learning - Academic Conferences Limited

Robert Lucas
origin at the centre of the cube, we can imagine the points of intersection of this plane with each
affected triangle.
Figure 3: Visualizing the cut points of a plane with a cube
This gives us the necessary insight to define a cut-point as consisting of two vertices and a number
representing how far from the first vertex toward the second is this cut-point. Thus a single cut point is
represented by the triple:
(VertexA, VertexB, lambda)
Where the cut occurs on this edge between VertexA and VertexB at the point
(VectorB-VectorA)* lambda
Where VectorA is the vector from the origin to VertexA and VectorB is the vector from the origin to
VertexB.
Thus the entire planar cut can then be represented by the ordered list:
{CutA, CutB,…,CutH}
It is clearly the case that this representation can be used on any arbitrary complicated mesh, just as
long as the mesh is composed of triangles. More importantly, we can now perceive how a cutting
algorithm might proceed by taking each cut point and creating a vertex at the point of the cut and
rebuilding each triangle that has a vertex on the edge containing the cut point. Finally we will need to
fill in the missing faces that the cut has created. In pseudo code (Lucas 2000) the algorithm is:
Begin
CutPoint1 = Cut[0]
VertexA = CreateVertex(CutPoint1)
For each remaining cut point do
CutPoint2 = Cuts[next];
VertexB = CreateVertex(CutPoint[next])
If both cut points share same first vertex then
CreateTriangle(VertexA, VertexB, SharedVertex)
Else
CreateTriangle(VertexA, VertexB, CutPoint1.StartVertex)
CreateTriangle(VertexA, CutPoint2.StartVertex, CutPoint1.StartVertex)
Endif
VertexA = VertexB
Repeat
End
437