259 research outputs found
On three soft rectangle packing problems with guillotine constraints
We investigate how to partition a rectangular region of length and
height into rectangles of given areas using
two-stage guillotine cuts, so as to minimize either (i) the sum of the
perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of
the rectangles. These problems play an important role in the ongoing Vietnamese
land-allocation reform, as well as in the optimization of matrix multiplication
algorithms. We show that the first problem can be solved to optimality in
, while the two others are NP-hard. We propose mixed
integer programming (MIP) formulations and a binary search-based approach for
solving the NP-hard problems. Experimental analyses are conducted to compare
the solution approaches in terms of computational efficiency and solution
quality, for different objectives
Survey of two-dimensional acute triangulations
AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual
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Shape Design and Optimization for 3D Printing
In recent years, the 3D printing technology has become increasingly popular, with wide-spread uses in rapid prototyping, design, art, education, medical applications, food and fashion industries. It enables distributed manufacturing, allowing users to easily produce customized 3D objects in office or at home. The investment in 3D printing technology continues to drive down the cost of 3D printers, making them more affordable to consumers.
As 3D printing becomes more available, it also demands better computer algorithms to assist users in quickly and easily generating 3D content for printing. Creating 3D content often requires considerably more efforts and skills than creating 2D content. In this work, I will study several aspects of 3D shape design and optimization for 3D printing. I start by discussing my work in geometric puzzle design, which is a popular application of 3D printing in recreational math and art. Given user-provided input figures, the goal is to compute the minimum (or best) set of geometric shapes that can satisfy the given constraints (such as dissection constraints). The puzzle design also has to consider feasibility, such as avoiding interlocking pieces. I present two optimization-based algorithms to automatically generate customized 3D geometric puzzles, which can be directly printed for users to enjoy. They are also great tools for geometry education.
Next, I discuss shape optimization for printing functional tools and parts. Although current 3D modeling software allows a novice user to easily design 3D shapes, the resulting shapes are not guaranteed to meet required physical strength. For example, a poorly designed stool may easily collapse when a person sits on the stool; a poorly designed wrench may easily break under force. I study new algorithms to help users strengthen functional shapes in order to meet specific physical properties. The algorithm uses an optimization-based framework — it performs geometric shape deformation and structural optimization iteratively to minimize mechanical stresses in the presence of forces assuming typical use scenarios. Physically-based simulation is performed at run-time to evaluate the functional properties of the shape (e.g., mechanical stresses based on finite element methods), and the optimizer makes use of this information to improve the shape. Experimental results show that my algorithm can successfully optimize various 3D shapes, such as chairs, tables, utility tools, to withstand higher forces, while preserving the original shape as much as possible.
To improve the efficiency of physics simulation for general shapes, I also introduce a novel, SPH-based sampling algorithm, which can provide better tetrahedralization for use in the physics simulator. My new modeling algorithm can greatly reduce the design time, allowing users to quickly generate functional shapes that meet required physical standards
Decomposing and packing polygons / Dania el-Khechen.
In this thesis, we study three different problems in the field of computational geometry: the partitioning of a simple polygon into two congruent components, the partitioning of squares and rectangles into equal area components while minimizing the perimeter of the cuts, and the packing of the maximum number of squares in an orthogonal polygon. To solve the first problem, we present three polynomial time algorithms which given a simple polygon P partitions it, if possible, into two congruent and possibly nonsimple components P 1 and P 2 : an O ( n 2 log n ) time algorithm for properly congruent components and an O ( n 3 ) time algorithm for mirror congruent components. In our analysis of the second problem, we experimentally find new bounds on the optimal partitions of squares and rectangles into equal area components. The visualization of the best determined solutions allows us to conjecture some characteristics of a class of optimal solutions. Finally, for the third problem, we present three linear time algorithms for packing the maximum number of unit squares in three subclasses of orthogonal polygons: the staircase polygons, the pyramids and Manhattan skyline polygons. We also study a special case of the problem where the given orthogonal polygon has vertices with integer coordinates and the squares to pack are (2 {604} 2) squares. We model the latter problem with a binary integer program and we develop a system that produces and visualizes optimal solutions. The observation of such solutions aided us in proving some characteristics of a class of optimal solutions
Hinged Dissections Exist
We prove that any finite collection of polygons of equal area has a common
hinged dissection. That is, for any such collection of polygons there exists a
chain of polygons hinged at vertices that can be folded in the plane
continuously without self-intersection to form any polygon in the collection.
This result settles the open problem about the existence of hinged dissections
between pairs of polygons that goes back implicitly to 1864 and has been
studied extensively in the past ten years. Our result generalizes and indeed
builds upon the result from 1814 that polygons have common dissections (without
hinges). We also extend our common dissection result to edge-hinged dissections
of solid 3D polyhedra that have a common (unhinged) dissection, as determined
by Dehn's 1900 solution to Hilbert's Third Problem. Our proofs are
constructive, giving explicit algorithms in all cases. For a constant number of
planar polygons, both the number of pieces and running time required by our
construction are pseudopolynomial. This bound is the best possible, even for
unhinged dissections. Hinged dissections have possible applications to
reconfigurable robotics, programmable matter, and nanomanufacturing.Comment: 22 pages, 14 figure
Fast Fencing
We consider very natural "fence enclosure" problems studied by Capoyleas,
Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a
set of points in the plane, we aim at finding a set of closed curves
such that (1) each point is enclosed by a curve and (2) the total length of the
curves is minimized. We consider two main variants. In the first variant, we
pay a unit cost per curve in addition to the total length of the curves. An
equivalent formulation of this version is that we have to enclose unit
disks, paying only the total length of the enclosing curves. In the other
variant, we are allowed to use at most closed curves and pay no cost per
curve.
For the variant with at most closed curves, we present an algorithm that
is polynomial in both and . For the variant with unit cost per curve, or
unit disks, we present a near-linear time algorithm.
Capoyleas, Rote, and Woeginger solved the problem with at most curves in
time. Arkin, Khuller, and Mitchell used this to solve the unit cost
per curve version in exponential time. At the time, they conjectured that the
problem with curves is NP-hard for general . Our polynomial time
algorithm refutes this unless P equals NP
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