Lecture computer graphics and virtual reality slides lesson 5 modeling transformation

Trang 4 Transformation Definition■ What is a transformation?□ P′=TP□Transform the coordinates / normal vectors of objects■ Uses□Moving the objects to the desired location in the environm

Lesson 5

Modeling Transformation

Trinh Thanh Trung School of ICT, HUST

Transformation

□ Virtual camera: parallel and perspective projections

Types of transformation

■ Object Transformation alters the coordinates of

each point according to some rule, leaving the

underlying coordinate system unchanged

■ Coordinate Transformation produces a

different coordinate system, and then represents all original points in this new system

Display

Coordinate System

MODELING

TRANSFORMS

Projection Coordinate System

PROJECTION

TRANSFORMS

Modeling

Transformation

Transformation

Transformation

Affine transformation

■ Whole collections of points may be transformed

by the same transformation T, e.g lines or circles

■ The image of a line, L, under T, is the set of all images of the individual points of L

■ For most mappings of interest, this image is still

a connected curve of some shape

■ For some mappings, the result of transforming a line may no longer be a line

lines, and are the most commonly-used

transformations in computer graphics

Affine transformation properties

endpoints of the original line and then draw a

straight line between them

Affine transformation properties

■ Preservation of proportional distances

2D transformations

Matrix representation

■ All affine transformations in 2D can be

generically described in terms of a generic equation

P d

c

b a

Q Q

■ A translation moves an object into a different position in a scene

Note: House shifts position relative to origin

y

x

0 1

x y

x

t

t P

P Q

t T

y

x P

Transformation matrix

■ Scaling multiplies all coordinates

□ Alters the size of an object

Note: House shifts position relative to origin

y

x

0 1

x P

x x

y

x

S P

S P

Q Q

x = r cos , y = r sin  x’ = r cos ( + ), y’ = r sin ( + ) x’ = r ( cos cos - sin sin )

Rotates all coordinates by a specified angle

x new = x old × cosθ – y old × sinθ

y new = x old × sinθ + y old × cosθ

■ Using the trigonometric relations, a point

rotated by an angle q about the origin is given by the following equations:

y

x P

sincos

y x

y x

y

x

P P

P P

Q Q

■ A shearing affects an object in a particular

direction (in 2D, it’s either in the x or in the y

■ A shear in the x direction would be as follows:

■ More generally: a simultaneous shear in both the

x and y directions would be

y

x

P

hP P

Q Q

y x

y

x

gP P

hP P

Q Q

x y

x

t

t P

P Q

Q

10

01

sin

sincos

y

x y

x

P

P θ θ

θ θ

1

y

x y

x

P

P g

h Q

Homogenous representation

■ An affine transformation is composed of a linear

combination followed by a translation

■ Unfortunately, the translation portion is not a

matrix multiplication but must instead be added as

an extra term, or vector – this is inconvenient

□ E.g pivotal point rotation

R (  ) ( − , − ) T ( dx , dy ) R (  ) T ( − dx , − dy ) H

Homogeneous Coordinates

■ A point (x, y) can be re-written in homogeneous

coordinates as (xh, yh, h)

■ The homogeneous parameter h is a

non-zero value such that:

■ We can then write any point (x, y) as (h.x, h.y, h)

■ We can conveniently choose h = 1 so that

y = h

Why use homogeneous coordinates

■ Allow scaling factors to be removed from

equations

■ All of the transformations we discussed

previously can be represented as 3*3 matrices

■ Allows us use matrix multiplication to calculate transformations

Homogeneous Coordinates

1 to both P and Q, and also a third row and column to M, consisting of zeros and a 1

y

x y

x y

x

P

P t

d c

t b a Q

1 0

0 1

y

x

t

t M

0 0

0 0

y

x

S

S M

0

0 cos sin

sin cos

0 1

0 1

g

h M

Translation by {tx, ty}

Scale by Sx, Sy Shear by g, h:

Rotate by :

Homogeneous Translation

■ The translation of a point by (dx, dy) can be

written in matrix form as:

0

1 0

0 1

dy dx

+ +

* 1

* 0

* 0

1

*

* 1

* 0

1

*

* 0

* 1

1 1

0 0

1 0

0 1

dy y

dx x

y x

dy y

x

dx y

x y

x dy

dx

Homogenous Coordinates

■ To make operations easier, 2-D points are

written as homogenous coordinate column vectors

100

00

00

y s

x s y

x s

s

y

x y

sin cos

1 1

0 0

0 cos

sin

0 sin

cos

y x

y x

R (  ) ( − , − ) T ( dx , dy ) R (  ) T ( − dx , − dy ) H

0 0

1 0

0 1

1 0

0

0 cos

sin

0 sin

cos

1 0

dx dy

REMEMBER: Matrix multiplication is NOT

commutative so order matters

v dy dx

T R

dy dx

T

Scale → Translate

Translate → Scale

3D transformations

Homogenous coordinates

■ Point representation in 3D

■ Homogenous transformation matrix

■ We are still able to use P’ = R.P

0 0

z y x

d j

h g

d f

e d

d c

b a

P P P

0 0

1 0 0

0 1 0

0 0 1

z y x

d d d

1 0

0 0

1 0 0

0 1 0

0 0 1

z y x

z y x

d z

d y

d x

z y x

d d d

0 0

0 0

0

0 0

0

0 0

0

z y

x

S S

1 0

0 0

0 0

0

0 0

0

0 0

0

z S

y S

x S

z y x

S S

S

z y x

z y

x

down to the coordinate center

x y

z

x y

z

x y

z

Every rotation around the origin can be decomposed into a rotation around the x- axis followed by a rotation around the y-axis followed by

a rotation around the z-axis.

Euler’s Theorem

0 0

0 cos

sin 0

0 sin

cos 0

0 0

0 1

0

0 1 0

0

0 0 cos

sin

0 0 sin

0 0

0 cos

0 sin

0 0

1 0

0 sin

0 cos









Any questions?

Lecture notes provided by School of Information and

Communication Technology, Hanoi University of Science and

Technology.

Composed by Huynh Quyet Thang, Le Tan Hung, Trinh Thanh

Trung and others

Edited by Trinh Thanh Trung

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