Quaternions

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Quaternions
Design and Creation of Virtual Environments
CISE 6930/4930 Section 6589/3146
Benjamin Lok
Fall 2003
Euler Angles
Orientation of an object in
computer graphics is often
described using Euler
Angles.
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Z
RxRyRz
Any axis order will work
and could be used.
Yaw, Pitch & Roll
Y
X
Problems with Euler Angles
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Rotations not uniquely defined
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Ex: (z, x, y) [roll, yaw, pitch] = (90, 45, 45) = (45, 0, -45)
takes positive x-axis to (1, 1, 1)
Cartesian coordinates are independent of one another, but Euler
angles are not
Remember, the axes stay in the same place during rotations
Gimbal Lock
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Term derived from mechanical problem that arises in gimbal
mechanism that supports a compass or a gyro
Second and third rotations have effect of transforming earlier
rotations, we lose a degree of freedom
ex: Rot x, Rot y, Rot z
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If Rot y = 90 degrees,
Rot z is equivalent to -Rot x
A Gimble is a hardware implementation of
Euler angles (used for mounting gyroscopes
and globes)
Another Problem with Euler Angles
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What if we want to produce a smooth animation
of a rotation from point P1 to Point P2 around
some axis.
The entire rotation is defined as RxRyRz but how
do we get an intermediate point?
Rotations are defined as rotations about the x, y
and z axes. Rotations about any other vector in
space have to be broken down into equivalent
rotations about the major axes.
Smooth Rotation
y
z
With Euler Angles, knowing the
transformations for the entire
rotation does not help us with
the intermediate rotations. Each
is a unique set of three
rotations about the x,y and z
axes
x
Quaternions
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Invented by Sir William Hamilton (1843)
Do not suffer from Gimbal Lock
Provide a natural way to interpolate
intermediate steps in a rotation about an
arbitrary axis
Are used in many position tracking
systems and VR software support systems
Quaternions
Definitions & Basic Operations
Remember Complex Numbers?
z = a + i b where i = (-1)
Can think of a complex number as having a real and an
imaginary part or as a vector in two-dimensional space
z = [a, b]
z1+z2 = (a1 + i b1) + (a2 + i b2) = (a1+a2) + i (b1+b2)
z1*z2 = (a1+i b1)*(a2+i b2) = (a1a2–b1b2) + i (a2b1+a1b2)
Quaternions
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You can think of quaternions as an extension of
complex numbers where there are three different
square roots of -1.
q = w + i x + j y + k z where
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i = (-1), j = (-1), k = (-1)
i*j = k, j*k = i, k*i = j
j*i = -k, k*j = -i, i*k = -j
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You could also think of q as a value in fourdimensional space, q = [w, x, y, z]
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Sometimes written as q = [w, v] where w is a
scalar and v is a vector in 3-space
Addition of Quaternions
q1 + q 2
= (w1+i x1+j y1+k z1) + (w2+i x2+j y2+k z2)
= w1+w2 + i (x1+x2) + j (y1+y2) +k (z1+z2)
q1 + q2
= [w1, v1] + [w2, v2]
= [w1+w2, v1+v2]
Multiplication of Quaternions
q1 * q2
= (w1+i x1+j y1+k z1) * (w2+i x2+j y2+k z2)
= w1w2 + i w1x2 + j w1y2 + k w1z2 +
-x1x2 + i w2x1 – j x1z2 + k x1y2 +
-y1y2 + i y1z2 + j w2y1 - k x2y1 +
-z1z2 - i y2z1 + j x2z1 + k w2z1
q1 * q2 = [w1, v1] * [w2, v2] =
Multiplication of Quaternions
q1*q2 = (w1+ix1+jy1+kz1)*(w2+ix2+jy2+kz2)
= w1w2 + i w1x2 + j w1y2 + k w1z2 +
-x1x2 + i w2x1 – j x1z2 + k x1y2 +
-y1y2 + i y1z2 + j w2y1 - k x2y1 +
-z1z2 - i y2z1 + j x2z1 + k w2z1
q1*q2 = [w1, v1] * [w2, v2] =
[w1w2 – v1°v2, w1v2 + w2v1+ v1v2]
Practice
q1 = [1.57 0 0 1], q2= [.78 0 1 0]
 q1 = [.25 3 4 4], q2= [.20 2 9 9]
 q1 + q2?
q1 + q2 = w1+w2 + i (x1+x2) + j (y1+y2) +k (z1+z2)
q1 + q2 = [w1+w2, v1+v2]
 q1 * q2?
q1 * q 2
= (w1+i x1+j y1+k z1) * (w2+i x2+j y2+k z2)
= w1w2 + i w1x2 + j w1y2 + k w1z2 +
-x1x2 + i w2x1 – j x1z2 + k x1y2 +
-y1y2 + i y1z2 + j w2y1 - k x2y1 +
-z1z2 - i y2z1 + j x2z1 + k w2z1
[w1w2 – v1°v2, w1v2 + w2v1+ v1v2]
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Other operations
||q|| = (w2 + x2 + y2 + z2)
||q|| = 1 is a unit quaternion
The inverse of a quaternion, q-1 =[w, -v]/ ||q||2
q*q-1 = [1,0,0,0]
Practice: Find ||q|| and q-1 of q1 = [1.57 0 0 1], q2 = [.25
3 4 4]
Quaternion multiplication is associate but not commutative
Find the inverse of q = [0,6,8,0]
q-1 =[w, -v]/ ||q||2
q-1 =[0,-6,-8,-0]/(0+36+64+0) = [0,-.06,.08,0]
Find the inverse of q = [0,6,8,0]
q-1 =[w, -v]/ ||q||2
q-1 =[0,-6,-8,-0]/(0+36+64+0) = [0,-.06,-.08,0]
To check, q*q-1 should give us the identity quaternion.
q*q-1
= [0-(6*-.06 + 8*-.08 + 0), [0,0,0]+ [0,0,0] +[0,0,0]]
= [1,0, 0, 0]
[w1w2 – v1°v2, w1v2 + w2v1+ v1v2]
Quaternions
Rotating with Quaternions
Rotation about an arbitrary axis
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Let q be a unit quaternion, i.e., ||q|| = 1
q = [w,x,y,z] describes a rotation through
an angle  where w = cos(/2) and [x,y,z]
is the axis of rotation.
(x,y,z)
Procedure
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To create a unit quaternion that
represents a rotation of  degrees about
an arbitrary axis v = [x,y,z].
q = [cos(/2), sin(/2)*v/||v||]
Find q, s/t q describes a rotation of
60 degrees about v = [3,4,0].
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Remember, we need a unit quaternion.
To get this:
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Compute cos (60/2) = cos30 = 1/2
Compute sin (60/2) = sin30 = 3/2
Compute v’ = v/||v|| = [3,4,0]/5 = [3/5,4/5,0]
q = [1/2, 3/2*[3/5,4/5, 0]]
= [1/2, 33/10, 43/10, 0]
||q|| = ¼ + 27/100 + 48/100
= (25+27+48)/100 = 1
To rotate a point, P = (px,py,pz)
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Protated = q[0,P]q-1
Remember this is quaternion multiplication
not matrix multiplication!
Example: P = (0,1,1) rotated 90
degrees about the vertical y-axis.
y
x
z
Rotated point should be (1,1,0)
Example: P = (0,1,1) rotated 90
degrees about the vertical y-axis.
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First, compute q
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Next compute q-1
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q = [cos(90/2), sin(90/2)[0,1,0]]
q = [cos45, 0, sin45, 0] = [2/2, 0, 2/2, 0]
q-1 = [2/2, 0, -2/2, 0]
Why?
Compute q[0,P]q-1
=[2/2, 0, 2/2, 0][0,0,1,1][2/2, 0, 2/2, 0]
Compute q[0,P]q-1
=[2/2, 0, 2/2, 0][0,0,1,1][2/2, 0, -2/2, 0]
Remember that q1*q2 = [w1w2 – v1°v2, w1v2 + w2v1+ v1v2] so
q[0, P] = [2/2, 0, 2/2, 0][0,0,1,1]
= [2/2*0 – [0,2/2,0]°[0,1,1], 2/2[0,1,1] + 0*v1 + v1 x v2 ]
= [-2/2, 2/2, 2/2, 2/2]
q[0, P] q-1 = [-2/2, 2/2, 2/2, 2/2][2/2, 0, -2/2, 0]
= [-2/2* 2/2 – [2/2, 2/2, 2/2]°[0, -2/2, 0], (-2/2)[0, -2/2, 0]
+(2/2)*[2/2, 2/2, 2/2] + v1 x v2
= [0, [1/2,1,1/2] + [1/2 ,0,-1/2] = [0, 1, 1, 0]
Transformed point is [1,1,0].
See notes page for ALL the details
Why use quaternions?
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If you want to apply multiple rotations,
say q1 and q2.
When?
Recall: q[0,P]q-1
Thus: q2 (q1 [0,P] q1-1 ) q2-1
(q2 q1) [0,P] (q1-1 q2-1 )
q 2 * q1
Why is a quaternion multiplication better than a matrix
multiply?
What about keeping rotations?
Smooth Interpolation
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SLERP – spherical linear
interpolation
slerp(t, a, b) = f(t) = (b.a-1)t a
What happens when t=0? t=1?
In actuality, we need to conserve
camera velocity (Spherical – Cubic
Interpolation) Why?
Where have you seen this before?
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D
A
Bezier/Hermite curves
slerp (t,a,c,d,b)
a, b – start points, c,d – define
curve
The curve touchs cd when t=0.5
B
C
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