Wk4 Basic Transf.pptcs4600/lectures/Wk4_Basic_Transf.pdf · Utah School of Computing 17 ... cos...

Utah School of Computing Fall 2015

Computer Graphics CS4600 1

Transformations I

CS4600 Computer Graphics

From: Rich RiesenfeldFall 2015

Javascript and Canvas

Utah School of Computing

translate (x,y)Moves the canvas and its origin on the grid. x indicates the horizontal distance to move, and y indicates how far to move the grid vertically.

rotate(angle)Rotates the canvas clockwise around the current origin by the angle number of radians.

scale(x, y)Scales the canvas units by x horizontally and by y vertically. Both parameters are real numbers. Values that are smaller than 1.0 reduce the unit size and values above 1.0 increase the unit size. Values of 1.0 leave the units the same size.

Order mattersConsider:translate (5,0);scale(2,2);

Order mattersConsider:scale(2,2);translate (5,0);

Utah School of Computing 7

Transformations and Matrices

• Transformations are functions

• Matrices are function representations

• Matrices represent linear transf’s

• We will use matrices for transf’s

2 2 Matrices 2 Linear Transf'sx D

Rocket

How to form a rocket?Model with triangles

Rocket

How to form (model) a rocket?

How to move a rocket?

What is a 2D Linear Transf ?

Recall from Linear Algebra:

: ( ) ( ) ( ) ,

for scalar and vectors and .

Def T ax y aT x T y

Look at a Diagram( )T

( ) ( )

aT x bT y

T ax T b y

, ax b y

( ), ( )T x T y

a b a b

Scale in x by 2: S2x(v )

0 1 0 1 0 1 0 1

0 0 1 1

Scale in , by 2, say:

2 , 2 2 ,

2 , 2 ,

x x y y x x y y

x y x y

Example:

Scale in x by S2x(v )

What is the graphical view?

), 00( yx ), 002( yx

), 11( yx ), 112( yx

Scale in x by 2: S2x(v )

), 002( yx

), 112( yx

yyxx 1010 ,22

), 00( yx

), 11( yx

yyxx 1010 ),(2

y yyxx 1010 ),(

yyxx 1010 ),(2

yyxx 1010 ),(

yyxx 1010 ),(2

Summary on Scale

• “Scale then add,” is same as

• “Add then scale”

Same results

Matrix Representation

Matrix Representation of S2y(v )

Scale in y by 2: S2y(v )

Matrix Representation S2(v)

2 0 22

Overall Scale by 2: S2(v)

Matrix Form of Same

Add and , then scale

Scale and , then add

0 1 0 1

2 2x y

x x x x

y y y y

What about Rotation?

Is it linear?

Rotate by

Rotate by : 1st Quadrant

cos ,sin

Rotate by : 1st Quadrant

)sin,(cos)0,1(

Rotate by : 2nd Quadrant

)cos,sin()1,0(

Summary of Rotation by

(1,0) (cos ,sin )

)cos,sin()1,0(

Summary (Column Form)

Using Matrix Notation

cossin

sin-cos

cossin

sin-cos

(Note that unit vectors simply copy columns)

General Rotation by Matrix

cos -sin cos -ysin

sin cos sin cos

What do the off diagonal elements do?

Off Diagonal Elements

y xb b y

Example 1

)4.0,1(

),()4.1,1(

Example 2

1( , )

xT x y

Example 2

0.6( , )

x yT x y

Example 2

( , ) 0.6x y

T x yy

)0,1()0,0(

(0.6,1) (1.6,1)

SummaryShear in x:

Shear in y:

xaShx 10

bShy 1

Double Shear: not commutative!

ab)(10

1 ab)(

“Lazy 1”

Translation in x

1 10 0 1

x xx x

Translation in y

0 0 1y y

y yd d

Homogeneous Coordinates

, for 0x

Homogeneous term affects overall scaling

0 10 1

x x xx

y y yy

Homogeneous CoordinatesAn infinite number of points correspond to (x,y,1).

They constitute the whole line (wx,wy,w).

(wx,wy,w)

(x,y,1) = (wx/w, wy/w, w/w)

What does a shear do?

Translation in w=1

1( , , )x y

(wx,wy,w) (wx,wy’,w)

Using Homogeneous Coord’s

• Shear in 3D

• Effects translation in 2D

• We have used a lineartransformation (shear) in 3D to effect a nonlinear transformation(translation) in 2D

Translation by : ( ) d T x x d

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

T u v u v d

T u T v u d v d

T u v d

T u v T u T v

Lots Going On Here!

We’ve got AffineTransformations:

Linear Map + Translation

Compound Transformations

• Build up compound transformations

by concatenating elementary ones

• Use for complicated motion

• Use for complicated modeling

Elementary Transformations

• Scale: Sx(v ), Sy

• Rotate: Rx(v ), Ry

• Translate: Tdx(v ), Tdy

• Shear: Shx(v ), Shy

• Reflect: Rfx(v ), Rfy(v )

Refection about y-axis

Reflection about y-axis

),( yx),( yx

)0,1()0,1(

Reflection about x-axis

1 0 0 0

0 -1 1 1

Reflection about x-axis

)1,0( ),( yx

),( yx )1,0(

Is Reflection “Elementary”?

• Can we effect reflection in an elementary way?

• (More elementary means scale, shear, rotation, translation.)

Reflection = Scale (-1)

),( ba

Ex: Advance clock hands

cos -sin cos -ysin

sin cos sin cos

),( ba

Ex: Advance clock hands: 30mins

What to do?

cos -sin cos -ysin

sin cos sin cos

),( ba

Rotate hr hand by ?Rotate min hand by -180

cos -sin cos -ysin

sin cos sin cos

),( ba

Rotate hr hand by -15Rotate min hand by -180

cos -sin cos -ysin

sin cos sin cos

),( ba

cos -sin cos -ysin

sin cos sin cos

),( ba

cos -sin cos -ysin

sin cos sin cos

),( ba

cos -sin cos -ysin

sin cos sin cos

),( ba

Clock Transformations

• Translate to Origin

• Move hand with rotation

• Translate hand back to clock

• Do other hand

),(12),(

),()(),(

twhere

baTtRbaTT

Translate Back Rotate About Origin Translate to Origin

1 0 1 0

0 1 0 1

1 1 1 1

cos( ) sin( ) 0

sin( ) cos( ) 0

0 0 0 0 0 0

xa t t a

b t t b y

Rocket Revisited

Map: [a,b] [0,1]

• Translate to Origin

• Map x to translated interval

ababaaba ,0,,

Map: [a,b] [0,1]

)( aT x

abS x1

Map: [a,b] [0,1]

• Normalize the interval

• Map x to normalized interval

abaaab

This is a homogeneous form for 1D

)( aT x

abS x1

Map: [a,b] [0,1]

Map: [a,b] [-1,1]

-1 +1x

Map: [a,b] [-1,1]

• Translate center of interval to origin

• Normalize interval to [-1,1]

a b a bx x

Map: [a,b] [c,d]

• First map [a,b] to [0,1]–(We already did this)

• Then map [0,1] to [c,d]

Map: [0,1] [c,d ]

• Scale [0,1] by [c,d ]

• Then translate by c

• That is, in 1D homogeneous form:

1 cxcdxcdc

All Together: Map: [a,b] [c,d ]

1 xaab

Now Map Rectangles

),(minmin yx

),(maxmax vu

),(minmin vu

),(maxmax yx

Transformation in x and y

minmax

,where

This is Viewport Transformation

• Good for mapping objects from one

coordinate system to another

• This is what we do with windows and

viewports

Example

3D Transformations

• Scale

• Rotate

• Translate

• Shear

)(),(),( dT zdT ydT x

)(),(),( dSh zdSh ydSh x

)(),(),( S zS yS x

)(),(),( R zR yR x

3D Scale in x

111000

3D Scale in y

111000

3D Scale in z

111000

Overall 3D Scale

111000

Overall 3D Scale

Same in x,y and z:

Positive Rotation in 3D?

• Sit at end of given axis

• Look at Origin

• CC Rotation is in Positive direction

3D Positive Rotations

3D Rotation about z-axis by

We have already done this:

00cossin

00sincos

)0,1,0(

)1,0,0(

3D Rotation about x-axis by

0cossin0

0sincos0

)0,0,1(

)1,0,0(

3D Rotation about y-axis by

0cos0sin

0sin0cos

3D Rotation about y-axis by

Example

• Cubev.js

Fall 2014 Utah School of Computing 111

Elementary Transformations• Scale: Sx

(v ), Sy(v )

• Rotate: Rx(v ), Ry

• Translate: Tdx(v ), Tdy

• Shear: Shx(v ), Shy

• Reflect: Rfx(v ), Rfy(v )

The End

Transformations I

Wk4 Basic Transf.pptcs4600/lectures/Wk4_Basic_Transf.pdf · Utah School of Computing 17 ... cos...

Documents

Transcript of Wk4 Basic Transf.pptcs4600/lectures/Wk4_Basic_Transf.pdf · Utah School of Computing 17 ... cos...

17. Numerical Filteringhomepage.ntu.edu.tw/~wttsai/fortran/ppt/17.Numerical_Filtering.pdf · = 0.5×sin()−0.3×cos(2)+0.7×cos(12)+0.5×sin(48)−0.6×cos(64) 0.5×sin() 0.5×sin()−0.3×cos(2)

· PDF file · 2014-02-04cos O sin 9 cos t) 0 o cos 9 —sin 0 0 ... 2.13 Prove the following: (Hint: ... — cos sin cos —sin 2.17 Given that A 3a + + az and B 5ap — 8az, find:

New · 2017-02-25 · 31 IF cm CDu (f stick fixed cos sm —sin v cosig T cos O sin 6 cos 9 — sin cos 0 sin 0 cos sin 0 —sin 0 COS O sine cos B sin 0 cos0 —sine cost) cos cp

Evaluate (a) sin 30°(b) sin 150° (c) sin 60°(d) sin 120° (e) cos 40°(f) cos 140° (c) cos 10°(d) cos 170°

WiFiStudy.com · 2019. 7. 21. · Simplify: cos - tan 9) + sin (I -cot 6) A)tan + cot C)sin 3 + cos ... (I -Sina + I)- C)cosee B)2 . sin 9-2 sin 3 Simplify: sin e + cos 2 ... cogee2

SECOND EDITION ADVANCED ENGINEERINGread.pudn.com/downloads652/ebook/2651962/Advanced...b/2 /2 Sin cos sin 9 sin 4) —Eo sin" T cos rab cos X sin Y sine — cos —z sin = COS —X

sin sin sin tan cos cos cos 2 2 2 ... - Utah State Universitycse.usu.edu/foraffiliates/2007/posters_07/FADS Industry Day Poster... · Utah State University 4130 Old Main Hill Logan,

s3-ap-southeast-1.amazonaws.com...sin 6 a + cos6 a — sin a + cos a sin2 a + cos a sin a + cos a sin a cos a Lòi gal sin 4 x + 6cos2 + 3cos4x+ cos + 6sin2 + 3sin4 b) 1 cot x cot

cos2 cos sin A A A 22 2cos 1 A 1 2sin A - Nicolson Institute Unit Practice.pdf22 2 2 sin sin cos cos sin cos cos cos sin sin sin2 2sin cos cos2 cos sin 2cos 1 1 2sin A B A B A B A

Vidyalankar Basic Mathematics Prelim Question Paper ...vidyalankar.org/file/diploma/prelim_paper_soln/SemI/1_Maths_Soln.pdf · put A = B = , cos 2 = cos cos sin sin = cos 2 sin 2

Tensor decomposition of polarized seismic waves · V TGR=[u, v 2]= 2 4 cos cos cos sin sin cos sin sin sin cos 3 5 If the complex envelope of the source signal is denoted by s(t),

1a 1b - sci.wsu.edu · 1a 1b Table of Trig Functions sin cos tan = sin cos sin cot = cos ( ) sec = 1 cos( ) csc = 1 sin( ) Obtain co-functions in right column from left column by

8-3 sin cos tan.pdf

New - saednews.com · 322F dx A = (sin x) + 1 sin Sin —1 sin e lim (COS (cos (cos (cos dx

Trigonometry. sin, cos, or tan ? 32m x 24 sin, cos, or tan ? x 13 35.

Complex Vibrations & Resonance...Now use the trigonometric identity: sin sin cos sin cos()A = BAB BA 22 2 2,sin cos cos sin x txt yxt A A pp pp lt lt =-æöæö æöæöçç ç ç÷÷

Answers to Chapter 3 - mrvahora.files.wordpress.com... (2.50c) i sin(2.50c)) i 2 √ __ 3 √ __ 7 (cos( 0.71c) i sin( 0.71c)) ... i sin 7___ 12) c 18 cos ... i sin 2___ 7 cos 4___

Subtraction Formula for Cosine - Noorduyn's Math …...Compound Angles cos(x+ y) = cos x cos y — sin x sin y cos(x — y) = cos x cos y + sin x sin y 1. Express each of the following

CHAPTER 5 Analytic Trigonometry · Section 5.1 Using Fundamental Identities 383 32. sec csc csc cot 1 cos 31. sin tan x x 1 sin sin cos sec x 33. 1 cos sin cos sin 1 sec sin tan 1

Trig Func HW€¦ · 3) Find the exact value of sin 110 ° cos 40 ° + cos 110 ° sin 40 ° 4) Find the exact value of sin 310 ° cos 70 ° - cos 310 ° sin 70 °. 5) The expression

Answers to Chapter 3 - mrvahora.files.wordpress.com... (2.50c) i sin(2.50c)) i 2 √ 3 √ 7 (cos( 0.71c) i sin( 0.71c)) ... i sin 7_ 12) c 18 cos ... i sin 2_ 7 cos 4___