Beveling

Here we provide some additional detail about the bevel operations and their implementation in Scotty3D. Each bevel operation has two components:

1. a method that modifies the connectivity of the mesh, creating new beveled elements, and
2. a method the updates the geometry of the mesh, insetting and offseting the new vertices according to user input.

The methods that update the connectivity are HalfedgeMesh::bevel_vertex, halfedgeMesh::bevel_edge, and HalfedgeMesh::bevel_face. The methods that update geometry are HalfedgeMesh::bevel_vertex_positions, HalfedgeMesh::bevel_edge_positions, and HalfedgeMesh::bevel_face_positions.

The methods for updating connectivity can be implemented following the general strategy outlined in edge flip tutorial. Note that the methods that update geometry will be called repeatedly for the same bevel, in order to adjust positions according to user mouse input. See the gif in the User Guide.

To update the geometry of a beveled element, you are provided with the following data:

• start_positions - These are the original vertex positions of the beveled mesh element, without any insetting or offsetting.
• face - This is a reference to the face currently being beveled. This was returned by one of the connectivity functions.
• tangent_offset - The amount by which the new face should be inset (i.e., “shrunk” or “expanded”)
• normal_offset - (faces only) The amount by which the new face should be offset in the normal direction.

Also note that we provide code to gather the halfedges contained in the beveled face, creating the array new_halfedges. You should only have to update the position (Vertex::pos) of the vertices associated with this list of halfedges. The basic recipe for updating these positions is:

• Iterate over the list of halfedges (new_halfedges)
• Grab the vertex coordinates that are needed to compute the new, updated vertex coordinates (this could be a mix of values from start_positions, or the members Vertex::pos)
• Compute the updated vertex positions using the current values of tangent_offset (and possibly normal_offset)
• Store the new vertex positions in Vertex::pos for the vertices of the new, beveled polygon only (i.e., the vertices associated with each of new_halfedges).

The reason for storing new_halfedges and start_positions in an array is that it makes it easy to access positions “to the left” and “to the right” of a given vertex. For instance, suppose we want to figure out the offset from the corner of a polygon. We might want to compute some geometric quantity involving the three vertex positions start_positions[i-1], start_positions[i], and start_positions[i+1] (as well as inset), then set the new vertex position new_halfedges[i]->vertex()->pos to this new value:

A useful trick here is modular arithmetic: since we really have a “loop” of vertices, we want to make sure that indexing the next element (+1) and the previous element (-1) properly “wraps around.” This can be achieved via code like

// Get the number of vertices in the new polygon
int N = (int)hs.size();

// Assuming we're looking at vertex i, compute the indices
// of the next and previous elements in the list using
// modular arithmetic---note that to get the previous index,
// we can't just subtract 1 because the mod operation in C++
// doesn't behave quite how you might expect for negative
// values!
int a = (i+N-1) % N;
int b = i;
int c = (i+1) % N;

// Get the actual 3D vertex coordinates at these vertices
Vec3 pa = start_positions[a];
Vec3 pb = start_positions[b];
Vec3 pc = start_positions[c];

From here, you will need to compute new coordinates for vertex i, which can be accessed from new_halfedges[i]->vertex()->pos. As a “dummy” example (i.e., this is NOT what you should actually do!!) this code will set the position of the new vertex to the average of the vertices above:

new_halfedges[i].vertex()->pos = ( pa + pb + pc ) / 3.; // replace with something that actually makes sense!

The only question remaining is: where should you put the beveled vertex? We will leave this decision up to you. This question is one where you will have to think a little bit about what a good design would be. Questions to ask yourself:

• How do I compute a point that is inset from the original geometry?
• For faces, how do I shift the geometry in the normal direction? (You may wish to use the method Face::normal() here.)
• What should I do as the offset geometry starts to look degenerate, e.g., shrinks to a point, or goes outside some reasonable bounds?
• What should I do when the geometry is nonplanar?
• Etc.

The best way to get a feel for whether you have a good design is to try it out! Can you successfully and easily use this tool to edit your mesh? Or is it a total pain, producing bizarre results? You be the judge!