A Quadrilateral Rendering Primitive

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A QuadrilateralRendering Primitive
Computer Graphics 2

• Federico Ponchio

TU Clausthal
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Problem parametrize 3d rotations

• Equivalent parametrize 3d orientation...

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Euler angles

• Rotation matrix can be written as a product of 3
  rotations MABC around 3 axis
• If the 3 axis are chosen in advance, 3 angles are
  enough.

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Euler angles

• The sequence is
• rotate around z-axis
• rotate around (the new) x-axis
• rotate around (the new) z-axis again

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Rotation matrix

• C cos(c) sin(c) 0 -sin(c) cos(c)
  0 0 0 1
• B 1 0 0 0
  cos(b) sin(b) 0 -sin(b) cos(b)
• A cos(a) sin(a) 0 -sin(a) cos(a)
  0 0 0 1

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Different choices

• One could also use a different sequence of axis
  (x,y, z) and get a different parametrization.
• But there are a few problems...
• ... the parametrization is not unique.

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Gimbal lock

• Gimbal lock is the phenomenon of two rotational
  axis of an object pointing in the same direction.
• The reason for this is that Eular angles evaluate
  each axis independently in a set order.
• The problem with gimbal lock occurs when you
  rotate the object down the Y axis, say 90
  degrees. Since the X component has already been
  evaluated it doesn't get carried along with the
  other two axis. What winds up happening is the X
  and Z axis get pointed down the same axis.

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Gimbal lock

• In other words a rotation in one axis could
  'override' a rotation in another, making you lose
  a degree of freedom.
• No matter the order of axis you pick gimbal lock
  will always occur...

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Rotations in 2D

• Really easy to parametrize...
• there is another interestingparametrization
  using complex numbers
• a ib eiq/2
• Where of course
• a2b2 1

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Algebra of 2D rotations

• Relate this to the matrix multiplication rule
• A cos(a) -sin(a) B cos(b) -sin(b)
• sin(a) cos(a) sin(b)
  cos(b)
• AB cos(ab) -sin(ab)
• sin(ab) cos(ab)
• AB cos(a)cos(b)-sin(a)sin(b)
  -cos(a)sin(b)-sin(a)cos(b)
• sin(a)cos(b)cos(a)sin(b)
  cos(a)cos(b)-sin(a)sin(b)
• so with a cos(a) and b sin(a) we get exactly
  the complex mutiplication.

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Quaternions

• Add to the real number i,j,k such that
• i2 j2 k2 ijk -1
• We can immediately derive (how?)
• ij k ji -k
• jk i kj -i
• ki j ik -j
• So what we get is not commutative anymore!

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Basic rules derivation

• Ijk -1 ijkk -k ij(-1) -k ij k
• ijj kj -i kj
• ijk -1 iijk -i jk i
• and so on and so forth...

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Quaternions

• Quaternions are a non-commutative extension of
  complex numbers.
• We can write a quaternion as
• q a bi cj dk
• where a, b, c, and d are real numbers.
• We can look at it as a vector space (like complex
  numbers but with 4 components).

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Quaternions

• Multiplication rules for the quaternions can be
  derived from the linearity and the multiplication
  rules for ijk.
• Example
• x 3ik
• y 2i j -2k
• xy ?
• xy ?
• yx ?

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Quaternions

• An alternative representation
• q a ib jc kd (a,u) with u a 3d vector.
• Then if p (b, v) we have
• p q (ab, uv)
• pq (ab uv, av bu u?v)

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Rotations and quaternions

• M a2-b2 -2ab
• 2ab a2-b2
• this is a rotation in 2d (with a2b21).
• We can build an analogous matrix in 3D
• R(q) q02q12-q22-q22 2q1q2-2q0q3
  2q1q3-2q0q2
• 2q1q2-2q0q3
  q02-q12q22-q22 2q2q3-2q0q1
• 2q1q3-2q0q2
  2q2q3-2q0q1 q02-q12-q22q22

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Quaternions and rotations

• You can verify that it is an orthogonal matrix
• colicolj 0 if i?j
• colicolj (q02q12q22q22)2
• so if we require q02q12q22q22 1 a point in
  the 3D sphere in R4, we get a rotation.
• Note q and -q results in the same rotation.

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Why quaternions?

• It turns out that if we multiply two matrices
• R(q1) and R(q2) the multiplication rules yield
• R(q1)R(q2) R(q1q2)
• If we use for q1 and q2 the quaternion
  multiplication.

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Geometric intuition

• We can describe a rotation with an axis and an
  angle.
• We can also write
• q (q0, q) (cos(q/2), nsin(q/2))
• where n is the unit axisof the rotation

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Geometry intuition

• Quaternion multiplication tell us how to compose
  rotations in the parameter space.
• An example lets k 1/?2
• A rotation of 90 degrees around x axis
• p (cos(p/4), xsin(p/4)), (k, k,0,0)

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Geometry intuition

• A rotation of 90 degrees aroun y axis
• q (cos(p/4), ysin(p/4)), (k, 0,k,0)
• What is the result of the composition pq?

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Example

• Remember the formula?
• pq (ab uv, av bu u?v)
• pq (k2, k2x k2y k2x?y)
• (1/2, (1/2)1,1,1 ) (cos(p/3), sin(p/3)n)

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A couple of useful formulas

• Conjugate of a quaternion
• p a ib jc kd
• p a ib -jc -kd
• Inverse of a quaternion
• p-1 p/(pp)
• Do someone of you remembers the same formula for
  the complex numbers?

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A couple of useful formulas

• If the quaternion is a rotation, the formulas is
  quite simple (just invert q...)
• q (q0, q) (cos(q/2), nsin(q/2))
• q-1 (q0, -q) (cos(-q/2), nsin(-q/2))

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Slerp

• Spherical linear interpolation
• a linear interpolation of p0 and p1
• on the circle. Linear in the
• sense of geodesic distance.
• Slerp(p0,p1,t)
• p0sin((1-t)a)/sin(a) p1sin(ta)/sin(a)

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Slerp woks on the sphere also

• Using the great circles we can interpolate on the
  spere also.
• Can we interpolate rotations?
• Not easy at all with euler angles or matrices...

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Direct interpolation

• What about
• q(t) (1-t)q0 tq1 ?

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Doesn't work because...

• q(t) ? 1
• that means it is not anymore a rotation...
• We could reproject it on the sphere in 4D...
• what could possibly go wrong?

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Equal speed...

• The speed of the interpolation would be greater
  near the endpoints...
• We have to use the SLERPfunction
• Slerp(q0,q1,t)
• q0sin((1-t)a)/sin(a) q1sin(ta)/sin(a)

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Crazy mathematicians

• Since pq composes rotations not pq we could
  write the interpolation as p(1-t)qt...
• Does that means anything at all?
• Well... we can define an exponential function
• eq 1 q q2/2 q3/6 ... qn/n!
• ea(cos(u) sin(u)u/u)
• We also have a logarithm function
• ln(q) ln(q) arcos(aq)/u/u

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Crazy mathematicians

• So we can define a power
• ax exln(a)
• lo and behold...
• Slerp(p,q, t) p(p-1q)t (qp-1)t p
• (We have to be a bit careful... it is not
  commutative!)