Geometric Constraints – Introduction.
OverOverView on Geometric Constraints
Orientation:
Let’s consider a square box of side ‘a’ (2d). Now imagine that we want to draw 3 circles
in this square without the circles overlapping each other or the boundaries of the square.
A desirable configuration would be something like this.
Basically what we want is to make these 3 circles as big as possible while still making
them fit into the box. That would translate to saying maximize
∑
r
i
where r
i
denotes the
radii of the circles.
Generally:
Pack(without overlapping) a disk D in 2d with given radius d with disks D1,D2,.
. Dn of
radius x1,x2, .
.,xn s.t.
∑
xi is maximized.
Task 1: Can this be expressed as a LP problem in the xi? (You are allowed to add more
variables.)
Task 1’: If no (answer to Task 1), express as simply as possibly. (Please note
“simply”
here. We’ll discuss it in detail later. )
Task 1:
Ideally we would like to capture all the constraints that have been implicitly
expressed in the above diagram and description in the form of equations. Write out one
such formulation. (Hint: Linear equations may not always work out). You are free to add
as many variables as necessary.
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View Full DocumentThe problem that was just described is known as a packing problem. And it is an example
of a geometric constraints problem. We have a lot of constraints and we would like to
solve for a possible configuration within those constraints. This particular problem under
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 Fall '08
 Staff
 Geometry, Convex hull, Delaunay triangulation, Convex geometry, Geometric Constraint Solving

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