Linear Algebra for FE
Developers
Elizabeth Ramirez
@eramirem
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About me
Ingeniera Electrónica
Applied Mathematician
Computational Science and Engineering.
Applied Scientist
Modeling complex systems on the planet, like agriculture
and transportation, using satellite data.
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Disclaimer
This presentation does not contain JS.
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H/T: @jmaquino
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https://xkcd.com/184/
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Computer Graphics: Computer
generated, do not come from sensors.
Computer Vision: Data comes from
sensors.
Matrix representation of scenes.
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Matrix Functions.
Scalar function , matrix , and
specifies to be a matrix of the same
dimensions as .
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Most common LA ops.
dot product: product of two vectors, used to
find angles.
cross product: product of two vectors, used to
find direction perpendicular to a plane.
matrix multiplication: represent matrix
functions of some geometric transformation.
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Coordinate Systems.
- Inertial: origin and axes don’t interfere
with observations.
- Body Fixed: Origin located at center of
mass. Accelerated.
- Euler Angles: Describe BFCS.
- Cartesian, Polar, Cylindrical, Spherical.
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Homogeneous Coordinates.
Represent 2D transformations exclusively
as matrix multiplications. Add an extra
dimension to Cartesian coordinate system.
Also account for the concept of infinity.
2D point:
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Transformations.
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Translation.
Move a 2D object to a different position in
the same coordinate system.
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Translation.
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Scaling.
Change the size of a 2D object.
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Scaling.
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Rotation.
Rotate a 2D object by some angle about
some axis. Preserve distance: isometric.
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Rotation.
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Fixed Point Rotation.
Rotate a 2D object by some angle about a
fixed point .
- Translate to the origin
- Rotate by angle
- Translate the origin back to
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Fixed Point Rotation.
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Reflection.
about x-axis about y-axis about y=x
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Shearing.
x-shearing y-shearing
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Other Transformations.
- Projection: Orthogonal, Oblique.
- Affine: Preserves parallelism.
- Similarity: rigid transformation
(reflection, rotation, translation)
followed by scaling.
- Rotation Quaternions: in 3D space
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Special Topic: Visual Effects.
Rely on a wide range of numerical methods
for Differential Equations.
- Water waves, using Navier-Stokes.
- Metals and granular materials like sand
and snow, using elastoplasticity.
- Hair and fabrics: using damped
spring-mass systems.
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Special Topic: Visual Effects.
Damped spring-mass system. General
form: