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Representations and

Transformations

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Objectives

•Derive homogeneous coordinate transformation matrices

• Introduce standard transformations- Rotations- Translation- Scaling- Shear

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Scalars, Points, Vectors

•Three basic elements from geometry:Scalars, Points, Vectors

•Scalars can be defined as members of a set which can be combined by the operations of addition and multiplication and obey the fundamental axioms:

associativity, commutivity, inversion•Examples include the real and complex numbers under the rules we are all familiar with

•Scalars alone have no geometric properties

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• A vector is a quantity with two attributes direction & magnitude and its own rules as we saw last lecture

• The set defines a vector space

Scalars, Points, Vectors

•But, vectors lack positionSame length and magnitude ->

•Vectors are insufficient to specify geometryWe need points

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5

•Points, we know, are locations in space

•Certain operations translate between points and vectors

- Point-point subtraction yields a vector

- Leads to equivalent to point-vector addition

P=v+Q

v=P-Q

Points and Vectors

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• Consider a basis v1, v2,…., vn

• A vector is written v=1v1+ 2v2 +….+nvn

• The list of scalars {1, 2, …. n} then is therepresentation of v with respect to the given basis

• And we write the representation as a row or column array of scalars

a=[1 2 …. n]T=

n

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1

.

Vector Spaces

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•Vector spaces do not represent points

• Instead, we work in an affine space and add a special point, the origin, to the basis vectors, this is called a frame

P0

v1

v2

v3

Affine Spaces

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Affine Spaces

•An affine space is both point and vector space

• It allows operations from vectors, points and scalars:

- Vector-vector addition

- Scalar-vector multiplication

- Point-vector addition

- All scalar-scalar operations

• Define our space with a coordinate “Frame”

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•A frame is determined by (P0, v1, v2, v3, ...)•Within this frame:

Every vector can be written as v=1v1+ 2v2 +….+nvn

And every point can be written asP = P0 + 1v1+ 2v2 +….+nvn

Affine Spaces

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Consider the point and the vector

P = P0 + 1v1+ 2v2 +….+nvn

v=1v1+ 2v2 +….+nvn

They appear to have the similar representations

p=[1 2 3] v=[1 2 3]

which confuse the point with the vector

But, a vector has no position v

pv

can be placed anywherefixed

Homogeneous Coordinates

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•An affine space distinguishes point and vector with

( 1 )( P ) = P

( 0 )( P ) = 0 (zero vector)

Homogeneous Coordinates

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With these rules, we can keep track of the difference:

v=1v1+ 2v2 +3v3 = [1 2 3 0 ] [v1 v2 v3 P0] T

P = P0 + 1v1+ 2v2 +3v3= [1 2 3 1 ] [v1 v2 v3 P0] T

Thus we obtain a four-dimensional representation for both:

v = [1 2 3 0 ] T

p = [1 2 3 1 ] T

Homogeneous Coordinates

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Homogeneous Coordinates

• Homogeneous coordinates are key to all computer graphics systems

• Hardware pipeline all work with 4 dimensional representations

• All standard transformations (rotation, translation, scaling) can be implemented by matrix multiplications with 4 x 4 matrices

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Homogeneous Coordinates

•Most generally, the form of homogeneous coordinates is

p=[x y z w] T

•We can return to a simple three dimensional point by

x'x/wy'y/wz'z/w

•But if w=0, the representation is that of a vector

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Transformations

•A transformation maps points to other points and/or vectors to other vectors

Q=T(P)

v=T(u)

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Affine Transformations

•Line preserving

•Characteristic of many physically important transformations

- Rigid body transformations: rotation, translation

- Non-rigid: Scaling, shear

• Importance in graphics is that we need only transform vertices (points) of line segments and polygons, then system draws between the transformed points

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Translation

•Move (translate, displace) a point to a new location

•Displacement determined by a vector d- Three degrees of freedom- P’=P+d

P

P’

d

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Moving objects

When we move a point on an object to a new location, to preserve the object, we must move all other points on the object in the same way

object translation: every point displacedby the same vector, d

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Translation Using Representations

Using the homogeneous coordinate representation in some frame

p=[ x y z 1]T

p’=[x’ y’ z’ 1]T

d=[dx dy dz 0]T

Hence p’ = p + d orx’=x+dxy’=y+dyz’=z+dz

note that this expression is in four dimensions and expressesthat point = vector + point

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Translation Matrix

We can also express translation using a 4 x 4 matrix T in homogeneous coordinatesp’=Tp where

T = T(dx, dy, dz) =

This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

1000

d100

d010

d001

z

y

x

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Rotation (2D)

• Consider rotation about the origin by degrees

- radius stays the same, angle increases by

What is this rotationabout the z axis?

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Rotation (2D)

• Consider rotation about the origin by degrees

- radius stays the same, angle increases by

x’=x cos –y sin y’ = x sin + y cos

x = r cos y = r sin

x = r cos (y = r sin (

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Rotation about the z axis

• Rotation about z axis in three dimensions leaves all points with the same z

- Equivalent to rotation in two dimensions in planes of constant z

- or in matrix notation (with p as a column)

p’=Rz()p

x’=x cos –y sin y’ = x sin + y cos z’ =z

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Rotation Matrix

1000

0100

00 cossin

00sin cos

R = Rz() =

Homogeneous Coordinates:

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Rotation about x and y axes

• Same argument as for rotation about z axis- For rotation about x axis, x is unchanged

- For rotation about y axis, y is unchanged

R = Rx() =

R = Ry() =

1000

0 cos sin0

0 sin- cos0

0001

1000

0 cos0 sin-

0010

0 sin0 cos

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Scaling

1000

000

000

000

z

y

x

s

s

s

S = S(sx, sy, sz) =

x’=sxxy’=syyz’=szz

p’=Sp

Expand or contract along each axis (fixed point of origin)

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Reflection

corresponds to negative scale factors

originalsx = -1 sy = 1

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

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Shear

•Helpful to add one more basic transformation•Equivalent to pulling faces in opposite directions

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Shear Matrix

Consider simple shear along x axis

x’ = x + y shy

y’ = yz’ = z

1000

0100

0010

00sh1 y

H() =

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Inverses

• Although we could compute inverse matrices by general formulas, we can also use simple geometric observations, for example:

- Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)

- Rotation: R -1() = R(-)

• Holds for any rotation matrix

• Note that since cos(-) = cos() and sin(-)= -sin()

R -1() = R T()

- Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

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Composite

•In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size•We apply an composite transformation to its vertices to

Scale OrientLocate

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Composite Transformations

• Scaling about a fixed point- Simply applying the scale transformation

also moves the object being scaled.

P

Q

P'

Q'

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Composite Transformations

• Scaling about origin does not move object• Origin is a fixed point for the scale transformation• We use composite transformations to create scale

transformations with different fixed points

P

Q

P'

Q'

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Composite Transformations

• Fixed point scale transformation• Move fixed point (px,py,pz) to origin• Scale by desired amount• Move fixed point back to original position

M = T(px, py, pz) S(sx, sy, sz) T(-px, -py, -pz)

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Composite Transformations

• Rotating about a fixed point- basic rotation alone will rotate about origin

but we want:

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Composite Transformations

• Rotating about a fixed point• Move fixed point (px,py,pz) to origin• Rotate by desired amount• Move fixed point back to original position

M = T(px, py, pz) Rx() T(-px, -py, -pz)

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Composite Transformations

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Composite transformations

A series of transformations on an object can be applied as a series of matrix multiplications

: position in the global coordinate

: position in the local coordinate

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Interpolation

• In order to “move things”, we need both translation and rotation

• Interpolating the translation is easy, but what about rotations?

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Interpolation of orientation

•How about interpolating each entry of the rotation matrix?

•The interpolated matrix might no longer be orthonormal, leading to nonsense for the in-between rotations

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Interpolation of orientation

Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y

Linearly interpolate each component and halfway between, you get this...

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Motivation

• Finding the most natural and compact way to present rotation and orientations

• Orientation interpolation which result in a natural motion

• A closed mathematical form that deals with rotation and orientations (expansion for the complex numbers)

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•Rotation matrix

•Euler angle

•Axis angle

•Quaternion

•Exponential map

Representing Rotation

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Rotational DOFs are widely used in character animation

3 translational DOFs

48 rotational DOFs

Each joint can have up to 3 DOFs

1 DOF: knee 2 DOF: wrist 3 DOF: arm

Joints and rotations

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Euler Angle Representation

• Angles used to rotate about cardinal axes

• Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about axes, ie. first about x, then y, then z

• Many possible orderings, don’t have to use all 3 axes, but can’t do the same axis back to back

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Euler Angles

• A general rotation is a combination of three elementary rotations: around the x-axis (x-roll) , around the y-axis (y-pitch) and around the z-axis (z-yaw).

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Euler Angles and Rotation Matrices

1000

0100

00cossin

00sincos

)(

1000

0cos0sin

0010

0sin0cos

)pitch(- y

1000

0cossin0

0sincos0

0001

)(

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3

22

22

211

111

yawz

rollx

1000

0

0

0

),,(213132131321

213132131321

23232

321 cccssscsscsc

csccssssccss

ssccc

R

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Gimbal Lock

• A 90 degree rotation about the y axis essentially makes the first axis of rotation align with the third.

• Incremental changes in x,z produce the same results – you’ve lost a degree of freedom

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Gimbal Lock

• Phenomenon of two rotational axis of an object pointing in the same direction.

• Result: Lose a degree of freedom (DOF)

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Gimbal Lock

A Gimbal is a hardware implementation of Euler angles used for mounting gyroscopes or expensive globes

Gimbal lock is a basic problem with representing 3D rotation using Euler angles or fixed angles

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Euler angles interpolation

z

xy

π

x-roll π

y

z

x

y

z

xz-yaw π

z

xy

π

z

xy

π

y-pitch π

R(0,0,0),…,R(t,0,0),…,R(,0,0)

t[0,1]

R(0,0,0),…,R(0,t, t),…,R(0,, )

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Euler Angles Interpolation Unnatural movement !

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•Rotation matrix

•Euler angle

•Axis angle

•Quaternion

•Exponential map

Representing Rotation

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Axis angle

• Represent orientation as a vector and a scalar

- vector is the axis to rotate about

- scalar is the angle to rotate by

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Angle and Axis

• Any orientation can be represented by a 4-tuple- angle, vector(x,y,z) where the angle is the amount to rotate

by and the vector is the axis to rotate about

• Can interpolate the angle and axis separately

• No gimbal lock problems!

• But, can’t efficiently compose rotations…must convert to matrices first!

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•Rotation matrix

•Euler angle

•Axis angle

•Quaternion

•Exponential map

Representing Rotation

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Quaternions

•Extension of imaginary numbers

•Requires one real and three imaginary components i, j, k

q=q0+q1i+q2j+q3k

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Quaternions

•Extension of imaginary numbers

•Requires one real and three imaginary components i, j, k

q = (w, x, y, z)

= cos + sin

x, y, z ] of axis

2

2

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Quaternion interpolation

• Interpolation means moving on n-D sphere

• Now imagine a 4-D sphere for 3-angle rotation

1-angle rotation can be represented by a unit circle

2-angle rotation can be represented by a unit sphere

•Quaternions can express rotations on unit sphere smoothly and efficiently.

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Quaternion interpolation

•Moving between two points on the 4D unit sphere

- a unit quaternion at each step - another point on the 4D unit sphere

- move along the great circle between the two points on the 4D unit sphere as an arc

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Quaternion interpolation

Spherical linear interpolation (SLERP)

Process:-Rotation matrix quaternion

-Carry out slerp/operations with quaternions

-Quaternion rotation matrix

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Matrix form

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Quaternions – points on a 4D unit hypersphere+ better interpolation+ almost unique- less intuitive

Summary

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Quaternion Math

Unit quaternion

Multiplication

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Quaternion Rotation

If is a unit quaternion and

then results in rotating about by

proof: see Quaternions by Shoemaker

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Choose a representation

•Choose the best representation for the task

- input:

- joint limits:

- interpolation:

- compositing:

- rendering:

quaternion or exponential map

Euler angles

orientation matrix ( quaternion can be represented as matrix as well)

quaternions or orientation matrix

Euler angles, quaternion (harder)