Computational geometry videos

Interactive Boundary Computation of Boolean Combinations of Solids

Boolean operations on solids are of the form UNION, DIFFERENCE, or 
INTERSECTION on the boundaries of regular solids. These operations allow 
for the design of complex objects. The algorithm presented is one that 
greatly reduces the computational power required to implement these 
operations.

			
OBB-Trees and Rejection sets of cuboids

The application investigated is that of the piston block of an engine.
The tree is a structure the traversal of which yields data aout the 
possible region of clearance for the engine block. The rejection set is 
reflective of the possibility of clearance. So, for instance, at the 
places where the engine block is in close contact with the walls of the 
piston, the rejection set is composed of a large number of small cuboids, 
while at the extremities, where the probability of collision is small, 
the rejection set is coarsely composed of large cuboids.

Real time representation of massive models

A massive model is one that is composed of more than 500,000 elements. 
Real time is defined to be rendering at approximately 20 frames/second.

The processing of the visibility of a large number of elements in the 
model is extremely intensive. Real time rendering requires some kind of 
simplification technique that allows for selective rendering. The method 
proposed is typified by the provision of simplification envelopes of 
different degres of detail which reduce each element in the model to a 
set of polygons. Culling techniques such as this greatly improve on 
rendering time. 

Radiosity in flatland made visibly simple.

Visibility is a critical factor.
Radiosity is the amount of light being reflected from each of the 
elements of the model. It is assumed that certain elements are diffuse 
reflectors of light. The technique outlined relies on the definition of a 
space and its dual which is a space of operators on the original vector 
space. A meshing technique is employed to simplify the rendering of the 
elements.

Four polytopes and a funeral!

A technique is outlined that allows for the projection of polytopes in 
hyperspaces to lower dimensional spaces

A package for triangulation.

Integer linear programming and graphs are used to achieve the optimal 
triangulation of a polygon. Two graphs A and B are termed adjacent if B 
differs from A in only one edge. A adjacency graph is created where 
each node in the graph is itself a graph and an edge exists between two 
nodes if and only if they are adjacent as graphs. A combination of 
integer LP and these graphs allow for optimal triangulation of arbitrary 
n-gons.

Testing control collision detection

A collision between two solids is viewed in terms of a finite number of 
rapid point collisions. The closest feature algorithm is used and the 
assumption of coherence of motion (continuous as opposed to discrete motion)
is used to maintain a knowledge of the closest points between the objects.
Furthermore, spatial partitioning is used to reduce the computation 
necessary to detect the regions that are most likely to be involved in a 
collision. 

Interactive mechanism analysis using configuration spaces.

HIPair software that allows for the interactive design of solids by 
analysing their contact constraints. 

3D modeling. 

Dealunay triangulations and Voronoi diagrams form a space/dual pair. 
Tetrahedra are filled into the contour spaces of adjacent sections of the 
solid. The tetrahedra that lie outside the delineated space of the object 
are selectively removed. The union of all the sections of the object 
allow for an accurate 3D rendition of the solid. 

Collison detection for mulitple moving polytopes.

Hull features are distinguished from concavity features. Coherence of 
motion is exploited to determine collision points and results.

Decomposition of convex boundaries into polygons.

Minimal surface decomposition technique.
Uses a stackless breadth first search technique, which for certain 
configuration of solids is highly inefficient. An alternative in these 
cases is the use of the 'flood and retreat' technique, which does provide 
optimal solutions.

Animating the offset polygon distance function.

Triangulation based object reconstruction method consists of a dense 
triangle mesh. The mesh is simplified and a process called A-patch 
fitting is carried out.