Basic Transformations

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Goals
• This unit is about the math for
these transformations
– Represent transformations using
matrices and matrix-vector multiplications.
• General Idea
– Object in model coordinates
– Transform into world coordinates
– Represent points on object as vectors
– Multiply by matrices
Simple Transformations
• Can be combined

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What is a Transformation?
• Maps points (x, y) in to
points (x', y')
• Just functions acting on
points
(x’, y’, z’) = F(x, y, z)
P’ = F(P)

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Transformations are used
• Modeling
– Position objects in a scene
– Change the shape of objects
– Create multiple copies of objects
– Efficiently represent hierarchical scenes
• Viewing
– World coordinates to camera coordinates
– Parallel / perspective projections from 3D to 2D
• Animations

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Cube Man (Instancing Hierarchy)

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How are Transforms Represented?
x′ = ax + by + c
ቊ
y′ = dx + ey + f

• Biểu diễn dưới dạng ma trận

𝑥′ 𝑎 𝑏 𝑥 𝑐
= 𝑦 + 𝑓
𝑦′ 𝑑 𝑒
P' = M P + T

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 Affine Transformations

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Affine Transformations
• Want transformations which preserve geometry
– lines, polygons, quadrics
• Affine = line preserving
– Rotation, translation, scaling
– Projection
– Concatenation (composition)

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Scale

𝑥 ′ = 𝑆𝑥 𝑥
൝ ′
𝑦 = 𝑆𝑦 𝑦
𝑥′ 𝑆𝑥 0 𝑥
= 0 𝑆𝑦 𝑦
𝑦′
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Shear

1 𝑎
ShearX =
0 1
𝑥′ 1 𝑎 𝑥
= 𝑦
𝑦′ 0 1
Rotations
• 2D simple, 3D complicated. [Derivation?
Examples?]
• 2D?
𝑥′ cos 𝜃 − sin 𝜃 𝑥
′ =
𝑦 sin 𝜃 cos 𝜃 𝑦

• Linear (Tuyến tính): R(X+Y) = R(X)+R(Y)

• Commutative (Giao hoán)

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Composing Transforms
• Often want to combine transforms
– E.g. first scale by 2, then rotate by 45 degrees
• Advantage of matrix formulation:
– All still a matrix

• Not commutative!! Order matters

– X2 = SX1
– X3 = RX2
– X3 = R(SX1) = (RS)X1
– X3  (SR)X1
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Inverting Composite Transforms
• Say I want to invert a combination of 3 transforms
• Option 1: Find composite matrix, invert
• Option 2: Invert each transform and swap order
– Obvious from properties of matrices
𝑀 = 𝑀1 𝑀2 𝑀3
𝑀−1 = 𝑀3−1 𝑀2−1 𝑀1−1
𝑀−1 𝑀 = 𝑀3−1 𝑀2−1 𝑀1−1 𝑀1 𝑀2 𝑀3 = 𝐼

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Rotations in 3D
• Rotations about coordinate axes simple
cos 𝜃 − sin 𝜃 0
• 𝑅𝑧 = sin 𝜃 cos 𝜃 0
0 0 1
1 0 0
• 𝑅𝑥 = 0 cos 𝜃 − sin 𝜃
0 sin 𝜃 cos 𝜃
cos 𝜃 0 sin 𝜃
• 𝑅𝑦 = 0 1 0
− sin 𝜃 0 cos 𝜃

• Always linear, orthogonal

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3D Rotation Around Arbitrary Axis
• Construct orthonormal frame
transformation F with p, u, v, w,
where p and w match the rotation
axis

• Apply the transform (F Rz(θ) F-1)

Interpretations (both valid):
– Move to Z axis, rotate, then move back
– Cast w-axis rotation in new coordinate frame

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Rodrigues’ Rotation Formula
• Rodrigues’ Rotation Formula
Rotation by angle α around axis n

• How to prove this magic formula?

– Matrix N computes a cross-product: Nx = n × x
– Assume orthonormal system e1, e2, n
R(n, ) = n
R(e1, ) = cos e1 + sin e2
R(e2, ) = -sin e1 + cos e2

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Axis-Angle: Putting it together
(𝑏\𝑎)𝑅𝑂𝑇 = (𝐼3×3 cos 𝛼 − 𝑎𝑎𝑇 cos 𝛼)𝑏 + (𝐴∗ sin 𝛼)𝑏
(𝑏 → 𝑎)𝑅𝑂𝑇 = (𝑎𝑎𝑇 )𝑏
𝑅(𝑎, 𝛼) = 𝐼3×3 cos 𝛼 + 𝑎𝑎𝑇 (1 − cos 𝛼) + 𝐴∗ sin 𝛼

1 0 0
𝑅 𝑎, 𝛼 = cos 𝛼 0 1 0 +
0 0 1
𝑥 2 𝑥𝑦 𝑥𝑧 0 −𝑧 𝑦
(1 − cos 𝛼) 𝑥𝑦 𝑦 2 𝑦𝑧 + sin 𝛼 𝑧 0 −𝑥
𝑥𝑧 𝑦𝑧 𝑧2 −𝑦 𝑥 0

(x y z) are cartesian components of a

 Homogeneous Coordinates

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Translation
• E.g. move x by + 5 units, leave y, z unchanged
• We need appropriate matrix. What is it?

𝑥′ 𝑥 𝑥+5
𝑦′ = ? 𝑦 = 𝑦
𝑧′ 𝑧 𝑧

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Homogeneous Coordinates
• Add a fourth homogeneous coordinate (w=1)
• 4x4 matrices very common in graphics, hardware
• Last row always 0 0 0 1 (until next lecture)

𝑥′ 1 0 0 5 𝑥 𝑥+5
𝑦′ 0 1 0 0 𝑦 𝑦
= =
𝑧′ 0 0 1 0 𝑧 𝑧
1 0 0 0 1 1 1

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Representation of Points (4-Vectors)
• Homogeneous coordinates 𝑥 𝑥/𝑤
– Divide by 4th coord (w) to get 𝑦 𝑦/𝑤
𝑃= =
(inhomogeneous) point 𝑧 𝑧/𝑤
𝑤 1
– Multiplication by w > 0, no effect

– Assume w ≥ 0.
For w > 0, normal finite point.
For w = 0, point at infinity (used for vectors to stop
translation)

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Advantages of Homogeneous Coords
• Unified framework for translation, viewing, rot…
• Can concatenate any set of transforms to 4x4 matrix
• No division (as for perspective viewing) till end
• Simpler formulas, no special cases
• Standard in graphics software, hardware

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General Translation Matrix
1 0 0 𝑇𝑥
0 1 0 𝑇𝑦 𝐼 𝑇
• 𝑇= = 3
0 0 1 𝑇𝑧 0 1
0 0 0 1

1 0 0 𝑇𝑥 𝑥 𝑥 + 𝑇𝑥
0 1 0 𝑇𝑦 𝑦 𝑦 + 𝑇𝑦
• 𝑃′ = 𝑇𝑃 = = =𝑃+𝑇
0 0 1 𝑇𝑧 𝑧 𝑧 + 𝑇𝑧
0 0 0 1 1 1

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Combining Translations, Rotations
• Order matters!! TR is not the same as RT
• General form for rigid body transforms
• We show rotation first, then translation (commonly
used to position objects) on next slide. Slide after
that works it out the other way

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Combining Translations, Rotations
• 𝑃′ = (𝑇𝑅)𝑃 = 𝑀𝑃 = 𝑅𝑃 + 𝑇

1 0 0 𝑇𝑥 𝑅11 𝑅12 𝑅13 0

0 1 0 𝑇𝑦 𝑅21 𝑅22 𝑅23 0
• 𝑀=
0 0 1 𝑇𝑧 𝑅31 𝑅32 𝑅33 0
0 0 0 1 0 0 0 1
𝑅11 𝑅12 𝑅13 𝑇𝑥
𝑅21 𝑅22 𝑅23 𝑇𝑦 𝑅 𝑇
= =
𝑅31 𝑅32 𝑅33 𝑇𝑧 0 1
0 0 0 1

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Combining Transforms: Scene
Graphs

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Hierarchical Scene Graph
Example Scene-Graphs
Drawing a Scene Graph
• Draw scene with pre-and-post-order traversal
– Apply node, draw children, undo node if applicable
• Nodes can carry out any function
– Geometry, transforms, groups, color, …
• Requires stack to “undo” post children
– Transform stacks in OpenGL
• Caching and instancing possible
• Instances make it a DAG, not strictly a tree
Skeleton - Linear Representation
head
torso
right upper arm
right lower arm
right hand
left upper arm
left lower arm
left hand
right upper leg
right lower leg
right foot
left upper leg
left lower leg
left foot

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Linear Representation
• Each shape associated with its own transform
A single edit can require updating many transforms
• E.g. raising arm requires updating transforms for
all arm parts

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Skeleton – Hierarchical Representation
body
torso
head
right arm
upper arm
lower arm
hand
left arm
upper arm
lower arm
hand
right leg
upper leg
lower leg
foot
left leg
upper leg
lower leg
foot
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Hierarchical Representation
Grouped representation (tree)
• Each group contains subgroups and/or shapes
• Each group is associated with a transform relative to
parent group
• Transform on leaf-node shape is concatenation of all
transforms on path from root node to leaf
• Changing a group’s transform affects all parts
– Allows high level editing by changing only one node
– E.g. raising left arm requires changing only one
transform for that group
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Skeleton - Hierarchical Representation

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Coordinate System Transform

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Coordinate System Transform

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Coordinate System Transform
• A new coordinate frame is defined by a point (origin)
and two vectors (axes)

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Coordinate System Transform
• Frame-to-world transform
– Reference frame defines matrix columns

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Transform Object or Camera?

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Case Study: Derive gluLookAt
• Defines camera, fundamental to how we view
images
gluLookAt(eyex, eyey, eyez, centerx,
centery, centerz, upx, upy, upz)
– Camera is at eye, looking at center, with the up
direction being up
Up vector

Eye

Center
Constructing a coordinate frame?

• We want to associate w with a, and v with b

• But a and b are neither orthogonal nor unit norm
• And we also need to find u
𝑎 Up vector
𝑤= b
𝑎
𝑏×𝑤
𝑢= Eye
𝑏×𝑤
a
𝑣 =𝑤×𝑢
Center
Constructing a coordinate frame
𝑎 𝑏×𝑤
𝑤= 𝑢= 𝑣 =𝑤×𝑢
𝑎 𝑏×𝑤
• We want to position Up vector
camera at origin, looking b
down–Z dir
• Hence, vector a is given Eye
by eye – center a
• The vector b is simply the
up vector Center
Geometric Interpretation 3D Rotations
• Rows of matrix are 3 unit vectors of new coord
frame
• Can construct rotation matrix from 3 orthonormal
vectors
𝑥𝑢 𝑦𝑢 𝑧𝑢
𝑅𝑢𝑣𝑤 = 𝑥𝑣 𝑦𝑣 𝑧𝑣 𝑢 = 𝑥𝑢 𝑋 + 𝑦𝑢 𝑌 + 𝑧𝑢 𝑍
𝑥𝑤 𝑦𝑤 𝑧𝑤
Steps
• gluLookAt(eyex, eyey, eyez, centerx,
centery, centerz, upx, upy, upz)
– Camera is at eye, looking at center, with the up
direction being up

• First, create a coordinate frame for the camera

• Define a rotation matrix
• Apply appropriate translation for camera (eye)
location
Translation
• gluLookAt(eyex, eyey, eyez, centerx,
centery, centerz, upx, upy, upz)
– Camera is at eye, looking at center, with the up
direction being up

• Cannot apply translation after rotation

• The translation must come first (to bring camera to
origin) before the rotation is applied
Combining Translations, Rotations
𝑃′ = (𝑅𝑇)𝑃 = 𝑀𝑃 = 𝑅(𝑃 + 𝑇) = 𝑅𝑃 + 𝑅𝑇

𝑅11 𝑅12 𝑅13 0 1 0 0 𝑇𝑥

𝑅 𝑅22 𝑅23 0 0 1 0 𝑇𝑦
Với: 𝑀 = 21
𝑅31 𝑅32 𝑅33 0 0 0 1 𝑇𝑧
0 0 0 1 0 0 0
𝑅3×3 𝑅3×3 𝑇3×1
=
01×3 1
gluLookAt final form
𝑥𝑢 𝑦𝑢 𝑧𝑢 0 1 0 0 −𝑒𝑥
𝑥𝑣 𝑦𝑣 𝑧𝑣 0 0 1 0 −𝑒𝑦
𝑥𝑤 𝑦𝑤 𝑧𝑤 0 0 0 1 −𝑒𝑧
0 0 0 1 0 0 0 1

𝑥𝑢 𝑦𝑢 𝑧𝑢 −𝑥𝑢 𝑒𝑥 − 𝑦𝑢 𝑒𝑦 − 𝑧𝑢 𝑒𝑧
𝑥𝑣 𝑦𝑣 𝑧𝑣 −𝑥𝑣 𝑒𝑥 − 𝑦𝑣 𝑒𝑦 − 𝑧𝑣 𝑒𝑧
= 𝑥 𝑦𝑤 𝑧𝑤 −𝑥𝑤 𝑒𝑥 − 𝑦𝑤 𝑒𝑦 − 𝑧𝑤 𝑒𝑧
𝑤
0 0 0 1