computer graphics recitation 2. 2 the plan today learn about rotations in 2d and 3d. representing...

Computer Graphics

Recitation 2

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The plan today

Learn about rotations in 2D and 3D. Representing rotations by quaternions.

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Rotation in 2D

Rotation about the origin by angle .

O

Positive angle meanscounter-clockwise direction.

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Rotation in 2D

Positive angle meanscounter-clockwise direction.

O

Rotation about the origin by angle .

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Rotation in 2D – matrix representation

Multiply (x, y) by the rotation matrix:

O

P (x, y)

P’ = R (P)

cossin

sincos

cossin

sincos

)(

yxy

yxx

y

x

y

x

PRP

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Interpolating rotations in 2D

O

We want to generate N intermediate positions:

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O

Divide the arc into N pieces of equal length

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Apply rotations for i = 1, 2, …, N

O

NiR

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O

NiR

10



O

NiR

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O

NiR

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O

NiR

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O

NiR

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O

NiR

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O

NiR

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Rotations mathematically

Rotation is a linear orthogonal transformation that doesn’t change the orientation.

It means: Let v, u be vectors (or points), R the rotation

<Ru, Rv> = <u, v> (length and angle preservation) => ||Ru|| = ||u||

In matrix representation: let R be n n matrix: RRT = I => R-1 = RT

det(R) = +1 The rows of R are orthonormal vectors (unit-length pairwise

orthogonal vectors). They form a “right-hand” basis of Rn.

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2D rotations as complex numbers multiplication

Represent points (x, y) as (x + iy) in C. Multiplying by ei is equivalent to rotation by angle :

ei = cos + i sinei (x + iy) = (cos + i sin) (x + iy) =

(x cos y sin) + i (x sin + y cos)

O

Re

Im

P = x + iy

P’ = ei (x + iy)

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Rotations in 3D

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Rotations in 3D

Euler’s theorem: any orientation can be obtained from a fixed reference orientation by a single unique rotation around an appropriate axis in space.

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Rotations in 3D

We express the rotation in terms of the rotation axis n = (nx , ny , nz), ||n|| = 1, and the rotation angle .

O

n P

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Rotations in 3D

We express the rotation in terms of the rotation axis n = (nx , ny , nz), ||n|| = 1, and the rotation angle .

O

n P

P’

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The formula for P’

Decompose v into to components:

v = v|| + v

v|| = <v, n> n => v|| || n

v = v – v|| => v nO

n

P

v

v

v||

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The rotation doesn’t affect the axis of rotation, therefore:

v’ = Rn, (v) = Rn, (v|| + v) =

= Rn, (v||) + Rn, (v) =

= v|| + Rn, (v)

We only need to rotate v by angle in the plane perpendicular to n.

O

n

P

v

v

v||

P’

v’

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O

n

P

v

v||

The rotation doesn’t affect the axis of rotation, therefore:

v’ = Rn, (v) = Rn, (v|| + v) =

= Rn, (v||) + Rn, (v) =

= v|| + Rn, (v)

We only need to rotate v by angle in the plane perpendicular to n.

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We rotate v in the plane perpendicular to n. The plane = Span{v, n v}. v, n v n

n v v

Therefore,

Rn, (v) = cos v + sin (n v) = cos v + sin (n v)

n v = n (v|| + v) = n v|| + n v = n v

n

v

v|| R (v)

v

n v

R (v) = (cos, sin)

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v’ = Rn, (v) = v|| + Rn, (v) =

= <n, v>n + cos v + sin (n v) =

= <n, v>n + cos (v <n, v>n) +

+ sin (n v) =

= cos v + (1 cos) <n, v> n +

+ sin (n v).

n

v

v||

P’

Rn, (v) = cos v + (1 cos ) <n, v> n + sin (n v).

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We are animating an object by specifying the key transformations (key frames) .

Suppose two consecutive key-frames are specified by rotations R1 and R2.

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How to define an interpolation??

We can try to interpolate the two rotation matrices:

R(t) = (1 t) R1 + t R2

Doesn’t work – the result is not a rotation matrix

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Solution

Use extension of ei to 3D => quaternions Interpolate the quaternions…

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Multiplication by ei in 3D - ???

Hamilton was searching for an extension of complex numbers that would represent 3D rotations.

He first tried to find a field of numbers of the form x +yi + zj, where x, y, z R and i2 = j2 = 1.

This is not a field (not closed under multiplication).

One day, Hamilton realized that he needs four scalars: s +xi + yj + zk ! He called these objects quaternions.

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Quaternions

The ring of quaternions = {s +xi + yj + zk } i2 = j2 = k2 = 1 ij = k, ji = k; jk = i, kj = i; ki = j, ik = j

We will denote q = (s, v) = s +xi + yj + zk where v = (x, y, z)

q1 q2 = (s1, v1) (s2, v2) =

= (s1 s2 <v1, v2 >, s1 v2 + s2 v1 + v1 v2)

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Conjugate and inner product

q = (s1, v1) = s xi yj zk

< q1 , q2 > = q1 q2 = s1 s2 + <v1, v2 >

||q||2 = <q, q> = s2 + x2 + y2 + z2

q1 = q / ||q||2

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Rotating with quaternions

Represent a vector v as vQ = (0, v). Rotating about axis n, ||n|| = 1, by angle :

Rn, (v) = qvQq

q = (cos( /2), sin( /2) n)

Proof:

sincossin2

cos1sin2

,,)(

)(sin,)cos1(cos,0(

)(sincos2,sin2)sin(cos,0(

)sin,)(cos,0)(sin,(cos

22

22

2222

22

22

2222

cbabcacba

vnnvnv

vnnvnv

nvnqvq

:factsbasicfollowingtheUsing

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Some facts about quaternions:

Rotating by q and by q has the same effect. “Rotating” by q such that ||q|| = has a scaling effect. So, we consider the unit quaternions only ||q|| =1 These quaterions live on the unit sphere in four

dimensions… So, we have 3 degrees of freedom for our rotation

quaterion (just like with n, rotation:

1 degree for + 2 degrees for n = 3 degrees of freedom.)

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Now, how do we interpolate?

Given two rotations q1 and q2

Simple interpolation q(t) = (1 t) q1+t q2 is not good enough – doesn’t give “constant speed”:

q1q2

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Now, how do we interpolate?

Given two rotations q1 and q2

Simple interpolation q(t) = (1 t) q1+t q2 is not good enough – doesn’t give “constant speed”

The unit quaternions live on the unit sphere, so we want to draw a great arc from q1 to q2 on that sphere (it gives the shortest path on the sphere).

q1

q2

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Spherical interpolation (slerp)

q1

q2

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21

,cos

]1,0[,sin

)sin(

sin

))1sin(()(

qq

tqt

qt

tq

thatsuchiswhere

q(t)

See you next time

Next week the lesson will be presented by Andrei, you will learn about SVD…

computer graphics recitation 2. 2 the plan today learn about rotations in 2d and 3d. representing...

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