Linear Algebra for FE Developers

Transcript

1. Linear Algebra for FE Developers Elizabeth Ramirez @eramirem
2. 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.
3. Disclaimer This presentation does not contain JS.
4. H/T: @jmaquino
5. https://xkcd.com/184/
6. Computer Graphics: Computer generated, do not come from sensors. Computer Vision: Data comes from sensors. Matrix representation of scenes.
7. Matrix Functions. Scalar function , matrix , and specifies to be a matrix of the same dimensions as .
8. 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.
9. 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.
10. 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:
11. Transformations.
12. Translation. Move a 2D object to a different position in the same coordinate system.
13. Translation.
14. Scaling. Change the size of a 2D object.
15. Scaling.
16. Rotation. Rotate a 2D object by some angle about some axis. Preserve distance: isometric.
17. Rotation.
18. 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
19. Fixed Point Rotation.
20. Reflection. about x-axis about y-axis about y=x
21. Shearing. x-shearing y-shearing
22. Other Transformations. - Projection: Orthogonal, Oblique. - Affine: Preserves parallelism. - Similarity: rigid transformation (reflection, rotation, translation) followed by scaling. - Rotation Quaternions: in 3D space
23. 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.
24. Special Topic: Visual Effects. Damped spring-mass system. General form:
25. Special Topic: Visual Effects.
26. Special Topic: Visual Effects.
27. © Disney/Pixar Thank you!
28. References Higham, Nicholas J. Functions of Matrices: Theory and Computation. https://graphics.pixar.com/library/CurlyHai rA/paper.pdf