cs380: introduction to computer graphics linear transformation...

Min H. Kim (KAIST) Foundations of 3D Computer Graphics, S. Gortler, MIT Press, 2012

CS380: Introduction to Computer GraphicsLinear Transformation

Chapter 2

Min H. KimKAIST School of Computing


GLSL PIPELINERECAP

2


GLSL Pipeline: Vertex Shader

• Vertices are stored in a vertex buffer.• When a draw call is issued, each of the vertices passes

through the vertex shader• On input to the vertex shader, each vertex (black) has

associated attributes.• On output, each vertex (cyan) has a value for gl_Position and

for its varying variables. 3


GLSL Pipeline: Rasterization

• The data in gl_Position is used to place the three vertices of the triangle on a virtual screen.

• The rasterizer figures out which pixels (orange) are inside the triangle and interpolates the varying variables from the vertices to each of these pixels.

4


GLSL Pipeline: Fragment Shader

• Each pixel (orange) is passed through the fragment shader, which computes the final color of the pixel (pink).

• The pixel is then placed in the frame buffer for display.

5


GLSL Pipeline: Fragment Shader

• By changing the fragment shader, we can simulate light reflecting of different kinds of materials.

6


Texture Mapping

• A simple geometric object described by a small number of triangles.

• An auxiliary image called a texture.• Parts of the texture are glued onto each triangle

giving a more complicated appearance. 7


LINEAR TRANSFORMATIONChapter 2

8


Point vs. Vector• Represent Points using coordinates• To perform geometric transformations to these

points• Vectors: 3D motion via linear transformations• Coordinate vector: the position of the point

9

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


Points Vector Coordinatesystem

Coordinate vector

Point vs. Coordinate Vector

1. Point (geometric object): notated as (tilde above the letter), non-numerical object.

2. Vector (motion): notated as (arrow above the letter), non-numerical object.

3. Coordinate system: denoted as (bold: column vector, t makes it transpose), non-numerical object basis for vector; frame for point

4. Coordinate vector: noted as (bold letter), numericalobject

10

!p

!v

c

!f t


Vector Space• A vector space is some set of elements • NB: Vector (motion) is NOT just a set of three

numbers!!!• If a set of vectors is not linearly dependent, we call

linearly independent.• If are linearly independent, all vectors

of can be expressed with coordinates of a basis of (a set of ).

• is the dimension of the basis/space11

V !v

!b1...!b2

V ciV

!v = ci!bi

i∑

n

!v

!bi


Vector Space• Free motion in space, 3 dimensional vector• In vector algebra notation:

• a vector • row basis vectors• column coordinate vector

12

c

!v = ci!bi

i∑ =

!b1!b2!b3

⎡⎣⎢

⎤⎦⎥

c1c2c3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

!v

!bt

!v =!btc


Linear Transformation• Linear transformation follows these two

properties:

• Vector transformation (such that the basis is linearly independent):

13

L(!v+ !u)=L(!v)+L(!u)L(α !v)=αL(!v).

!v⇒ L(!v)=L ci

!bi

i∑⎛

⎝⎜

⎞

⎠⎟= ciL(

!bi )

i∑


3-by-3 Transformation• Rewrite the linear transform

• is actually a linear combination of the original basis vectors.

14

!b1!b2!b3

⎡⎣⎢

⎤⎦⎥

c1c2c3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⇒ L(!b1) L(

!b2) L(

!b3)

⎡⎣⎢

⎤⎦⎥

c1c2c3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

L(b1)

L(b1!"!)= b1

!"!b2!"!

b3!"!⎡

⎣⎢⎤⎦⎥

M1,1

M2,1

M3,1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥


3-by-3 Transformation• 3-by-3 matrix:

• Putting all together:

• A matrix to transform one vector to another:

15

L(b1!"!) L(b2

!"!) L(b3

!"!)⎡

⎣⎢⎤⎦⎥= b1!"!

b2!"!

b3!"!⎡

⎣⎢⎤⎦⎥

M1,1 M1,2 M1,3

M2,1 M2,2 M2,3

M3,1 M3,2 M3,3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

.

b1

b2

b3⎡

⎣⎤⎦

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥⇒ b1

b2

b3⎡

⎣⎤⎦

M1,1 M1,2 M1,3

M 2,1 M 2,2 M 2,3

M 3,1 M 3,2 M 3,3

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

c1c2c3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

v= b tc⇒ b tMc


Linear transform of a vector• A vector undergoes a linear transformation

• The matrix M depends on the chosen linear transformation.

16

v= b tc⇒ b tMc

v!⇒ L(v

!)


Inverse transform• Identity matrix

• In 3D graphics, while moving objects around in space, it will seldom make sense to use an non-invertible transform.

17

I =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

MM −1 = M −1M = I .


Linear transform of a basis• A basis undergoes a linear transformation

• Valid to multiply a matrix times a coordinate vector

• change a basis of a vector to

18

b t

⇒ b tM

v= b tc = a tM −1c . a

t= b tM ,

b t

a t


Dot product• Input: two vectors• Output: a real number• dot product = the squared length

• The angle between the two vectors:

19

v⋅w

v 2:= v⋅v

cosθ = v

⋅w

vw .

θ ∈[0...π ]


3D orthogonal basis• Orthogonal vectors:• A right-handed orthogonal coordinate system.

The z axis comes out of the screen (OpenGL). • A left-handed orthogonal coordinate system.

The z axis goes into the screen (DirectX).

20

v⋅w= 0

right-handed

left-handed


Cross product• Input: two vectors• Output: a vector

• where is a unit vector that is orthogonal to the plane spanned by and

• forms a right-handed basis

21

[v,w,n]

n

v

w

v×w:= vwsinθn,


Cross product• In a right-handed orthogonal basis

• We can compute a cross-product as

22

(b tc)× (b

td) =

c2d3 − c3d2c3d1 − c1d3c1d2 − c2d1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

b t


2D Rotation• Let be a 2D right-handed orthonormal

(orthogonal and unit vectors) basis

• Rotated vector

23

v= b1

b2⎡

⎣⎤⎦

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥.

b t

x ' = xcosθ − ysinθy ' = xsinθ + ycosθ .

x 'y '

⎡

⎣⎢⎢

⎤

⎦⎥⎥= cosθ −sinθ

sinθ cosθ⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢⎢

⎤

⎦⎥⎥.


3D Rotation • Every rotation fixes an axis of rotation and

rotates by some angle about that axis.• Rotation around the z axis:

24

b1

b2

b3⎡

⎣⎤⎦

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⇒ b1

b2

b3⎡

⎣⎤⎦

cosθ −sinθ 0sinθ cosθ 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


3D Rotation• Rotation around the x axis

• Rotation around the y axis

25

1 0 00 cosθ −sinθ0 sinθ cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cosθ 0 sinθ0 1 0

−sinθ 0 cosθ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


xyz-Euler angle rotation• Axis of rotation

• xyz-Euler angle rotation matrix

where26

kx2v + c kxkyv − kzs kxkzv + kys

kykxv + kzs ky2v + c kykzv − kxs

kzkxv − kys kzkyv + kxs kz2v + c

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

k= kx ,ky ,kz⎡⎣ ⎤⎦

t

c := cosθ , s := sinθ , v := 1− c.


Scales• Scaling operations

27

b1

b2

b3⎡

⎣⎤⎦

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⇒ b1

b2

b3⎡

⎣⎤⎦

α 0 00 β 00 0 γ

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

xyz

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

cs380: introduction to computer graphics linear transformation...

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