2D Transformations
CMPT 361
Introduction to Computer Graphics
Torsten Möller
© Machiraju/Zhang/Möller
Graphics Pipeline
Hardware
Modelling
Transform
Visibility
Illumination +
Shading
Perception,
Color
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Interaction
Texture/
Realism
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Reading
• Chapter 3 of Angel
• Chapter 5 of Foley, van Dam, …
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Schedule
•
•
•
•
•
•
•
•
Geometry basics
Affine transformations
Use of homogeneous coordinates
Concatenation of transformations
3D transformations
Transformation of coordinate systems
Transform the transforms
Transformations in OpenGL
© Machiraju/Zhang/Möller
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Basic transformations
• The most basic ones
–
–
–
–
–
Translation
Scaling
Rotation
Shear
And others, e.g., perspective transform, projection, etc.
• Basic types of transformations
– Rigid-body: preserves length and angle
– Affine: preserves parallel lines, not angles or lengths
© Machiraju/Zhang/Möller
– Free-form: anything
goes
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Translation in 2D
P =P +T
x = x + dx
y = y + dy
x
y
⇥
=
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x
y
⇥
+
dx
dy
⇥
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Scaling in 2D
P =S
P
x = sx
x
y = sy
y
• Uniform sx = sy
• Non-uniform sx ⇥= sy
x
y
⇥
=
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sx
0
0
sy
⇥
x
y
⇥
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Rotation about origin
• Positive angles:
counterclockwise
• For negative angles
cos(
sin(
) = cos( )
) = sin( )
P =R
x = x cos( ) y sin( )
y = x sin( ) + y cos( )
⇥
⇥
x
cos( )
sin( )
=
y
sin( ©)Machiraju/Zhang/Möller
cos( )
x
y
⇥
P
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Derivation of rotation matrix
• Make use of polar coordinates: (r, )
x = r cos( )
y = r sin( )
x = r cos(⇥ + )
= r cos(⇥) cos( )
(x, y)
x = r cos(⇥ + )
y = r sin(⇥ + )
(r, )
(x, y)
r sin(⇥) sin( )
= x cos( ) y sin( )
y = r sin(⇥ + )
= r sin(⇥) cos( ) + r cos(⇥) sin( )
= y cos( ) + x sin( )
© Machiraju/Zhang/Möller
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Shearing in 2D
P = SHy
x
y
P = SHx
x
y
⇥
=
1 a
0 1
⇥
P
x
y
⇥
⇥
=
1 0
b 1
⇥
P
x
y
⇥
x = x + ay
y =y
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Schedule
•
•
•
•
•
•
•
•
Geometry basics
Affine transformations
Use of homogeneous coordinates
Concatenation of transformations
3D transformations
Transformation of coordinate systems
Transform the transforms
Transformations in OpenGL
© Machiraju/Zhang/Möller
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Homogeneous coordinates
• Only translation cannot be expressed as
matrix-vector multiplications
• But we can add a third coordinate –
Homogeneous coordinates for 2D points
– (x, y) turns into (x, y, W)
– if (x, y, W) and (x', y', W’ ) are multiples of one
another, they represent the same point
– typically, W ≠ 0.
– points with W = 0 are points at infinity
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2D Homogeneous Coordinates
• Cartesian coordinates of the homogenous point
(x, y, W): x/W, y/W (divide through by W)
• Our typical homogenized points: (x, y, 1)
• Connection to 3D?
– (x, y, 1) represents a 3D
point on the plane W = 1
– A homogeneous point is a
line in 3D, through the origin
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Transformation in
homogeneous coordinates
• General form of affine (linear) transformation
• E.g., 2D translation in
homogeneous coordinates
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Homogeneous Coords (3)
• 2D line equation
– Duality of 3D points to a line
a
• 3D plane equation:
– Duality of 4D points to a plane
• Homog. Coords in 1D?
ax + by + c = 0
⇤
⌅
x
⇥
b c ⇧ y ⌃=0
1
ax + by + cx + d = 0
⇤
⌅
x
⇥⌥ y
a b c d ⌥
⇧ z ⌃=0
1
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Basic 2D transformations
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Inverse of transformations
• Inverse of T(dx, dy) = T(−dx, −dy)
T(−dx, −dy) = T(dx, dy)−1
• R(θ)−1 = R(−θ)
• S(sx, sy)−1 = S(1/sx, 1/sy)
• Sh(ax)−1 = Sh(−ax)
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Schedule
•
•
•
•
•
•
•
•
Geometry basics
Affine transformations
Use of homogeneous coordinates
Concatenation of transformations
3D transformations
Transformation of coordinate systems
Transform the transforms
Transformations in OpenGL
© Machiraju/Zhang/Möller
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Compound transformations
• Often we need many transforms to build
objects (or to “direct” them)
– e.g., rotation about an arbitrary point/line?
• Concatenate basic transforms sequentially
• This corresponds to multiplication of the
transform matrices, thanks to homogeneous
coordinates
© Machiraju/Zhang/Möller
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Compound translation
• What happens when a point goes through
T(dx1, dy1) and then T(dx2, dy2)?
• Combined translation: T(dx1+dx2, dy1+dy2)
⇥
⇥
⇥
1 0 dx2
1 0 dx1
1 0 dx1 + dx2
⇤ 0 1 d y2 ⌅ ⇤ 0 1 d y1 ⌅ = ⇤ 0 1 d y1 + d y2 ⌅
0 0 1
0 0 1
0 0
1
Concatenation of transformations: matrix multiplication
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Compound rotation
• What happens when a point goes through
R(θ) and then R(φ)?
• Combined rotation: R(θ + φ)
• Scaling or shearing (in one direction) is similar
• These concatenations are commutative
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Commutativity
• Transformations that do commute:
–
–
–
–
–
Translate – translate
Rotate – rotate (around the same axis)
Scale – scale
Uniform scale – rotate
Shear in x (y) – shear in x (y), etc.
• In general, the order in which transformations
are composed is important
• After all, matrix multiplications are not
commutative © Machiraju/Zhang/Möller
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Commutativity
• the order in which we compose matrices is
important: matrix multiplication does not
y
y
commute
X then Z
y
x
z
z
x
x
z
y
y
Z then X
z
x
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z
x
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Rotation about an
arbitrary point
• Break it down into easier problems:
– Translate to the origin: T(−x1, −y1)
– Rotate: R(q)
– And translate back: T(x1, y1)
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Rotation about an
arbitrary point
• Compound transformation (non-commutative):
T(x1, y1) • R(q) • T(− x1, − y1)
– P’ = T(x1, y1)R(q)T(− x1, − y1)P = AP
– Why combine these matrices into a single A?
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Another example
• Compound transformation:
T(x2, y2) • R(90o) • S(sx, sy) • T(−x1, −y1)
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Rigid-body transformation
• A transformation matrix
• where the upper 2 × 2 submatrix is
orthonormal, preserves angles and lengths
– called rigid-body transformation
– any sequence of translation and rotation give a
rigid-body transformation
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Affine transformation
• Preserves parallelism, not lengths or angles
Unit cube
Rot(−45o)
Scale in x
• Product of a sequence of translation, scaling,
rotation, and shear is affine
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Types of Transforms
• Continuous/Discrete, one to one ; invertible
• Classification
– Euclidean (preserves angles)
• translation, rotation, uniform scale
– Affine (preserves parallel lines)
• Euclidean plus non-uniform scale, shearing
– Projective (preserves lines)
– Isometry (preserves distance)
• Reflection, rigid body motion
– Nonlinear (lines©become
curves)
Machiraju/Zhang/Möller
• Twists, Bends, warps, morphs
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Window-To-Viewport
Transformation
• Important example of compound transform
• “World coords”: space users work in
• need to map world to the screen
– window in world-coordinate space and
– viewport in screen coordinate space
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Window-To-Viewport
Transformation (2)
• one can specify non-uniform scaling
• results in distorted images
• What property of window and viewport
indicates whether the mapping will distort the
images?
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Window-To-Viewport
Transformation (3)
• Specify transform in matrix form
Mwv = T
umin
vmin
⇥
⇥S
umax
xmax
vmax
ymax
umin
xmin
vmin
ymin
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⇥
⇥T
xmin
ymin
⇥
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