Things left over from last time
Concept of “World Coordinates” vs. Local Coordinates
Intermediate co-ordinate systems
Push / Pop Again (planets, show common prefixes)
3D
Any 3 vectors
Orthogonal Bases – right hand rule
Cross product = |a| |b| sin, A x B = YaZb – ZbYa, ZaXb – XaZb, XaYb – Ya Xb
X cross Y = Z
Constructing a basis from 2 vectors
Construct from 1 vector (largest magnitude component trick)
Rotations and bases (much easier to build basis than to do sin/cos)
3D Transforms
3D homogeneous coordinates = 4x4 matrices
Translate – same
Scale – same
nuScale (can be any axis)
Shears (any axis, or pair of axis – decomposed into one axis at a time)
Rotations – a whole topic unto itself
3D Rotations (an intro)
Rotations (motions) vs. Orientations (configurations) vs. Rotations (relative orientations)
3 D.O.F. (point in direction, rotate around it)
Since loops around space is complex – SO(3) – group of positive, orthonormal 3x3 matrices
Cannot map nicely to R^3 – will be a singularity
Cannot map nicely to R^4 - redundancy
Rotations about an axis – always will be an axis – Basis axes, arbitrary axis
Euler Theorems
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Any rotation can be represented by a single rotation about an arbitrary axis (axis-angle form)
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Any rotation can be represented by 3 rotations about fixed axes (Euler Angle form)
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XYZ, XZX, any non-repeating set works
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Each set is different (gets different singularities)
Building rotations
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Pick a vector (for an axis)
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Pick another perpindicular vector (or make one w/cross product)
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Get third vector by cross product
“Euler Angles” – horrible term
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Pick convention
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Are axes local or global?
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Local: roll, pitch, yaw
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What order?
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Apply 3 rotations
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Good news: 3 numbers
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Bad news:
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Can’t add, can’t compose
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Many representations for any rotation
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Singularities
Rot X ~~ Rot Y Rot Z - same place, 2 ways – long way from one to the other!
Axis Angle – harder to think about (for most people)
Rotation Vector – multiply by angle (scale length)
Lots of different 360 degree rotations. Shells at different radii
Scale by sin(theta)? (makes math messy)
Still can’t compose or interpolate
Quaternion
4D
Unit Sphere
Encoding rotation vectors cos, sin V
Compose by multiplication
Interpolate by careful formula