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Linear Algebra for FE Developers

Linear Algebra for FE Developers

Elizabeth Ramirez

November 16, 2018
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  1. 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.
  2. Computer Graphics: Computer generated, do not come from sensors. Computer

    Vision: Data comes from sensors. Matrix representation of scenes.
  3. Matrix Functions. Scalar function , matrix , and specifies to

    be a matrix of the same dimensions as .
  4. 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.
  5. 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.
  6. 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:
  7. Rotation. Rotate a 2D object by some angle about some

    axis. Preserve distance: isometric.
  8. 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
  9. Other Transformations. - Projection: Orthogonal, Oblique. - Affine: Preserves parallelism.

    - Similarity: rigid transformation (reflection, rotation, translation) followed by scaling. - Rotation Quaternions: in 3D space
  10. 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.
  11. References Higham, Nicholas J. Functions of Matrices: Theory and Computation.

    https://graphics.pixar.com/library/CurlyHai rA/paper.pdf