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Assignment 2: Simplification and Subdivision Surfaces
The goal of this assignment is to try your hand at various mesh
processing tasks. A halfedge mesh adjacency data structure is
provided, along with a very simple .obj mesh parser. As you modify
the triangle mesh, it is critical that the manifold properties of the
input surface are maintained, and that the adjacency data structure
remains consistent. Throughout this assignment you are asked to
consider the efficiency of various mesh operations: is it quadratic,
linear, logarithmic, constant, etc? Since this assignment is rather
long, it's ok if you don't always implement the most efficient
strategy (which may require additional data structures), but please
discuss the tradeoffs in your README.txt file.
Warning: This assignment is probably way too long. Start early,
and just get the basics working first. Spend a reasonable
amount of time coding and write about what you've learned in your
README.txt file.
Tasks
 Download the provided source code and compile it in your favorite
C++ environment. You should be able to load simple .obj files and
turn wireframe visualization on & off by pressing "w". Try these
command lines:
mesher input cube.obj size 300
mesher input cube.obj size 300 wireframe
mesher input open_box.obj size 300
mesher input open_box.obj size 300 wireframe
Notice that the adjacency data structure detects boundaries in the
mesh and draws them with thicker red lines in the wireframe visualization:
 Now you're ready to start coding. First make use of the
adjacency data structure provided to implement gouraud
shading. In gouraud shading, average normals are computed for
each vertex, these normals are used for lighting, and the shaded
values automatically interpolated in GL. This shading option should
be enabled by specifying gouraud on the command line, or pressing
"g".
mesher input bunny_1k.obj size 500
mesher input bunny_1k.obj size 500 gouraud
Describe in your README.txt file any helper functions you
created to implement this shading. Comment on the efficiency of your
implementation.
 Next, let's experiment with edge collapses, the key mesh
manipulation operation from the Progressive Mesh framework. When you
press the "d" key (for decimate, a.k.a. simplification) the number of
triangles in your mesh should be reduced by approximately 10 percent.
First implement the topology of an edge collapse. Identify a
target edge to collapse. For now, use the function
Bag::ChooseRandom() to pick a random edge in the mesh. Using
the adjacency data structure, identify which two triangles will be
removed permanently from the model, and which other triangles will
change. Choose a simple location for the remaining vertex (e.g., it's
old position, or a position averaged with the deleted vertex). Note:
in the provided adjacency data structure you will delete and then
recreate triangles rather than modify them. Initially you may want
to just collapse one edge each time the "d" key is pressed. In your
README.txt file, describe any helper functions you create.
mesher input bunny_1k.obj size 500 wireframe
Even though a single random collapse works great, you will
probably quickly run into problems: 1) the mesh looks bad with small
triangles and big triangles that don't represent the highresolution
model very well, 2) it creates self intersections of the surface (it
turns itself inside out and you can see the blue side), and 3)
(worstofall) it sometimes crashes with an error like:
assertion "opposite == NULL" failed: file "edge.h", line XXX
Think about what that error might indicate is wrong with your
implementation. (If you get a different error, of course, try to
explain that one.) Certain edges should not be collapsed
because it would cause the surface to become nonmanifold. Write code
to check for these conditions. NOTE: This code can be very
complicated. It's ok if you only just catch some of these conditions
and your program still crashes occasionally. Don't spend too long
trying to make it perfect. Discuss this in your README.txt
file.
Now implement something to make a smarter choice for which edge to
collapse (this might also minimize the occurance of lingering bugs
from the previous part). Collapsing the shortest edge first (or
actually make that the shortest legal edge) will usually do a
very good job at preserving overall surface shape. Done naively,
repeatedly selecting the shortest edge for collapse can be an
expensive operation. That's fine for now, you can implement something
very inefficient. Discuss these performance issues in your
README.txt file and what common data structures would help.
 Finally, implement the Loop Subdivision Surface scheme. When
the "s" key is pressed, your implementation should perform one
iteration of subdivision. First you want to get the topology correct.
Add a vertex at the center of each edge, then replace each triangle
with 4 smaller triangles. Make sure you share the new vertices across
the edges and don't create multiple copies of the vertex. Consider
how to do this efficiently and discuss various options in your
README.txt file. The provided VertexParent class
should be very helpful for this task. You can verify correct vertex
sharing by visualizing the wireframe and looking for the red boundary
edges. If you have extra red lines on your mesh (not at the
boundaries) you have not properly shared the vertices.
mesher input open_box.obj size 300 wirefame
Once you have the correct topology and vertex sharing, you can
adjust vertex positions according to the Loop Subdivision rules. To
handle general shapes you must implement the rules for regular
vertices (valence = 6, i.e., 6 edges meeting at a vertex),
extraordinary vertices (valence != 6), and boundary conditions. Here
are the Loop
Subdivision Rules from the SIGGRAPH
2000 course notes  Subdivision for Modeling and Animation (page 70).
Finish off your subdivision surface code by implementing the simple
binary (infinitely sharp or infinitely smooth) crease method described
in: "Piecewise Smooth Surface Reconstruction."
H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, W. Stuetzle.
ACM SIGGRAPH 1994.
mesher input creased_cube.obj size 300 wirefame
Ideas for Extra Credit
Include a short paragraph in your README.txt file describing
your extensions.
 Implement gouraud shading with a proper average weighted by angle
for the triangles meeting at each vertex (if you haven't done this
already). Find or create a mesh which illustrates the difference.
 Implement a "better" edge weight selection method. First define
your goal for simplification and construct an edge weight function
that matches that goal. E.g., if your goal is to create equaliateral
triangles than the shortest edge collapse is a very good choice.
Discuss in your README.txt.
 Implement a better positioning method for the remaining vertex in
an edge collapse. Discuss in your README.txt.
 Implement the geomorph option to smoothly transition
between two different resolutions within a progressive mesh.
 Improve the efficiency of your implementation by using the right
data structures for various operations (hash tables, priority queues,
etc.)
 Implement the integer crease weight control described by "Subdivision
Surfaces in Character Animation", DeRose, Kass & Truong, SIGGRAPH
1998 to perform n iterations of crease subdivision
rules followed by infinite iterations of smooth subdivision.
 Implement the floating point weight control by interpolating
between the nearest integer iterations.
 Implement the butterfly interpolating subdivision surface scheme.
 Basic Code (vectors.h, matrix.h, matrix.cpp, argparser.h, camera.h, camera.cpp,
glCanvas.h, glCanvas.cpp, main.cpp, Makefile)
Similar to Assignment 1.
 Mesh Library (boundingbox.h,
boundingbox.cpp, vertex.h, triangle.h, edge.h, edge.cpp, mesh.h,
mesh.cpp, vertex_parent.h)
An implementation of the halfedge data structure. The load
function will parse simple .obj files.
 templated Bag class (bag.h, utils.h)
This class allows efficient (constant time expected) retrieval and
removal of elements, implemented with a hash table. The "extract"
function is used to hash elements. It is used by the Mesh code to
efficiently find pairings between opposite edges at load time.
 .obj files (cube.obj, creased_cube.obj, open_box.obj, l.obj,
torus.obj, bunny_200.obj, bunny_1k.obj, bunny_40k.obj,
complex.obj)
Please read the Homework information page again before submitting.
