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## Efficient Distance Computation Between NonConvex Objects

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Title: Efficient Distance Computation Between NonConvex Objects

1
Efficient Distance Computation Between Non-Convex
Objects
• By Sean Quinlan
• Reviewed by Mehboob A. Nazarani

2
Overview
• Model objects as the union of a set of convex
components
• Construct a hierarchical bounding representation
based on sphere from this model
• Simple search routine that uses the hierarchical
bounding representation of each object and
determines pairs of components to compare with a
convex distance algorithm.
• Compute the distance between pairs of convex
components using preexisting techniques
• Accept a relative error in the returned result

3
Assumptions
• Underlying model is a surface representation
consisting of a set of convex polygons
• Collision not detected when one object completely
contains another object.

4
Basic Structure
• Bounding representation is based on Spheres.
• Representation consists of a balanced binary tree
• Each node of the tree contains a single sphere

5
Tree Properties
• The Union of leaf spheres completely contains the
surface of the object.
• The sphere of each node completely contains the
spheres of its descendant leaf nodes.

6
Idea Behind Bounding Representation
• Leaf spheres closely approximate the surface of
the object.
• Interior nodes represent an approximation of
descendant leaf spheres
• Nodes determine the lower bound for the distance
to any of the descendant leaf nodes

7
Building the Tree First Step
• Cover surface with leaf nodes
• Regular grid of equal-sized spheres covers the
polygon with the center of each sphere lying in
the plane of the polygon
• Label each sphere with the polygon for which it
was created.

8
Building the Tree Second Step
• Build interior nodes through divide and conquer
technique
• Divide set of leaves into two subtrees
• Build tree for each subset and combine them into
a single tree by creating a new node with each
subtree as children
• Build subtrees through recursion until the set
consists of single leaf node

9
The Bounding Tree for an Object
10
Splitting
• No optimal methods for splitting known
• Objective is to split set of leaf nodes into two
subsets so that the bounding sphere will be small
• Bounding Rectanguloid

11
Splitting, contd.
• Determine Min and Max value for 3 coordinates for
the position vectors
• Select axes along which bounding box is longest
and divide leaf nodes using the average value
along these axes as the discriminating node
• Build a tree for each subset
• Determine a bounding sphere

12
Computing the Sphere
• First Heuristic Method
• Find bounding sphere that contains spheres of 2
children nodes.
• Works well near the leaves.
• Second Heuristic Method
• Directly consider the leaf spheres. Select a
center by using the average position of the
centers of the leaf spheres.
• Examine each descendant leaf sphere to determine
• Works well closer to the root.

13
Execution Time
• Depth logn
• Expected execution time O(nlogn)
• O(n2), worst case
• Precomputation

14
Computing the Distance
• d is the distance between two objects
• Goal is to show
• Objects are d distance apart, or
• Objects intersect

15
General Outline of Algorithm
• d 8
• Examine the pair of nodes in DFS manner starting
with the root nodes of the two trees
• if the distance between the two nodes is ? d
• then do nothing
• else
• Examine the children of the nodes

16
Algorithm, contd.
• Case 1 Both nodes are from interior
• Split one of the nodes into two children
• Recursively split the pair
• Case 2 One interior and one leaf node
• Split the interior node
• Recursively search the pair
• Case 3 Two leaf spheres less than d distance
apart
• Compute the distance between the underlying
polygon
• Return new d

17
Relative Error
• Relative Error ?
• d such that d lt d
• And d d lt ?d

18
Typical Configuration of Chess Pieces
19
Search Size v. Relative Error