geometric algebrageometry.mrao.cam.ac.uk/wp-content/uploads/2015/10/ga... · 2015-10-14 ·...
TRANSCRIPT
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Geometric Algebra
Dr Chris Doran
ARM Research
2. Geometric Algebra in 3 Dimensions
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Three dimensions
Introduce a third vector.
These all anticommute.
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Bivector products
The product of a vector and a bivector
can contain two different terms.
The product of two perpendicular
bivectors results in a third bivector.
Now define i, j and k. We have
discovered the quaternion algebra
buried in 3 (not 4) dimensions
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Unification
Quaternions arise naturally in the geometric
algebra of space.
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The directed volume element
Has negative square Commutes with vectors Swaps lines and planes
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3D Basis
Grade 0
1 Scalar
Grade 1
3 Vectors
Grade 2
3 Plane / bivector
Grade 3
1 Volume / trivector
A linear space of dimension 8
Note the appearance of the binomial coefficients - this is general
General elements of this space are called multivectors
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Products in 3D
We recover the cross product from duality:
Can only do this in 3D
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Unification
The cross product is a disguised from of the
outer product in three dimensions.
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Vectors and bivectors
Decompose vector
into terms into and
normal to the plane
A vector lying in the
plane
Product of three orthogonal vectors,
so a trivector
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Vectors and bivectors
Write the combined product:
Lowest grade Highest grade
Inner product is antisymmetric, so define
This always returns a vector
With a bit of work, prove that
A very useful result. Generalises the
vector triple product.
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Vectors and bivectorsSymmetric component of
product gives a trivector:
Can defined the outer product
of three vectors
Vector part does not contribute
The outer product is associative
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Duality
Seen that the pseudoscalar
interchanges planes and vectors in 3D
Can use this in 3D to understand
product of a vector and a bivector
Symmetric part is a trivector Antisymmetric part is a vector
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Reflections
See the power of the geometric
product when looking at operations.
Decompose a into components into
and out of the plane.
Form the reflected vector
Now re-express in terms of the
geometric product.
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Rotations
Two reflections generate a rotation.
Define a rotor R. This is formed from
a geometric product!
Rotations now formed by
This works for higher grade objects
as well. Will prove this later.
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Rotors in 3D
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Rotors are even grade,
so built out of a scalar
and the three bivectors.
These are the terms
that map directly to
quaternions.
Rotors are normalised.
Reduces the degrees of
freedom from 4 to 3.
This is precisely the
definition of a unit
quaternion.
Rotors are elements of
a 4-dimensional space
normalised to 1.
They live on a 3-sphere.
This is the GROUP
MANIFOLD.
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Exponential form
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Use the following
useful, general result.
Polar decomposition
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Exponential form
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Sequence of two reflections gives
a rotation through twice the angle
between the vectors
Also need to check
orientation
Useful result when
vector a lies in the
plane B
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Rotors in 3D
Can work in terms of Euler angles, but best avoided:
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The rotor for a rotation through |B| with handedness of B:
In terms of an axis:
Decompose a vector
into terms in and out of
the plane
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Unification
Every rotor can be written as
Rotations of any object, of any grade, in any
space of any signature can be written as
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Unification
Every finite Lie group can be realised as a group of rotors.
Every Lie algebra can be realised as a set of bivectors.
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Resources
geometry.mrao.cam.ac.uk
@chrisjldoran
#geometricalgebra
github.com/ga
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