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Week 5 -Wednesday
What did we talk about last time? Transforms
- Translation
- Rotation
- Scaling
- Shearing
- Rigid-body transforms and their inverses
Week 5 -Wednesday What did we talk about last time? Transforms - - PowerPoint PPT Presentation
Week 5 -Wednesday What did we talk about last time? Transforms - - PowerPoint PPT Presentation
Week 5 -Wednesday What did we talk about last time? Transforms Translation Rotation Scaling Shearing Rigid-body transforms and their inverses Normal transforms For normals and other things, we need to be able to
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For normals and other things, we need to be able to compute
inverses
- Rigid body inverses were given before
- For a concatenation of simple transforms with known parameters,
the inverse can be done by inverting the parameters and reversing the order:
▪ If M = T(t)R(ϕ) then M-1 = R(-ϕ)T(-t)
- For orthogonal matrices, M-1 = MT
- If nothing is known, use the adjoint method
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We can describe orientations from some
default orientation using the Euler transform
The default is usually looking down the –z
axis with "up" as positive y
The new orientation is:
- E(h, p, r) = Rz(r)Rx(p)Ry(h)
- h is head, like shaking your head "no"
▪ Also called yaw
- p is pitch, like nodding your head back and
forth
- r is roll… the third dimension
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One trouble with Euler angles is that they
can exhibit gimbal lock
In gimbal lock, two axes become aligned,
causing a degree of freedom to be lost
Euler angles can describe orientations in
multiple ways, however the wrong choice
- f rotations may cause gimbal lock
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Sometimes you want to rotate around some arbitrary axis r To do so, create an orthonormal basis r, s, and t as follows
- Take the smallest component of r and set it to 0
- Swap the two remaining components and negate the first one
- Divide the result by its norm, making it a normal vector s
- Let t = r x s
This basis can be made into a matrix M The final transform X transforms the r-axis to the x-axis, does the rotation by α and then
transforms back to r
X = MTRx(α)M
=
T T T
t s r M
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Quaternions are a compact way to represent orientations Pros:
- Compact (only four values needed)
- Do not suffer from gimbal lock
- Are easy to interpolate between
Cons:
- Are confusing
- Use three imaginary numbers
- Have their own set of operations
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A quaternion has three imaginary
parts and one real part
We use vector notation for
quaternions but put a hat on them
Note that the three imaginary
number dimensions do not behave the way you might expect
w v w z y x w v
q q kq jq iq q + = + + + = = q q q ) , ˆ ( ˆ
k ji ij j ik ki i kj jk k j i = − = = − = = − = − = = = 1
2 2 2
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Multiplication Addition Conjugate Norm Identity
) , ( ˆ ˆ
v v w w v w v w v v
r q q r r q r q r q r q ⋅ − + + × = ) , ( ˆ ˆ
w w v v
r q + + = + r q r q ) , ( ˆ *
w v q
q q − =
2 2 2 2
) ˆ (
w z y x
q q q q n + + + = q ) 1 , ( ˆ i =
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Inverse: One (useful) conjugate rule: Note that scalar multiplication is just like scalar vector
multiplication for any vector
Quaternion quaternion multiplication is associative but not
commutative
For any unit vector u, note that the following is a unit
quaternion:
* 2 1
ˆ ) ˆ ( 1 ˆ q q q n =
−
) cos , (sin ˆ φ φu q=
* * *
ˆ ˆ ) ˆ ˆ ( q r r q =
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Take a vector or point p and pretend its four coordinates make
a quaternion
If you have a unit quaternion
the result of is p rotated around the u axis by 2ϕ
Note that, because it's a unit quaternion, There are ways to convert between rotation matrices and
quaternions
The details are in the book
p ˆ
) cos , (sin ˆ φ φu q=
1
ˆ ˆ ˆ
−
q p q
* 1
ˆ ˆ q q =
−
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Short for spherical linear interpolation Using unit quaternions that represent orientations, we can
slerp between them to find a new orientation at time t = [0,1], tracing the path on a unit sphere between the orientations
To find the angle ϕ between the quaternions, you can use the
fact that cosϕ = qxrx + qyry + qzrz + qwrw r q r q s ˆ sin ) sin( ˆ sin )) 1 ( sin( ) , ˆ , ˆ ( ˆ φ t φ φ t φ t + − =
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If we animate by moving rigid bodies around each other, joints
won't look natural
To do so, we define bones and skin and have the rigid bone
changes dictate blended changes in the skin
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The following equation shows the effect that each bone i (and
its corresponding animation transform matrix Bi(t) and bone to world transform matrix Mi ) have on the p, the original location of a vertex
Vertex blending is popular in part because it can be computed
in hardware with the vertex shader
∑ ∑
− = − = −
= =
1 1 1
1 where ) ( ) (
n i i n i i i i
w t w t p M B u
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Finish vertex blending and morphing Projections Work day to finish Project 1
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