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Figure 4.20. Computing e for a flat angle arc: (a) edge, normals, and arc orientation;
(b) flat-angle arc direction; (c) flipped normal order; and (d) flipped edge direction, which
is only free in the flat-angle case.
simply assume e =
according to
the right-hand rule. See Figure 4.20(b) for an example. If we switch n 0 and n 1
and keep the direction of e , we'll effectively flip the arc (see Figure 4.20(c) ) . If
we flip the direction of e , we'll also effectively flip the arc (see Figure 4.20(d) ) .
Thus, it is always possible to choose the direction of e that works with our arc-arc
test. Just keep in mind the direction of the arc from n 0 to n 1 : it must be consis-
tent with right-hand-rule rotation with regard to direction e , and that is all that is
needed for the arc-arc test to work.
Notice that the arc-arc test does not work with arcs of over 180 . Neither does
the origin-in-tetrahedron test work with such arcs. Fortunately, polytope SAT
does not have to deal with such cases, as all normals of the collision shape must
point “outwards” of the shape. In nonpolytope cases, special care must be taken.
For example, the one edge in a capsule shape maps to a great circle on
norm ||·|| ( v 1
v 0 ) and choose the sign
2 ,sowe
must employ a different kind of edge-edge culling.
Figure 4.21. SAT traits: (a) SAT can skip vertex-vertex pairs for penetrating shapes;
(b) SAT search space is inherently concave: the typical and simple Minkowski sum may
have six (or more) local minima (corresponding penetrating distance vectors shown).
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