis
100
01
001
ysh
100
010
01 xsh
yy
yshxx x
'
'
38. Changing Coordinate System
P is described as (x,y) in the x-y coordinate system. We wish to
describe P in the x’-y’ coordinate system. What is (x’,y’)?
First, translate by (x,y) by (-x0,-y0)
Then, rotate by - q
y axis
x axis
y’ axis
x’ axis
P
y’
x’
x
y
q
y0
x0
39. Changing Coordinate Systemy axis
x axis
y’ axis
x’ axis
P
y’
x’
x
y
q
y axis
y’ axis
x’ axis
P
y’
x’q
y axis/
y’ axis
x axis/x’ axis
P
x’
y’
40. 2D Viewing
Which part of a picture is to be displayed
clipping window or world window or viewing
window
Where that part will be displayed on the display
device
viewport
41. 2D Viewing
Clipping window selects what we want to see
Viewport indicates where it is to be viewed on
the output device
Usually, clipping windows and viewport are
rectangles
43. 2D Viewing Pipeline
2D viewing pipeline
Construct world-coordinate scene using modeling-
coordinate transformations
Convert world-coordinates to viewing coordinates
Transform viewing-coordinates to normalized-
coordinates (ex: between 0 and 1, or between -1 and
1)
Map normalized-coordinates to device-coordinates.
45. Transform the square and
the models to the device
coordinates in the display
viewport.
in the modeling
coordinates:
Specify a window Transform the window
and the models to the
area to be displayed
normalized coordinates.
Clip against the square
Xmodeling
Ymodeling
Xnormalized
Ynormalized
Xdevice
Ydevice
The 2D Viewing Pipeline
S(2/width, 2/height);
T(-center);
Trans(models);
// normalized models
// glOrtho()
Clipping();
// clipped models
T(Center);
S(With/2, Height/2);
Trans(models);
// device models
//glViewport()
46. ViewingTransformation
46
Viewing transformation is the mapping of a part of a world-
coordinate scene to device coordinates.
In 2D (two dimensional) viewing transformation is simply
referred as the window-to-viewport transformation or the
windowing transformation.
Mapping a window onto a viewport involves converting from
one coordinate system to another.
If the window and viewport are in standard position, this just
involves translation and scaling.
if the window and/or viewport are not in standard, then
extra transformation which is rotation is required.
50. Window-To-ViewportCoordinateTransformation
50
xv - xvmin = xw - xwmin
xvmax - xvmin xwmax - xwmin
yv – yvmin = yw - ywmin
yvmax – yvmin ywmax - ywmin
From these two equations we derived
xv = xvmin + (xw – xwmin)sx
yv = yvmin + (yw – ywmin)sy
where the scaling factors are
sx = xvmax – xvmin sy = yvmax - yvmin
xwmax – xwmin ywmax - ywmin
51. Window-To-ViewportCoordinateTransformation
51
The sequence of transformations are:
1. Perform a scaling transformation using a
fixed-point position of (xwmin,ywmin) that
scales the window area to the size of the
viewport.
2. Translate the scaled window area to the
position of the viewport.
52. Window-To-ViewportCoordinateTransformation
52
Relative proportions of objects are maintained if the scaling
factors are the same (sx = sy). Otherwise, world objects will be
stretched or contracted in either x or y direction when displayed
on output device.
How about character strings when map to viewport?
maintains a constant character size (apply when
standard character fonts cannot be changed).
If character size can be changed, then windowed will be
applied like other primitives.
For characters formed with line segments, the mapping
to viewport is carried through sequence of line
transformations .
53. 2001. 7. 13 53
Two-Dimensional Viewing Functions (1/2)
• Definition about a viewing reference system
– evaluateViewOrientationMatrix (x0, y0, xV, yV, error, viewMatrix)
• Setting up the elements of a window-to-viewport mapping
matrix
– setviewRepresentation (ws, viewIndex, viewMatrix, viewMap-
pingMatrix, xclipmin, xclipmin, xclipmin, xclipmin, clipxy)
• Storing combinations of viewing and window-viewport
mappings for various workstations in a viewing table
– evaluateViewMappingMatrix (xwmin, xwmax, ywmin, ywmax,
xvmin, xvmax, yvmin, yvmax, error, viewMappingMatrix)
54. 2001. 7. 13 54
Two-Dimensional Viewing Functions (2/2)
• Selection of a paticular set of options from
the viewing table
– setViewIndex (viewIndex)
• Selection of a workstation window-
viewport pair
– setWorkstationWindow (ws, xwsWindmin,
xwsWindmax, ywsWindmin, ywsWindmax)
– setWorkstationViewport (ws, xwsVPortmin,
xwsVPortmax, ywsVPortmin, ywsVPortmax)
55. 2001. 7. 13 55
Clipping Operations
• Clipping
– Any procedure that identifies those portions of a picture
that are either inside or outside of a specified region of
space
• Applied in World Coordinates
• Adapting Primitive Types
– Point
– Line
– Area (or Polygons)
– Curve, Text (omit!!)
56. 2001. 7. 13 56
Point Clipping
• Assuming that the clip window is a
rectangle in standard position
• Saving a point P=(x, y) for display
• Appling Fields
– Particles (explosion, sea foam)
maxmin
maxmin
ywyyw
xwxxw
57. 2001. 7. 13 57
Line Clipping (1/3)
a) Before Clipping b) After Clipping
• Line clipping against a rectangular clip window
58. 2001. 7. 13 58
Line Clipping (2/3)
• Parametric representation of Line segment
with endpoints (x1, y1) and (x2, y2)
• Exterior of the window
– Intersection with outside the range u
• Interior of the window
– Intersection with inside the range u
),yu(yyy
)xu(xxx
121
121
1u0
63. Algorithm
Input vertex P
First Point YesNo
F=P
Does SP intersect
E?
No
Yes
Compute
Intersection Point
I
Output
vertex I
S=P
Is S on left
side of E?
Exit
NO
Yes
Output
vertex S
Close Polygon entry
Does SF
intersect E?
Compute
Intersection I
Yes
Output
vertex I
Exit
No
