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Figure 4.3. Contact reduction and stability. For stability, the center-of-mass projection
(the × ) should be within the contact manifold. (a) Stable full-contact manifold. (b) Unsta-
ble reduced-contact manifold. (c) and (d) Stable reduced-contact manifold.
Feature (vertex, edge, or face)
Ta b l e 4 . 1 . Notation guide.
time. However, more contact points will often mean more-precise simulations, so
it's a trade-off between quality and speed. If speed is of higher priority, then
reducing the contact manifold is probably a win. The solver will generally go
through the contact points multiple times and execute more complex code than
contact reduction. So it's rather safe to assume that the time spent reducing con-
tact points will be a net win.
In case of concave contact areas, we only need to keep the convex hull of
contact points as the contact manifold, as in Figure 4.1. But even then, we may
reduce the contact points further, as in Figure 4.3(d).
Three or more contact points are necessary to keep an object statically stable.
As a rule of thumb, the center of mass should project along the gravity vector ( not
along the contact normal) well inside the contact manifold formed by them, as
in Figure 4.3(a), (c), and (d) (obviously, only when it's possible). It's convenient
to choose those points among the unreduced contact manifold corners, as shown
in Figure 4.3(c), but it's not always stable. Instead we may opt to leave four or
more points, like in Figure 4.3(a) and (d), of the originally computed manifold,
potentially using some heuristic to reduce other points, like in Figure 4.3(d).
One heuristic is to sort manifold vertices by their angle and reduce those with
the highest angle (updating the neighboring vertex angles). We need to always
keep the contact point with the most penetration, or otherwise compensate for the
penetration by assigning high penetration to other points to resolve it.
Resampling the contact manifold is another possible heuristic. If we choose
to resample every 90 , we are guaranteed not to miss manifold vertices with 90
or sharper angle. See Figure 4.14.
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