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7.3. COMPONENTS 93
of the corresponding matrix, when a row and a column is added to or removed from the matrix.
The factorization classes are used to implement the internals of constraint and LCP solvers.
7.3.2 Geometry
Geometry library abstracts collision geometries and supports and implements collision detection
according to [10] and [22]. Collision geometries are represented by CollisionGeometry superclass
providing access to the collision geometry’s geometry hull (an axes aligned bounding box),
an infrastructure for attaching geometries to parent geometries and rigid bodies and mapping
of the local collision geometry spaces, the geometries are defined at, to the parent spaces (local
spaces of the parent geometries or the world space). This allows to create geometry hierarchies
and compose complex collision geometries from more primitive geometries. The library handles
the following geometry types and implements classes to perform low-level collision detection
among these types,
• Convex polyhedra
Convex polyhedra are volumes defined as the intersection of a set of half-spaces determined
by their boundary planes. Convex polyhedra are represented by the instances of Polytope
class, which store the boundary planes and other related information distilled from the
polytope definitions, such as lists of polytope faces, edges and vertices together with the
topology information, utilized by the collision detection code.
• Spheres
Spheres are defined as volumes with centers c and radii r that are parametrized by points
p, where p − r ≤ r. They are represented by the instances of Sphere class.
• Capped cylinders
Capped cylinders are volumes defined by two end points a and b and radii r, parametrized
by points p so that dist(p, −→ ab) ≤ r, where dist(p, −→ ab) denotes the distance of point p from
line −→ ab. Capped cylinders are usual cylinders with radii r and their principal axes pointing
along the directions of −→ ab with the addition of two sphere “caps” at their end points.
Capped cylinders can be seen as volumes swept by spheres with radii r when they are
translated from a to b along straight lines. Capped cylinders are represented by instances
of CappedCylinder class.
• Spaces
Spaces are containers for other geometry types. They represent volumes corresponding to
the unions of the (appropriately transformed) owned geometries and are implemented by
Space class.
CollisionDetection class manages collision geometries and implements higher-level and
lower-level collision detection. Higher-level collision detection operates upon geometry hulls of
collision geometries attached to rigid bodies and utilizes the sort and sweep algorithm to determine
all overlapping geometry hull pairs. Geometry hulls are represented by bounding boxes
with their sides aligned with the world space axes.
Lower-level collision detection invokes geometry type pair specific algorithms, encapsulated
by the subclasses of CollisionDetection::Collider class, that test for penetration and compute
contacts if the tested geometries do not penetrate. Collider classes often rely on the fact that contacts
need not be computed on penetrations (in fact it would be hard to define what the contacts
should be if geometries already penetrated), which allows to use more efficient algorithms.