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need to find at most one face aligned with the contact plane. Then intersect that
face with the other shape's face—and we have our contact manifold. High-poly
objects (like a curvy sports car) may need to have a lot of faces aligned with the
contact plane, so this simple method needs to be adjusted. But let's examine the
toy cubes.
Usually we can recover a shape feature (vertex, edge, or face) that is closest
to the support contact plane from our collision algorithm. If we have a degenerate
case like edge-face, face-face, or a pair of parallel edges, we'll usually have a
vertex-face or edge-edge pair to start with, but we want to expand both features to
larger features (edge-edge
face-face). From the smaller shape feature (vertex
or edge), we can examine all adjacent edges and faces and choose the one (edge
or face) that is most aligned with the support contact plane within some tolerance
α . That is, we want to find a face that is tilted at most by angle α from s .Ifno
face is aligned, we want an edge. And if no edge is aligned enough, we'll have
to live with one vertex. Whatever feature we find, let's call it the contact feature
(contact vertex, edge, or face).
To find a contact face, we just need to compute the alignment of the edges be-
cause for a face to be aligned with the contact plane, its edges need to be aligned.
Another important reason is more subtle: to compute how aligned the edges are,
we compute the dot product of the edge vector and the contact normal. When it's
close to zero, the value of the dot product is approximately the length of the edge
multiplied by the angle between the edge and the contact plane. That's a good
number to compare to the threshold. Equivalently, we could compute the differ-
ence between the heights of the edge vertices above the contact plane—a value
we may already have cached when we computed the support function, if we deal
with really low-poly shapes like cubes.
If we compute a face's alignment with the contact plane using n
s ,we'll
end up with the cosine of the face tilt angle relative to s . That value needs to
be close to 1 . 0 for the face to be aligned, and it isn't very sensitive to the face
tilt at small angles, but it is very sensitive to round-off errors. That is because
unlike the dot product of the edge vector with the normal, this cosine is changing
as 1
·
o ( x 2 ) instead of o ( x ).Thevariable x is the very small angle between the
face (or edge) and the contact plane, so x 2 isn't a good value to compare to a very
G ( F )
Gauss map of feature F
n 0 n 1
Angle between vectors n 0 and n 1
S 2
Unit sphere: { n R 3 : || n || =1 }
Spherical cap on S 2 : { x S 2 :( s x ) ≤ α}
C ( α, s )
The border of C ,thecircle: { x S 2 :( s x )= α}
δC
Ta b l e 4 . 5 . Notation guide.
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