For the written part of your first homework you will do some exercises that will help familiarize you with basic descriptions and representations of combinatorial surfaces (simplicial surfaces, adjacency matrices, halfedge meshes), which will help prepare you to work with such surfaces as we continue through the course. (If any of this stuff seems abstract right now, don’t worry: we’ll use it over and over again to implement “real” algorithms starting in just a couple weeks!)

You must complete **8 out of 15 exercises** in the *Written Exercises* section of Chapter 2 of the course notes. You may choose any set of 8 exercises you like, but if you do **more** than 8, please mark clearly on your submission which ones you would like us to grade.

The assignment is due on **February 8, 2020 at 5:59:59pm Eastern** (*not* at midnight!). Remember to turn in the whole assignment via a single email including both the written exercises (as a PDF file) and the code (in a ZIP file). Further hand-in instructions can be found on this page.

For definition of next map $\rho$ in half edge, the note says

$\rho(ij) = jk \ \forall ijk \in K$.

Is it required that $i\neq j, j \neq k, i \neq k$?

If not, then in the cone example given on page 19, why $\rho(h_1) = h_2$, and not $\rho(h_1) = h_0, h_2$?

I now see below this it says “oriented triangle ijk”, implying ijk are different vertices. However, it still leaves me confused about exercise 2.10, “surface as permutation”, as there are many possible “next” map.

For example, one way to obtain the surface in 2.10 is by connecting two triangles with two edges, and gluing the two connecting edge together. Then the top triangle could contain edges 0->1->2, or 1->2->4.

You are correct. With special (almost degenerate) cases like these, the combinatorics does not completely fix the geometry. If you give a valid permutation, you will be marked as correct!

Thanks!

Is there any definition for polygonal disk and polygonal sphere ?

A polygonal disc is a closed disc, meaning it has $\chi = 1$. It is compact with boundary. A polygonal sphere on the other hand has $\chi = 2$ and is compact without boundary. You may think of a polygonal disc as a polygonal sphere with a face removed.

What is $\chi$?

The Euler Characteristic.

Double checking: the unique identifier to put in submission email title for this assignment is DDG20A0 ?

Yep!

Would there be a problem if I sent two emails (one with the incorrect title DDG20A1) for this assignment?

That’s fine. Just make sure you send one with the correct header.

For the written part, can we assume that we only need to consider triangular meshes?

Unless otherwise specified sure. However some of the problems do involve other types of meshes(ex: quad meshes).