Game Development Reference
In-Depth Information
The reference system consists of three sets of reference surfaces S U , S V , S W , as
labeled in Figure 13e. For a better understanding, the reference system has been
chosen uniformly distributed. Notice that, usually in a real case, as shown in
Figure 13c, the reference grid is non-uniformly distributed. The reference grid
is defined by the intersection points between the three sets of reference surfaces
S U , S V , S W , as given by Equation (7).
∑∑∑
RG
=
S
,
S
,
S
(7)
U
V
W
U
V
W
∑∑∑
()
()
()
CW
=
C S
,
C S
,
C S
(8)
U
V
W
U
V
W
The discrete position ( u , v , w ) of a reference grid point represents the indices of
the reference surfaces { S U , S V , S W } intersecting at that point, while the
coordinate ( x , y , z ) of a reference grid point is equal to the coordinate of the
computed intersection point.
There is a constraint imposed on the reference surfaces, however. They must
be chosen in such a way that the reference surfaces from one set do not intersect
each other, but intersect the reference surfaces from the other sets. To obtain
the connectivity-wireframe, the T RI S CAN method performs the contouring of the
object in each of the reference surfaces S U , S V , S W . Any intersection between
two contours defines a vertex. The connectivity-wireframe consists of the set
of all vertices generated by the intersections between contours, and the
connectivity between these vertices. A mathematical definition of the connec-
tivity-wireframe is given by Equation (8).
In the general case, the connectivity-wireframe is heterogeneous and can be
seen as a net of polygonal shapes ranging from triangles to heptagons, where,
except for the triangles, the other polygons may not be planar. Therefore, to
triangulate the connectivity-wireframe in a consistent way, a set of connectivity
rules has been designed especially for that purpose, as explained in Salomie
(2002a; Salomie, 2002b).
As one might have noticed, there exists a relationship between the vertices and
the reference grid, since a vertex is the intersection point between two contours,
therefore, belonging to two reference surfaces from different sets. This relation-
ship can be followed in the 2D cross-section (see Figure 14), inside a reference
surface, intersecting the object. Any vertex (label 4), lying on a contour of the
object (label 5), is located on a reference grid line (label 1) in between two
reference grid points, one inside the object (label 3) and one outside the object
(label 2). Notice that a reference grid line is the intersection curve of two
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