w = COI EYE view direction vector u = w ( 010,, ) cross product with y-axis v = w u up vector

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29

. ^{(EQ 1)}

w = COI–EYE view direction vector u = w×(0 1 0, , ) cross product with y-axis

v = w×u up vector

30

x z

center of interest

up vector

view vector

observer

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31

FIGURE 7. Object to world space transformation and the view frustum in world space.

Object Space

World Space

Object transformed into world space

Center of Interest

Observer

Angle of View Hither Clipping Distance

Yon Clipping Distance

Up V ector

z

x y

z

x y

View Vector

View Frustum

32

FIGURE 8.Display pipeline showing transformation between spaces.

Object Space

World Space

x y

z

y

z

x

y

Eye Space

x

z y

Image Space eye at negative infinity

Screen Space x z

parallel lines of sight lines of sight

emanating from observer

x y

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33

.

FIGURE 9. Transformation through spaces using ray casting.

Object Space

World Space y

z

y

x

Screen Space x

z

virtual frame buffer

ray constructed through pixel center

34

(EQ 2)

. (EQ 3)

(EQ 4)

(EQ 5)

. (EQ 6)

(EQ 7)

(EQ 8)

(EQ 9)

x w---- y

w---- z w----

 , , 

  = [x y z w, , , ]

x y z, ,

( ) = [x y z 1, , , ]

x y z 1

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

=

x y z 1

x y z 1 = x y z 1

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

P' = M₁M₂M₃M₄M₅M₆P M = M₁M₂M₃M₄M₅M₆

P' = MP

P' = PM₆^TM₅^TM₄^TM^T₃M₂^TM₁^T M^T = M₆^TM₅^TM^T₄M^T₃M₂^TM₁^T

P' = PM^T

x' y' z' 1

a b c d e f g h i j k m 0 0 0 1

x y z 1

=

x+t_x y+t_y z+t_z

1

1 0 0 t_x 0 1 0 t_y 0 0 1 t_z 0 0 0 1

x y z 1

=

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35

(EQ 10)

(EQ 11)

(EQ 12)

(EQ 13)

(EQ 14)

S_x⋅x Sy⋅y S_z⋅z 1

S_x 0 0 0 0 Sy 0 0 0 0 S_z 0 0 0 0 1

x y z 1

=

S x⋅ S y⋅ S z⋅ 1

x y z 1 S---

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 S---

= =

x y z 1

x' y' z' 1

1 0 0 0

0 cosθ –sinθ0 0 sinθ cosθ 0

0 0 0 1

x y z 1

=

x' y' z' 1

θ

cos 0 sinθ 0

0 1 0 0

θ sin

– 0 cosθ 0

0 0 0 1

x y z 1

=

x' y' z' 1

θ

cos –sinθ0 0 θ

sin cosθ 0 0

0 0 1 0

0 0 0 1

x y z 1

=

36

FIGURE 10. Desired Position and Orientation.

z

y

x

z

y

x

(20, -10, 35)

(23,-14,40)

a) Object space definition b) World space position and orientation of aircraft up vector

Computer Animation: Algorithms and Techniques

37

.

FIGURE 11. Projection of desired orientation vector onto y-z plane.

Y

Z

(-4,5) -4

5 ψ

38

FIGURE 12. Projection of desired orientation vector onto x-z plane.

Z

X (3,5)

5

3 φ

Computer Animation: Algorithms and Techniques

39

FIGURE 13. Global coordinate system and unit coordinate system to be transformed.

z

y

x

Z

X Y

x,y,z - global coordinate system X,Y,Z - deisred orientation defined by

unit coordinate system

40

(EQ 15)

(EQ 16)

(EQ 17)

X = M x⋅ X_x

X_y X_z

M

= 1 0 0

⋅

Y = M y⋅ Y_x

Y_y Y_z

M 0 1 0

⋅

=

Z = M z⋅ Z_x

Z_y Z_z

M 0 0 1

⋅

=

X_xY_xZ_x X_yY_yZ_y X_z Y_z Z_z

M

1 0 0 0 1 0 0 0 1

⋅

=

X_x Y_xZ_x X_y Y_yZ_y X_z Y_z Z_z

M

=

x' y' z' 1

A₁₁ A₁₂ A₁₃ A₁₄ A₂₁ A₂₂ A₂₃ A₂₄ A₃₁ A₃₂ A₃₃ A₃₄

0 0 0 1

x y z 1

=

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41

.

FIGURE 14. Translation of moon out to its initial position on the x-axis.

(r,0,0) x

y

z z

x y

42

FIGURE 15. Rotation by applying incremental rotation matrices to points.

(r,0,0)

1 2 3

R_dy= y-axis rotation of 5 degrees for each point P’ of the moon { record a frame of the animation

P’ = R_dy*P’

for each point P of the moon { P’ = P

z

x y

repeat until (done) {

} } }

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43

FIGURE 16. Rotation by incrementally updating the rotation matrix.

(r,0,0) 1 2

3 R = identity matrix

R_dy= y-axis rotation of 5 degrees

R = R*R_dy

record a frame of the animation P’ = R*P

for each point P of the moon { y

x z

repeat until (done) {

}

}

44

FIGURE 17. Rotation by forming the rotation matrix new for each frame.

(r,0,0) 1 2 3

y = 0

record a frame of the animation P’ = R*P

for each point P of the moon {

R = y-axis rotation matrix of ‘y’ degrees

y = y+5 y

x z

repeat until (done) {

} }

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45

.

FIGURE 18. Orthonormalization.

Step 1: Normalize one of the vectors

Step 2:

Form vector perpendicular (orthogonal) and to one of the other two original vectors Normalize it.

Step 3:

Form the final orthogonal vector by taking the cross product of the two just generated.

Normalize it.

to the vector just normalized by taking cross product of the two.

the original unit orthogonal vectors each other due to repeated transformations have ceased to be orthogonal from

46

FIGURE 19. Direct interpolation of transformation matrix values can result in nonsense.

0 0 1 0 1 0

1 – 0 0

0 0 –1 0 1 0 1 0 0

a) Positive 90 degree y-axis rotation b) Negative 90 degree y-axis rotation

c) Half way between orientation representations 0 0 0

0 1 0 0 0 0

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47

.

FIGURE 20. Fixed angle representation.

z

y

x

z

y

x

48

FIGURE 21. Fixed angle representation of (0,90,0).

x y

z

x y

z

a) Original definition b) (0,90,0) orientation

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49

.

FIGURE 22. Effect of slightly altering values of fixed angle representation (0,90,0).

x y

z (+/-ε,90,0) orientation

x y

z

(0,90+/-ε,0) orientation

x y

z

(0,90,+/-ε) orientation

a) b) c)

50

FIGURE 23. Example orientations to interpolate.

x y

z

(0,90,0) orientation

x y

z

(90,45,90) orientation; the object lies in the y-z plane

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51

.

FIGURE 24. Euler angle representation.

X Y

Z

x y

z

Global coordinate system

Local Coordinate system attached to object yaw

pitch roll

52

(EQ 18)

(EQ 19)

R_y'( )Rβ _x( )α = R_x( )Rα _y( )Rβ _x( )Rα _x(–α)= R_x( )Rα _y( )β

R_z''( )Rγ _y'( )Rβ _x( )α = R_x( )Rα y( )Rβ _z( )Rγ _y(–α)Rx(–β)R_y( )Rβ _x( )α = R_x( )Rα y( )Rβ _z( )γ

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53

FIGURE 25. Euler’s Rotation Theorem implies that, for any two orientations of an object, one can be produced from the other by a single rotation about an arbitrary axis.

X Y

Z

X Y

Z

θ

orientation A orientation B

angle and axis of rotation

54

!

FIGURE 26. Interpolating axis-angle representations.

X Y

Z

θ1

Α₁

Α₂ θ2

B=A₁xA₂

φ φ cos^–1 A₁•A₂

A₁ A₂ ---

 

 

=

A_k = R_B(k⋅φ) A₁ θk = (1–k) θ⋅ 1+k⋅θ2

B = A₁×A₂

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55

.

(EQ 20)

(EQ 21)

(EQ 22)

(EQ 23)

(EQ 24)

(EQ 25)

(EQ 26)

(EQ 27)

(EQ 28)

27.

s₁,v₁

[ ]⋅[s₂,v₂] = [s₁⋅s₂–v₁•v₂,s₁⋅v₂+s₂⋅v₁+v₁×v₂]

0 v, ₁

[ ]⋅[0 v, ₂] = [0 v, ₁×v₂] iff v1•v₂ = 0

q^–1 = (1⁄ q )²⋅[s,–v] where q = s²+x²+y²+z²

q⁄ q

q⁄( q )

v' = Rot v( ) = q^–1⋅ ⋅v q

Rot_q(Rot_p( )v) = q^–1⋅(p^–1⋅ ⋅v p)⋅q ( )pq ^–1⋅ ⋅v ( )pq

( )

=

Rot_pq( )v

=

Rot^–1(Rot v( )) = q⋅(q^–1⋅ ⋅v q)⋅q^–1 = v

q = Rot_{θ,(x y z, , )} = [cos(θ⁄2),sin(θ⁄2)⋅(x y z, , )]

q

– = Rot_{–θ,–(x y z, , )} θ – ⁄2

( )

cos ,sin(( )–θ ⁄2)⋅(–(x y z, , ))

[ ]

=

θ⁄2

( )

cos ,–sin(θ⁄2)⋅(–(x y z, , ))

[ ]

=

θ⁄2

( )

cos ,sin(θ⁄2)⋅x y z, ,

[ ]

=

Rot_{θ,(x y z, , )}

= q

=

56