2.8. BINARY SPACE PARTITIONING TREES
Exploration of BSP trees
Overview of binary space-partitioning trees
BSP Tree Hierarchies
A binary space-partitioning (BSP) tree recursively partitions space into
pairs of subspaces with respect to dividing planes of arbitrary position and
orientation. The two partitions are referred to as the positive and negative
halfspaces (respectively in front of and behind the dividing plane).
A BSP tree is a versatile data structure, capable of performing the same
tasks as a k-d tree or octree (but not vice versa).
The BSP tree can
also be used to
provide
boundary/solid
representations
of arbitrary
polyhedral
scenes.
BSP Tree Hierarchies
BSP trees are very versatile in collision detection as they can provide both
a spatial partitioning (a nonboundary representation) and a volume
representation (a boundary representation, for a solid object). As a volume
representation, BSP trees can
be used to represent and
distinguish the interiors of
polygons and polyhedra from
their exteriors.
BSP Tree Hierarchies
The left figure provides an example of spatial partitioning (with a typical
lookup cost of O(log n)). The right figure shows how a BSP tree can map
out the exterior/internal of a polygonal object.
When, as illustrated, the dividing planes are selected to match the faces of
the input geometry it is termed autopartitioning (or polygon aligned). In
contrast, dividing planes coplanar to the xy, xz, or yz planes are called axis
aligned. Dividing planes that have no restrictions put on them (nor on the
partitionings they form) are called arbitrary or general.
Types of BSP Tree: Leaf storing
A leaf-storing (or leaf-based) BSP tree stores geometry in the leaves of the
tree (rather than in the internal nodes). Each internal node only contains
the dividing plane and references to the child subtrees. The dividing planes
of leaf-storing trees can be either arbitrary or autopartitioning.
An example leaf-storing tree is
shown using arbitrary positioning.
Types of BSP Tree: Solid Leaf
Solid-leaf BSP trees are built to represent the solid volume occupied by the
input geometry, i.e. dividing planes are selected to separate the solid
volume from the exterior of the object. No geometry is stored in the tree,
with leaf nodes only indicating if the area in front and behind of the
dividing plane is empty or solid.
Consider the example in which a solid-leaf BSP tree is constructed to
represent the interior of a concave (dart-shaped) polygon.
Solid-leaf BSP trees
are useful for collision
detection as there is
no need to perform
specific polygon tests
(no polygons are
explicitly stored).
Overview of how a BSP tree can be constructed
Building the BSP tree
Building a BSP tree (either from scratch or recomputing parts of a tree) is
computationally expensive, i.e. BSP trees are typically pre-computed and
hold static background geometry (moving objects are handled using some
other approach). Building a BSP tree involves three steps.
1. Selection of a partitioning plane.
2.Partitioning input geometry into the
positive and negative halfspaces of
the dividing plane. Geometry that
straddles the plane is split to the
plane before partitioning.
3. Recursively repeat the above steps
for each subtree until some
termination condition is reached.
Building the BSP tree
Recursively build a leaf storing BSP using the input list of polygons (depth initially equal to o)
BSPNode BuildBSPTree(List<Polygon> polygons, int depth)
{
if (polygons.empty()) return NULL; If no more polygons are available, then return
if (depth >= MAX_DEPTH || polygons.Count <= MIN_LEAF_SIZE) || etc. )
return new BSPNode(polygons); If leaf creation criteria is met, then create leaf node
For a leaf-storing tree, coplanar
Select the partitioning plane
Plane splitPlane case
= PickSplittingPlane(polygons);
COPLANAR_WITH_PLANE: polygons can be sent to either side, to
based on the input geometry
the front in this implementation.
case IN_FRONT_OF_PLANE:
frontList.AddAtBack(polygons[i]);
List<Polygon> frontList
= new List<Polygon>();
break;
List<Polygon> backList
= new List<Polygon>();
case
BEHIND_PLANE:
for (int i = 0; i < numPolygons; i++) {
backList.push_back(poly);
switch (ClassifyPolygonToPlane(polygons[i],
splitPlane)) {
break;
// -> ... Testcase
Straddling
are split, list(s)
each polygon
against the dividing plane, adding
them polygons
to their forwarding
STRADDLING_PLANE:
}
Polygon frontPart, backPart; with a part sent to each side
}
SplitPolygon(polygons[i],
out node
frontPart, out backPart);
Recursively build child subtrees andsplitPlane,
return the combined
BSPNode frontTree =frontList.AddAtBack(frontPart);
BuildBSPTree(frontList, depth + 1);
BSPNode backTree = backList.AddAtBack(backPart);
BuildBSPTree(backList, depth + 1);
break;
return new BSPNode(frontTree, backTree);
}
Building the BSP tree
The PickSplittingPlane() function is non-trivial if a good partitioning plane
is to be selected. The condition under which tree construction halts
depends on the type of tree.
• For a solid-leaf tree construction proceeds until the set of remaining
input polygons becomes empty.
• A leaf-storing BSP tree is typically stopped when:
oThe leaf contains less than some preset
number of polygons.
oA fixed cutoff depth has been reached.
oA good dividing plane cannot be found.
Selecting Dividing Planes
The dividing planes selected during tree construction greatly effects the
size and shape of the resulting tree (and hence performance and memory).
Typically good results can be obtained by trying a number of candidate
dividing planes at each level of the tree, picking the one that scores best
according to an evaluation function
If using autopartitioning then the search is restricted to the supporting
planes of the faces in the input geometry. Whilst simple to implement,
autopartitioning may produce more splitting than desired in leaf-storing
trees – e.g. consider the autopartitioning of a polygon sphere (all convex
faces will lie on the same
side of the plane, with a
resultant tree depth equal
to the number of sphere
faces!)
Selecting Dividing Planes
A good approach of handling detailed objects within a scene, is to use
bounding volume approximations of the objects and autopartition through
the faces of the bounding volumes before performing any splitting
involving the geometry within the bounding volumes.
Alternative candidate dividing planes can include planes in a few
predetermined directions (e.g. coordinate axes) or evenly distributed
across the extents of the input geometry (i.e. forming a grid across the
geometry).
Arbitrary planes can also be selected through a
hill-climbing approach whereby, repeatedly, a
number of slightly different planes to a
candidate plane are tested to determine the
new candidate plane.
Evaluating Dividing Planes
Evaluation criteria of use to collision detection includes picking planes so
as to minimize splitting of geometry (least-crossed stragegy) or to attempt
to balance the geometry equally on both sides of the splitting plane.
The approaches tend not to be complementary, with a strong focus on one
criteria likely to provide poor performance in the other criteria. A weighted
linear combination of the two is often used in practice (with reduced
splitting often weighted higher than balancing).
Line A minimises
split polygons;
Line B balances
the number of
polygons; Line C
represents a
compromise.
Classifying Polygons with Respect to a Plane
When building a BSP tree, all polygons need to be
partitioned with respect of a dividing plane into those that:
• lie in front of the dividing plane.
• lie behind the dividing plane.
• straddle the dividing plane.
• lie coincident with the dividing plane
Whilst mathematically straightforward, inaccuracies in floating-point
arithmetic entail that a split polygon may still be found to straddle the
dividing plane.
A practical solution to this problem is to use thick planes defined as n • (X −
P) < ε (for some ε), i.e. all points within a small distance of the plane are
considered lying on the plane.
Splitting Polygons Against a Plane
During BSP tree
construction, when a
polygon is found
straddling a dividing
plane it must be split in
two. Clipping a polygon
against a plane is done
using the Sutherland–
Hodgman algorithm. The
algorithm considers one
polygon edge at a time
and returns a set of points
based on an intersection
test between the edge
and plane.
Splitting Polygons Against a Plane
Whilst polygons are split during the construction of the tree, this need not
mean that the polygon fragments ending up in the leaves must be the
polygons output in the final tree.
For collision detection, it is better to store a reference to the original
polygon with each fragment generated during splitting. In the leaves, the
reference is output, and not the fragment.
Overview of how a BSP tree can be used
Testing a Point Against a Solid-leaf BSP Tree
Determining if a point lies in empty or solid space of a solid-leaf BSP tree
can be done by evaluating the point with respect to a node’s dividing
plane. The side on which the point falls determines which child node is next
visited. Traversal continues until a leaf node is reached. The value of the
leaf node answers the query.
If the tested point lies on the boundary of a
solid volume represented by the BSP tree
then both subtrees should be explored.
When both subtrees are traversed if
different results are returned (e.g. solid and
empty) then the point lies on the boundary
(a matching return from both subtrees is
simply returned).
Testing a Point Against a Solid-leaf BSP Tree
HitType PointInSolidSpace(BSPNode node, Point p)
{
Determine the distance the point is from the plane
while (!node.IsLeaf) {
float dist = Dot(node.Plane.n, p) – node.Plane.d;
if (dist > EPSILON) {
If the point lies in front of the plane, then visit the
node = node.Child[0];
first subtree, else if behind the plane, then visit
} else if (dist < -EPSILON) { the second subtree
node = node.Child[1];
} else {
If the point lies on the dividing plane then visit both subtrees
HitType front = PointInSolidSpace(node.Child[0], p);
HitType back = PointInSolidSpace(node.Child[1], p);
If the returned results differ, then return that the point is on the boundary
return (front == back) ? front : HitType.OnBoundary;
}
}
If a leaf node has been reached then return it’s recorded type
return node.IsSolid ? HitType.Inside : HitType.Outside;
}
Intersecting a Ray Against a Solid-leaf BSP Tree
When testing for intersection of a ray or segment against the tree care
must be exercised to ensure that the tested ray or segment is not
repeatedly subdivided against the splitting plane (as repeated subdivision
can introduce drift due to accumulated floating point inaccuracies). A
better approach is to determine the intersection ‘time’ of the ray against
the plane (R(t) = P + td, tmin ≤ t ≤ tmax). The time thit of the first intersection
with a solid leaf is returned (should an intersection be found).
Aside: If thick planes are
used, then an epsilon term
needs to be added to the
plane intersection tests, with
both sides explored given
plane intersection.
Testing a Ray Against a Solid-leaf BSP Tree
HitType RayIntersect(BSPNode node, Point p, Vector d,
float tmin, float tmax, out float thit)
{
Stack<BSPNode> nodeStack; Define two stacks to hold
Stack<float>
timeStack; ‘recursive’ information
Determine if the ray (or segment)
intersects with solid geometry. If
available, also return the ‘time’ of
intersection thit (where R(t) = p +
t*d, tmin <=t <=tmax).
while (true) {
if (node.IsLeaf == false) {
Find the side of the splitting
float dist = node.Plane.d - Dot(node.Plane.n, p);
plane on which the ray starts
int nearIndex = dist > 0.0f ? 1 : 0;
Determine the direction of the ray relative to the plane (a value of 0.0 denotes a
parallel ray – in which case the side of the plane the ray lies will be followed)
float denom = Dot(node.Plane.n, d);
if (denom != 0.0f) {
Determine the ray ‘time’ of intersection
float t = dist / denom;
A straddling ray has been detected, i.e.
if (0.0f <= t && t <= tmax) {
if (t >= tmin) {
push far side onto stack and visit near side
nodeStack.Push(node.Child[1 ∧ nearIndex]);
timeStack.Push(tmax);
tmax = t;
} else nearIndex = 1 ∧ nearIndex; If 0 <= t < tmin, then visit far side
}
}
node = node.Child[nearIndex];
}
Testing a Point Against a Solid-leaf BSP Tree
else { A leaf node has been reached. If it is solid, then record a hit at tmin
if (node.IsSolid) {
thit = tmin;
return HitType.Inside;
}
If no more subtrees remain to be
if (nodeStack.empty()) break; considered, then exit, otherwise pop the
topmost node from the stack and continue.
tmin = tmax;
node = nodeStack.Top(); nodeStack.Pop();
tmax = timeStack.Top(); timeStack.Pop();
}
}
return HitType.Outside; If reaching here, no solid leaf node could be found.
}
Directed mathematical reading
Directed reading
• Read Chapter 8 of Real Time
Collision Detection (pp349-381)
• Related papers can be found from:
http://realtimecollisiondetection.net/books/rtcd/references/
Summary
Today we
explored:
 How to build
and use a BSP
tree