Graphics Reference
In-Depth Information
polygons are considered co-planar and can be merged. This method works for many
applications, such as general model simplification [Hinker93]. However, for collision
detection purposes this approach is fundamentally flawed. Consider two polygons,
A
1
and
B
1
, at a fixed angle
to each other, requiring a plane thickness of
d
1
of
a representative plane for all vertices to be included in the plane, as illustrated in
Figure 12.14. Let the polygons be scaled up in size, giving
A
2
and
B
2
. Even though
θ
θ
remains constant, the required thickness
d
2
of the new representative plane increases
without bound as the polygons are scaled up in size.
Clearly, the angle between the polygon normals is only a
relative
measurement of
the co-planarity of the polygons. For an
absolute
error measurement, the thickness
of the representative plane must be considered. A better solution to testing two
polygons for co-planarity is therefore to perform the merging of the two polygons,
conceptually, and see if the resulting polygon is considered planar by a polygon
planarity test (presented in the next section). This approach directly extends to the
merging of an arbitrary number of polygons.
To use a relative measurement of co-planarity for controlling merging of polygons
for use with a collision detection system, the vertices of the merged polygons must be
snapped to the representative plane of the merged polygon. This bounds the deviation
of the polygon vertices from the plane, allowing the distance error to remain less than
the tolerance value used to enforce thickness of the polygon for robust intersection
tests.
However, with welded vertices snapping the vertices of one or more polygons to a
representative plane inevitably introduces nonplanarity errors in faces sharing vertices
with these polygons. Trying to address this would likely cause a ripple effect involving
all vertices of the object. A reasonable solution is to let a subsequent triangulation
pass take care of triangulating any faces that have been made more nonplanar than
the given tolerance allows. Unfortunately, nothing guarantees that there are not more
faces created during this triangulation than are removed through the initial merging
process!
A
1
B
1
d
1
A
2
B
2
d
2
Figure 12.14
Testing the angle between the normals of the planes of two polygons is a
relative
measurement of their co-planarity, unaffected by scaling up the polygons. Testing
the thickness required for the best-fit plane to contain all polygon vertices is an
absolute
measurement of the co-planarity of the polygons.
