lecture 4: transformation cont.lingqi/teaching/...• transformation matrix?-translate (center to...

Transformation Cont.Lecture 4:

CS180 / 280: Introduction to Computer GraphicsWinter 2021, Lingqi Yan, UC Santa Barbara

http://www.cs.ucsb.edu/~lingqi/teaching/cs180.html

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara

Announcement• Homework 1 will be released TODAY

- Start early

• The next 2 lectures will be difficult :)

2


Last Lecture

• Transformation- Why study transformation

- 2D transformations: rotation, scale, shear

- Homogeneous coordinates

- Composite transform

3


Today• Recap of useful math

• 3D transformations

• Viewing transformation- View / Camera transformation

- Projection transformation

- Orthographic projection

- Perspective projection

4


Recap: Orthogonal Matrix

5


Recap: “Extracting a column”

6

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D E F

G H I

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CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara7 Ren NgCS184/284A

3D Transformations

Use homogeneous coordinates again:

• 3D point = (x, y, z, 1)T

• 3D vector = (x, y, z, 0)T

Use 4×4 matrices for affine transformations�

⇧⇧⇤

x�

y�

z�

1

⇥

⌃⌃⌅ =

�

⇧⇧⇤

a b c txd e f tyg h i tz0 0 0 1

⇥

⌃⌃⌅ ·

�

⇧⇧⇤

xyz1

⇥

⌃⌃⌅

In general, (x, y, z, w) (w != 0) is the 3D point:

(x/w, y/w, z/w)

CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara8Ren NgCS184/284A

3D Transformations

Scale

Translation

Coordinate Change(Frame-to-world)

T(tx, ty, tz) =

�

⇧⇧⇤

1 0 0 tx0 1 0 ty0 0 1 tz0 0 0 1

⇥

⌃⌃⌅

S(sx, sy, sz) =

�

⇧⇧⇤

sx 0 0 00 sy 0 00 0 sz 00 0 0 1

⇥

⌃⌃⌅

F(u,v,w,o) =

u v w o0 0 0 1

�


3D Transformations

Rotation around x-, y-, or z-axis

Rx(�) =

�

⇧⇧⇤

1 0 0 00 cos � � sin � 00 sin� cos � 00 0 0 1

⇥

⌃⌃⌅

Ry(�) =

�

⇧⇧⇤

cos � 0 sin� 00 1 0 0

� sin � 0 cos � 00 0 0 1

⇥

⌃⌃⌅

Rz(�) =

�

⇧⇧⇤

cos � � sin� 0 0sin � cos � 0 0

0 0 1 00 0 0 1

⇥

⌃⌃⌅

y

z

x

Rotation around x-axis

Anything strange about Ry?


3D Rotations

Compose any 3D rotation from Rx, Ry, Rz?

• So-called Euler angles

• Often used in flight simulators: roll, pitch, yaw

Rxyz(�,⇥, ⇤) = Rx(�)Ry(⇥)Rz(⇤)


Rodrigues’ Rotation Formula

Rotation by angle α around axis n

How to prove this magic formula?

• Matrix N computes a cross-product: N x = n × x

• Assume orthonormal system e1, e2, n

R(n,↵) = cos(↵) I + (1� cos(↵))nnT + sin(↵)

0

@0 �nz ny

nz 0 �nx

�ny nx 0

1

A

| {z }N

Rn = nRe1 = cos � e1 + sin� e2

Re2 = � sin � e1 + cos � e2

Check out the supplementary material on the course website!


Today

12







View / Camera Transformation

• What is view transformation?

• Think about how to take a photo- Find a good place and arrange people (model transformation)

- Find a good “angle” to put the camera (view transformation)

- Cheese! (projection transformation)

13



• How to perform view transformation?

• Define the camera first- Position

- Look-at / gaze direction

- Up direction (assuming perp. to look-at)

14

Position

Up direction

Look-at direction

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View / Camera Transformation• Key observation

- If the camera and all objects move together, the “photo” will be the same

15

==

• How about that we always transform the camera to- The origin, up at Y, look at -Z

- And transform the objects along with the camera

Y

X

-Z(0, 0, 0)



16

• Transform the camera by- So it’s located at the origin, up at Y, look at -Z

• Mview in math?- Translates e to origin

- Rotates g to -Z

- Rotates t to Y

- Rotates (g x t) To X

- Difficult to write!

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Y

X

-Z

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17

• in math?- Let’s write

- Translate e to origin

- Rotate g to -Z, t to Y, (g x t) To X

- Consider its inverse rotation: X to (g x t), Y to t, Z to -g

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WHY?

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1 0 0 �xe

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775

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R�1view =

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18

• Summary- Transform objects together with the camera

- Until camera’s at the origin, up at Y, look at -Z

• Also known as ModelView Transformation

• But why do we need this?- For projection transformation!


Today

19







Projection Transformation• Projection in Computer Graphics

- 3D to 2D



20

Fig. 7.1 from Fundamentals of Computer Graphics, 4th Edition

Orthographicprojection

Perspectiveprojection


Projection Transformation• Perspective projection vs. orthographic projection

21

https://stackoverflow.com/questions/36573283/from-perspective-picture-to-orthographic-picture

https://stackoverflow.com/questions/36573283/from-perspective-picture-to-orthographic-picture


Projection Transformation• Why is the pigeon so big?

22

Pigeon

Wall

People


Orthographic Projection• A simple way of understanding

- Camera located at origin, looking at -Z, up at Y (looks familiar?)

- Drop Z coordinate

- Translate and scale the resulting rectangle to [-1, 1]2

23


Orthographic Projection• In general

- We want to map a cuboid [l, r] x [b, t] x [f, n] to the “canonical” cube [-1, 1]3

24

x

z

y

l r

t

bn

f

x

z

Translate

y

x

z

y

Scale


Orthographic Projection• Slightly different orders (to the “simple way”)

- Center cuboid by translating

- Scale into “canonical” cube

25

x

z

y

l r

t

bn

f

x

z

Translate

y

x

z

y

Scale


Orthographic Projection• Transformation matrix?

- Translate (center to origin) first, then scale (length/width/height to 2)

26

x

z

y

l r

t

bn

f

x

z

Translate

y

x

z

y

Scale

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CS180 / 280, Winter 2021 Lingqi Yan, UC Santa Barbara27

Ren NgCS184/284A

3D Transformations

Use homogeneous coordinates again:

• 3D point = (x, y, z, 1)T

• 3D vector = (x, y, z, 0)T

Use 4×4 matrices for affine transformations�

⇧⇧⇤

x�

y�

z�

1

⇥

⌃⌃⌅ =

�

⇧⇧⇤

a b c txd e f tyg h i tz0 0 0 1

⇥

⌃⌃⌅ ·

�

⇧⇧⇤

xyz1

⇥

⌃⌃⌅

What’s the order? Linear Transform first or Translation first?

Recall: The Implicit Order


Orthographic Projection• Caveat

- Looking at / along -Z is making near and far not intuitive (n > f)

- FYI: that’s why OpenGL (a Graphics API) uses left hand coords.

28

x

z

y

l r

t

bn

f

x

z

Translate

y

x

z

y

Scale

Thank you!

lecture 4: transformation cont.lingqi/teaching/...• transformation matrix?-translate (center to...

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