Author: olga

• Week 1: lectures only ( October 16 – 17 )
  • Planar curves. Parametrizations, geometric quantities, length variation, turning angles. Discrete planar curves. Discrete arc length, curvature.
  • Discrete space curves. Frames. Frenet-Serrat formula. Smooth surfaces. Metric. Curvature.
  • Lecture notes

• Week 2: lectures only ( October 23 – 24 )
  • Geometry of Surfaces. Curvature of Surfaces.
  • Introduction to Exterior Calculus – Vectors and 1-forms.
  • Introduction to Exterior Calculus – Differential forms and the wedge product.
  • A Quick and Dirty Introduction to Exterior Calculus — Part III: Hodge Duality
  • A Quick and Dirty Introduction to Exterior Calculus — Part IV: Differential Operators
  • Additional Material on Exterior Calculus

• Week 3: tutorials only ( October 30 – 31 )
  • Houdini Workflow, lights, camera, rendering.
  • Working on common examples.

• Week 4: lecture and tutorial ( November 6 – 7 )
  • Exterior algebra and exterior calculus, from the notes.
  • “A Quick and Dirty Introduction to Exterior Calculus — Part V: Integration and Stokes’ Theorem”
  • Houdini working with attributes and shaping curves.

• Week 5: tutorial and lecture ( November 13 – 14 )
  • Houdini time control. Large render tasks. Solvers.
  • Variational Derivations of Curvature.

• Week 6: lecture and tutorial ( November 20 – 21 )
  • Discrete Exterior Calculus. See also this quick and dirty introduction.

• Week 7: lecture and tutorial ( November 27 – 28 )
  • Discrete Exterior Calculus, Take Two: the dual complex. See also paragraphs 1-6 and 9 from this publication on DEC by Desbrun et al.
  • An example of the complications in building exterior derivative operators in the presence of boundaries. Using circumcentric subdivision one can identify ‘dual edges’ with ’90 deg. rotated edges’ and build a ‘boundary operator for dual 2d-cells’ ${\partial }_2^{\ast }$ as the transpose of the primal boundary operator for edges ${\partial }_1$, with the following relation ${\partial }_2^{\ast }=-{\partial }_1$. However such a relation cannot be set up for ${\partial }_1^{\ast }$  since one would need to add the midpoints of the boundary edges as vertices, and the sizes of ${\partial }_1^{\ast }$ and ${\partial }_2^T$ would no longer match.

• Week 8: lectures only ( December 4 – 5 )
  • Whitney Elements.
  • Differential Equations on Manifolds. Physical Units.
  • The Heat Equation – Derivation and properties of the solution.
  • Finite Elements: Weak and Strong formulations of the heat equation.

• Week 9: lectures only ( December 11 – 12 )
  • Discrete Finite Elements. Time Discretization/Stability. Poisson equation.
  • Conformal maps.

• Week 10: tutorials only ( December 18 – 19 )
  • Derivation of the formula for discrete Gauss curvature.
  • Waves on surfaces.

• Holidays: Please be happy!

• Week 11: lectures and tutorial ( January 8 – 9 )
  • Hodge Decomposition, Homology Generators and Harmonic Bases

• Week 12: No lectures ( January 15 – 16 )
• Week 13: No lectures ( January 22 – 23 )

• Week 14: lectures only ( January 29 – 30 )
  • Connections and Parallel Transport
  • Paper descriptions.

• Week 15: Tutorials only ( February 5 – 6 )
  • Boundary conditions: Dirichlet and Neumann

• Week 16: Tutorial and Project presentation ( February 12 – 13 )