learning - Academic Conferences Limited

Robert Lucas
Occasionally we will also refer to 2 vectors that are used to map textures onto surfaces. In both cases
vectors are represented by a bold letter, such as V1 in any formula or sentence.
3. Basic data structures
In the Unity (Unity 2011) world a mesh consists of an array of triangles. This is a list of indices into
another array of vertices where each vertex is a 3-vector. This defines the position of the vertex in 3D
space. Each consecutive three entries of the triangles array define a triangle. To make this tangible
figure 2 shows how a cube is represented by twelve triangles where each triangle consists of three
vertices. The ‘triangles’ array that corresponds to this contains: 3, 1, 2, 2, 1, 0, 5, 3, 4, 4, 3, 2, 7, 5, 6,
6, 5, 4, 1, 7, 0, 0, 7, 6, 5, 7, 3, 3, 7, 1, 2, 0, 4, 4, 0, 6. In fact, it is not quite as simple as this as there is
a good deal of redundancy in the vertices array with the same point being represented as many times
as it appears in the definition of a triangle. The very first operation applied in our cutting process is to
remove this redundancy to make the triangles array as defined above. This makes it very much easier
to compare vertices as the vertex number in the triangles array uniquely defines a point in space.
Thus if two entries in the triangles array are the same, then they must refer to the same point in 3D
space.
Figure 2: The vertices of a cube drawn as a mesh of triangles
4. Developing a cutting algorithm
Developing software for complex operations can sometimes be rather akin to the old joke: when you
ask an Irishman the way, he replies ‘Well I wouldn’t want to start from here’. This is because the
space the problem spans can just be too large to get to grips with. However much we like to pretend
in the software world that all things can be achieved by a top-down design approach (Lucas 2000), in
practice we commonly use a combination of top-down and bottom up design approaches, if for no
other reason than it can prevent us going insane from the complexity of what we are trying to deal
with. The bottom-up approach does allow us to start from ‘somewhere else’ which will indeed make it
easier to get to our destination.
4.1 Representing a planar cut as a data structure
In this case it seems sensible to start with a definition of a cut as a data structure that will give us a
handle on actually performing the cut. Then we can address the issue of how we might create such a
cut data structure at a later time when we are entirely happy that our representation has made the
problem tractable. This concentrates the mind wonderfully, what do we mean by a cut? What do we
want to be the result of the cut? These questions come to mind as soon as we address the problem of
representing the cut. For simplicity we will consider planar cuts. Elementary geometry tells us that a
plane can be represented by a normal to the plane and a single point on the plane, so there is
certainly no problem in representing that; two 3-vectors will suffice. If we look at Figure 3 and imagine
a planar cut caused by a blade occupying a plane containing the x and y axes passing through the
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