GEOMETRIC TRANSFORMATIONS

Yingcai Xiao

Roadmap to Geometric Transformations

Start with 2D GT Extend to 3D GTPreview mathematicsExpress GT in four type of

notations◦Algebraic◦Matrix◦Logical◦Visual

 Review of Mathematical Preliminaries

Mathematical NotationsPoints:

0

00

yx

P

1

11 y

xP

Vectors: directional lines

P1(x1,y1)

P0(x0,y0)

01

0110 yy

xxPPV

Vy

Vx

yx

V

1. A vector has a direction and a length: Length: |V|=(x2 + y2)1/2 A unit vector: |V| = 1 e.g.:

Normalize a vector:

2. Add two vectors

01

x

10

y

VV

v

v

yx

V

w

w

yx

W

wv

wv

yyxx

WV

3. Scalar Multiplication

yx

yx

V

4. Dot product of two vectors scalar,

wvwv yyxxWV

WVWV

)cos(

V

W

•

V W if = 90 => cos = 0 => V W = 0. if < 90 => cos > 0 => V W > 0. if > 90 => cos < 0 => V W < 0.

5.Normal: a unit vector perpendicular to a surface

. Lines y = mx+b e.g.: y = x;  ax+by+c=0

Let f(x,y) = ax + by + cA point p(xp,yp) is on the line if f(xp,yp) = 0.When b < 0:p(xp,yp) is above the line if f(xp,yp) < 0.p(xp,yp) is below the line if f(xp,yp) > 0.When b > 0:p(xp,yp) is above the line if f(xp,yp) > 0.p(xp,yp) is below the line if f(xp,yp) < 0.

Parametric Form: P(t)=P0+t(P1-P0); 0 <= t <= 1

P1(x1,y1)

P0(x0,y0)

P(t)=P0+t(P1-P0)

y(t)x(t)

y0

x0

y1

x1 t *(

y0

x0

)(*)(*

)()(

01

01

0

0

yytxxt

yx

tytx

Where,x(t) = x0 + t * (x1 – x0)y(t) = y0 + t * (y1 – y0)0 <= t <= 1

x(t) = (1-t) * x0 + t * x1 y(t) = (1-t) * y0 + t * y10 <= t <= 1

2D Transformations Translate a point The algebraic representation of translation of point P(x,y) by D(dx,dy) is x’= x + dx y’= y + dy Its “matrix” representation is

x’

y’

d x

d y

x

y

Its logical representation is

P’ = P + D Its visual representation is

 

P’ (x’,y’)

P(x,y) D(dx,dy)

To move a shape: translate every vertex of the shape

0 5 10

5

10

X

Y

Before Translation

(4,5) (7,5)

After Translation

0 5 10

5

10

X

Y

(7,1)(10,1)

Scaling (relative to the origin) Scale a point P(x,y) by S(sx, sy)

Algebraic: x’=sx* x

y’=sy* y

Matrix:

s

s

y

x

0

0

S

yx

ss

yx

y

x

00

''

Logic: P’ = SP or P’ = S(sx, sy)P

Scale a line P0P1 (scale each point) P0’=SP0

P1’=SP1

Scale a shape: scale every vertex of the shape. Visual representation

0 5 10

5

10

X

Y

Before Scaling

(4,5) (7,5)

0 5 10

5

10

X

Y

After Scaling

(2,5/4)

(7/2,5/4)

Uniform Scaling: sx=sy

Rotate (around the origin)Positive angles are measured counterclockwise from x axis to y axis.  Rotate point (x,y) around the originAlgebraic representation:

x’ = x * cos - y * siny’ = x * sin + y * cos

Matrix representation:

Logic representation:

P’ = R P

Visual representation

0 5 10

5

10

X

Y

0 5 10

5

10

X

Y

Rotate a shape: rotate every vertex of the shape

After Before

0 5 10

5

10

X

Y

(2.1, 4.9)

(4.9, 7.8)

0 5 10

5

10

X

Y

(5,2) (9,2)

Before Rotation After Rotation

Summary Translation: P’ = P + D Scaling: P’ = S P Rotation: P’ = R P

1yx

wywxw

yx

Homogeneous Coordinates: P(x,y,w) (x,y) to (x,y,w) :

(x,y,w) to (x,y)

1

/

/

w y

w x

w

y

x

),(

1001001

dydx

dydxT

' P T P

1 0 0

1 0

0 1

dy

dx

1

y

x

1

dy y

dx x

y’= y + dyw’=1x’= x + dx

P’’ = T(dx2 , dy2)P’ = T(dx2 , dy2) T(dx1 , dy1) P = T(dx2+dx1, dy2+dy1) P

Scaling P’=S(Sx, Sy)P

11000000

W''

yx

ss

yx

y

x

P’’ = S(Sx2 , Sy2)P’ = S(Sx2 , Sy2)S(Sx1 , Sy1) P = S(Sx2Sx1, Sy2Sy1)P

P’’=R(2)R(1)PP”=R(2+1)P

Rotation   P’=R()P

1 1 0 0

0 cos sin

0 sin cos

1

' '

y

x

y

x

Shear Transformation: SHx(a) and SHy(b) P’=SHx(a)P

110001001

'''

yxa

wyx

Shear in x against y by a (or an angle). x’ = x+ay y’ = y w’=1

P’=SHy(b)P

110001001

'''

yx

bwyx

Shear in y against x by b (or an angle). x’=x y’=y + bx w’=1.

0 5 10

5

10

X

Y

0 5 10

5

10

X

Y

Sheared in x Sheared in y

0 5 10

5

10

X

Y

Before Shear

Rigid-body Transformation: T and R.change: location, orientation; not change: size, angle between elements.Affine Transformation: S, SHchange: size, location, angle; not change: line (parallelism).Note: uniform scaling is between the two. It

changes size but not angle.So far, our S, R, SH are all around the origin.

Composition of 2D Transformation

Rotate the house around P1For every vertex (P) on the object: P’ = T(P1)R()T(-P1)Py y

x

After translation

of P1 to origin

y

xAfter rotation

y

x

P1

After translation

To P1

Scale and rotate the house around P1 and move it to P2y

xP1

Original

house

y

xP1

Translate P1

to originy

x

Scale

y

x

Rotate

y

x

P2

Translate to

final position P2

For every vertex (P) on the object: P’ = T(P2) R()S(Sx,Sy)T(-P1)P. Swap the order of operations: not permitted, except uniform scaling (sx=sy) can be swapped

with rotation.

Window-to-Viewport Transformationworld-coordinate: inches, feet etc & screen-coordinate: pixels

y

World coordinates

Windowx

y

Screen coordinatesWindow

x

y

World coordinatesWindow x

y

Screen coordinates

Viewport 1 x

Viewport 2

Window in

world coordinates

Window translated

to origin

Window scale to

size of viewportTranslated by (u,v)

to final position

Maximum range of

screen coordinates

Matrix Representation of 3D Transformation

x

y

z

Translation:

1 0 00 1 0

( , , )0 0 10 0 0 1

dxdy

T dx dy dzdz

Scaling: 0 0 00 0 0

( , , )0 0 00 0 0 1

x

yx y z

z

ss

S s s ss

Rotation, around Z-axis:

cos sin 0 0sin cos 0 0

( )0 0 1 0 0 0 0 1

Rz

Rotation, around X-axis:

1 0 0 00 cos sin 0

( )0 sin cos 0 0 0 0 1

Rx

Rotation, around Y-axis:

cos 0 sin 00 1 0 0

( )-sin 0 cos 0 0 0 0 1

Ry

Projection: Project 3D objects on to a 2D surface

(0,0,0)

d z

z

P( x, y, z )x

P’(xp, yp, zp)

xp

Projection Plane (View Plane)

( / ) 1p

p

x x d x xxd z d z d z d

( / ) 1p

p

y y d y yyd z d z d z d

xp= x’ / w’ = x / (z/d + 1)yp= y’ / w’ = y / (z/d + 1)

1 0 0 0'0 1 0 0'0 0 0 0'

1' 10 0 1

x xy yz zw

d

''' 0' ( / ) 1

x xy yzw z d

 Foreshortening: The size of the projected object becomes

smaller when the object moves away from the eye.

z

P’1

P’2

P1 P2

Perspective Projection: the projection that has the foreshortening effect.

' '

1 0 0 00 1 0 0

; ' ;0 0 0 010 0 1

per perM P M P

d

0

For computer-aided design (CAD), we can not have foreshortening. A meter long object should always measures to 1 meter regardless where it is.

Parallel Projection: d , lines of sight become parallel

View Plan

z

P1

P2

(0,0,0)

0

1 0 0 00 1 0 0

;0 0 0 00 0 0 1

par p pM x x y y

Summary2D GTHomogeneous coordinates3D GTRigid body transformationAffine transformation Perspective Projection &

Foreshortening Parallel Projection