Transformations 2D Transformations - Drexel CCIdavid/Classes/CS536/Lectures/L-18... · 2016. 11. 16. · 2D Transformations 4 2D Affine Transformations All represented as matrix operations

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CS 536 Computer Graphics

Transformations Week 9, Lecture 18

David Breen, William Regli and Maxim Peysakhov Department of Computer Science

Drexel University

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2D Transformations

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2D Affine Transformations

All represented as matrix operations on vectors! Parallel lines preserved, angles/lengths not

•  Scale •  Rotate •  Translate •  Reflect •  Shear

Pics/Math courtesy of Dave Mount @ UMD-CP 5

2D Affine Transformations

•  Example 1: rotation and non uniform scale on unit cube

•  Example 2: shear first in x, then in y Note: –  Preserves parallels –  Does not preserve

lengths and angles 1994 Foley/VanDam/Finer/Huges/Phillips ICG

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2D Transforms: Translation

•  Rigid motion of points to new locations

•  Defined with column vectors: as

1994 Foley/VanDam/Finer/Huges/Phillips ICG

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•  Stretching of points along axes: In matrix form: or just:

2D Transforms: Scale


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•  Rotation of points about the origin

Positive Angle: CCW Negative Angle: CW

Matrix form: or just:

2D Transforms: Rotation


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2D Transforms: Rotation •  Substitute the

1st two equations into the 2nd two to get the general equation


rcos(θ + φ) rcos(φ)

€

x'= x cos(θ) − y sin(θ)y'= x sin(θ) + y cos(θ)

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Homogeneous Coordinates

•  Observe: translation is treated differently from scaling and rotation

•  Homogeneous coordinates: allows all transformations to be treated as matrix multiplications

Example: A 2D point (x,y) is the line (x,y,w), where w is any real #, in 3D homogenous coordinates. To get the point, homogenize by dividing by w (i.e. w=1)


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Recall our Affine Transformations

Pics/Math courtesy of Dave Mount @ UMD-CP

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Matrix Representation of 2D Affine Transformations

•  Translation:

•  Scale:

•  Rotation:

•  Shear: Reflection: Fy=

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Composition of 2D Transforms •  Rotate about a point P1

–  Translate P1 to origin –  Rotate –  Translate back to P1


P’ = T ✴ P T

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Composition of 2D Transforms •  Scale object around point P1

–  P1 to origin –  Scale –  Translate back to P1

–  Compose into T


P’ = T ✴ P 15

Composition of 2D Transforms •  Scale + rotate object

around point P1 and move to P2 –  P1 to origin –  Scale –  Rotate –  Translate to P2


P’ = T ✴ P

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•  Be sure to multiple transformations in proper order!

Composition of 2D Transforms

P’ = T ✴ P

P’ = ((T✴ (R✴ (S ✴ T)))✴ P)

P’ = (T✴(R✴(S✴ (T✴ P))))

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The Window-to-Viewport Transformation

•  Problem: Screen windows cannot display the whole world (window management)

•  How to transform and clip: Objects to Windows to Screen

Pics/Math courtesy of Dave Mount @ UMD-CP

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Window-to-Viewport Transformation

•  Given a window and a viewport, what is the transformation from WCS to VPCS?

Three steps: •  Translate •  Scale •  Translate


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Transforming World Coordinates to Viewports

•  3 steps 1.  Translate 2.  Scale 3.  Translate

Overall Transformation:


€

P'= MwvP

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3D Transformations

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Representation of 3D Transformations

•  Z axis represents depth •  Right Handed System

–  When looking “down” at the origin, positive rotation is CCW

•  Left Handed System –  When looking “down”, positive

rotation is in CW –  More natural interpretation for

displays, big z means “far” (into screen)


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3D Homogenous Coordinates

•  Homogenous coordinates for 2D space requires 3D vectors & matrices

•  Homogenous coordinates for 3D space requires 4D vectors & matrices

•  [x,y,z,w] 27

3D Transformations: Scale

•  Scale –  Parameters for each

axis direction

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3D Transformations: Translate

•  Translation

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3D Transformations: Rotation

•  One rotation for each world coordinate axis

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Rotation Around an Arbitrary Axis

•  Rotate a point P around axis n (x,y,z) by angle θ

•  c = cos(θ) •  s = sin(θ) •  t = (1 - c)

Graphics Gems I, p. 466 & 498 €

R =

tx 2 + c txy + sz txz − sy 0txy − sz ty 2 + c tyz + sx 0txz + sy tyz − sx tz2 + c 00 0 0 1

#

$

% % % %

&

'

( ( ( (

•  Also can be expressed as the Rodrigues Formula

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Prot = P cos(ϑ ) + (n× P)sin(ϑ ) + n(n ⋅ P)(1− cos(ϑ ))

Rotation Around an Arbitrary Axis

Improved Rotations

•  Euler Angles have problems – How to interpolate keyframes? – Angles aren’t independent –  Interpolation can create Gimble Lock, i.e.

loss of a degree of freedom when axes align

•  Solution: Quaternions!

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€

p = (0, x )

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slerp – Spherical linear interpolation Need to take equals steps on the sphere

A & B are quaternions

What about interpolating multiple keyframes?

•  Shoemake suggests using Bezier curves on the sphere

•  Offers a variation of the De Casteljau algorithm using slerp and quaternion control points

•  See K. Shoemake, “Animating rotation with quaternion curves”, Proc. SIGGRAPH ’85

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3D Transformations: Reflect

•  Reflection:

about x-y plane

about y-z plane Fx =

−1 0 0 00 1 0 00 0 1 00 0 0 1

"

#

$$$$

%

&

''''

Reflection

corresponds to negative scale factors

E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

original sx = -1 sy = 1

sx = -1 sy = -1 sx = 1 sy = -1

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3D Transformations: Shear

•  Shear: (function of y)

Hx =

1 shx 0 00 1 0 00 0 1 00 0 0 1

!

"

####

$

%

&&&&

x ' = x +shx ⋅ y

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3D Transformations: Shear

€

X 'Y 'Z '1

"

#

$ $ $ $

%

&

' ' ' '

=

1 Shxy Shx

z 0Shy

x 1 Shyz 0

Shzx Shz

y 1 00 0 0 1

"

#

$ $ $ $

%

&

' ' ' '

∗

XYZ1

"

#

$ $ $ $

%

&

' ' ' '

X ' = X + ShxyY + Shx

zZ

Y ' = ShyxX +Y + Shy

zZ

Z ' = ShzxX + Shz

yY + Z

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Example: Composition of 3D Transformations

•  Goal: Transform P1P2 and P1P3


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Example (Cont.)

•  Process 1.  Translate P1 to (0,0,0) 2.  Rotate about y 3.  Rotate about x 4.  Rotate about z

(1)

(2-3)

(4)


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Final Result •  What we’ve really

done is transform the local coordinate system Rx, Ry, Rz to align with the origin x,y,z


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Example 2: Composition of 3D Transformations

•  Airplane defined in x,y,z •  Problem: want to point it

in Dir of Flight (DOF) centered at point P

•  Note: DOF is a vector •  Process:

–  Rotate plane –  Move to P


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Example 2 (cont.)

•  Zp axis to be DOF •  Xp axis to be a horizontal

vector perpendicular to DOF –  y x DOF

•  Yp, vector perpendicular to both Zp and Xp (i.e.Zp x Xp)


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Transformations to Change Coordinate Systems

•  Issue: the world has many different relative frames of reference

•  How do we transform among them? •  Example: CAD Assemblies & Animation Models

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Transformations to Change Coordinate Systems

•  4 coordinate systems 1 point P

)45()8.1,7.6(

)5.0,5.0()3,2()2,4(

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RTMSTM

TM

•=

•=

=

←

←

←


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Coordinate System Example (1)

•  Translate the House to the origin

€

M1←2 = T(x1,y1)

M2←1 = M1←2( )−1

= T(−x1,−y1)


The matrix Mij that maps points from coordinate system j to i is the inverse of the matrix Mji that maps points from coordinate system j to coordinate system i.

P1

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Coordinate System Example (2) •  Transformation

Composition:

€

M5←1 = M5←4 •M4←3 •M3←2 •M2←1


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World Coordinates and Local Coordinates

•  To move the tricycle, we need to know how all of its parts relate to the WCS

•  Example: front wheel rotates on the ground wrt the front wheel’s z axis: Coordinates of P in wheel coordinate system:

1994 Foley/VanDam/Finer/Huges/Phillips ICG €

P(wo) = T(αr,0,0) ⋅ Rz(α) ⋅ P(wh )

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P'(wh )= Rz (α) ⋅ P(wh )

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Properties of Transformation Matrices

•  Note that matrix multiplication is not commutative

•  i.e. in general M1M2 ≠ M2M1

•  T – reflection around y axis •  T’ – rotation in the plane

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Transformations 2D Transformations - Drexel CCIdavid/Classes/CS536/Lectures/L-18... · 2016. 11. 16. · 2D Transformations 4 2D Affine Transformations All represented as matrix operations