This homework focuses on getting familiar with 3D triangular meshes and the computation of discrete structures on them using linear algebra, including HalfEdge data structure, Gauss curvature, Euler number, and the fundamental discrete Gauss-Bonnet theore
computer science
Description
Homework 2
This homework focuses on getting familiar with 3D triangular meshes and the computation of discrete structures on them using linear algebra, including HalfEdge data structure, Gauss curvature, Euler number, and the fundamental discrete Gauss-Bonnet theorem. It builds the foundation for future 3D research and applications for both undergraduate and graduate students.
All homework is submitted to Moodle system. The source code and a corresponding report in .pdf format
are zipped as one file. The source documents such as Word, Latex, handwriting, etc., are unnecessary to be
turned in. Students are responsible for keeping track of the original version of homework.
Formatting: For project assignment, you should write a report to summarize your implementation and
the experimental results, which contains the following:
1. Title, student name, ID, and date.
2. A description of the program.
3. A readme to illustrate how to run the program.
4. At least one worked case for each operation (with snapshot illustration and the values you obtained).
5. A summary: done/on-going/to-do things; what you learned and the problems you met during implementation; remaining questions to be settled.
Project: Triangular Mesh Operations
1. Apply HalfEdge data structure for all the following triangular mesh operations. (30 pt)
2. Load a 3D model in .obj format (or other formats) and store it as a mesh object in program (including vertices, edges, and faces). (10 pt)
3. Verification (see Figure 1)
(a) Equation (1) gives two formulae to compute the Euler number χ. Please verify whether they are consistent. Left-hand side: compute the numbers of vertices, edges, faces, v, f, e; Right-hand side: visually obtain genus g; then compare the values of both sides. (20 pt)
(b) Equation (3) states the discrete Gauss-Bonnet theorem. Please verify the equality of both sides.
Left-hand side: compute the discrete Gauss curvatures for all the vertices by Eqn. (2) where αi
is the interior angle at vertex vi
in the neighboring triangle fi
. Right-hand side: apply the χ
obtained previously; then compare the values of both sides. (40 pt)
