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Figure 11.2. Tetrahedron (triangle) intersecting the world geometry.
Figure 11.3. Stick intersecting the world geometry in two different ways.
In an analogous way, collisions that take place on a face of the tetrahedrons
or even inside the tetrahedron will require moving three or all four particles to fix
the penetration. Let p be the penetration point on the tetrahedron and q be the one
on the obstacle. To handle any type of collision, follow the procedure described
First, express p as a linear combination of the four particles that make up the
tetrahedron: p = c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 such that the weights sum to one:
c 1 + c 2 + c 3 + c 4 =1(this calls for solving a small system of linear equations).
After finding d = q
p , compute the value
c 1 + c 2 + c 3 + c 4
λ =
( λ is a so-called Lagrange multiplier). The new particle positions are then given
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