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Non-Uniform Offsetting and its Applications in Laser Path Planning of Sterolithography Machine
Laser path planning is an important step in solid freeform fabrication processes such as
Stereolithography (SLA). An important consideration in the laser path planning is to compensate
the shape of laser beam. Currently the compensation is divided into two steps, Z-compensation
and X-Y compensation, and the shape of laser beam is assumed to be uniform for the whole
platform. In this research, we present a sampling based non-uniform offsetting method which
accounts for the different shapes of laser beam at various locations. We discuss the related steps
and algorithms. We demonstrate its effectiveness by using various test cases. Besides
improving the accuracy of SLA machine, non-uniform offsetting can also be applied to address
other accuracy issues caused by thermal and structural variationsMechanical Engineerin
Fuchsian convex bodies: basics of Brunn--Minkowski theory
The hyperbolic space \H^d can be defined as a pseudo-sphere in the
Minkowski space-time. In this paper, a Fuchsian group is a group of
linear isometries of the Minkowski space such that \H^d/\Gamma is a compact
manifold. We introduce Fuchsian convex bodies, which are closed convex sets in
Minkowski space, globally invariant for the action of a Fuchsian group. A
volume can be associated to each Fuchsian convex body, and, if the group is
fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be
studied in the same manner as convex bodies of Euclidean space in the classical
Brunn--Minkowski theory. For example, support functions can be defined, as
functions on a compact hyperbolic manifold instead of the sphere.
The main result is the convexity of the associated volume (it is log concave
in the classical setting). This implies analogs of Alexandrov--Fenchel and
Brunn--Minkowski inequalities. Here the inequalities are reversed
Minkowski Functional Description of Microwave Background Gaussianity
A Gaussian distribution of cosmic microwave background temperature
fluctuations is a generic prediction of inflation. Upcoming high-resolution
maps of the microwave background will allow detailed tests of Gaussianity down
to small angular scales, providing a crucial test of inflation. We propose
Minkowski functionals as a calculational tool for testing Gaussianity and
characterizing deviations from it. We review the mathematical formalism of
Minkowski functionals of random fields; for Gaussian fields the functionals can
be calculated exactly. We then apply the results to pixelized maps, giving
explicit expressions for calculating the functionals from maps as well as the
Gaussian predictions, including corrections for map boundaries, pixel noise,
and pixel size and shape. Variances of the functionals for Gaussian
distributions are derived in terms of the map correlation function.
Applications to microwave background maps are discussed.Comment: 24 pages with 2 figures. Submitted to New Astronom
Complexity in surfaces of densest packings for families of polyhedra
Packings of hard polyhedra have been studied for centuries due to their
mathematical aesthetic and more recently for their applications in fields such
as nanoscience, granular and colloidal matter, and biology. In all these
fields, particle shape is important for structure and properties, especially
upon crowding. Here, we explore packing as a function of shape. By combining
simulations and analytic calculations, we study three 2-parameter families of
hard polyhedra and report an extensive and systematic analysis of the densest
packings of more than 55,000 convex shapes. The three families have the
symmetries of triangle groups (icosahedral, octahedral, tetrahedral) and
interpolate between various symmetric solids (Platonic, Archimedean, Catalan).
We find that optimal (maximum) packing density surfaces that reveal unexpected
richness and complexity, containing as many as 130 different structures within
a single family. Our results demonstrate the utility of thinking of shape not
as a static property of an object in the context of packings, but rather as but
one point in a higher dimensional shape space whose neighbors in that space may
have identical or markedly different packings. Finally, we present and
interpret our packing results in a consistent and generally applicable way by
proposing a method to distinguish regions of packings and classify types of
transitions between them.Comment: 16 pages, 8 figure
Part Repair using a Hybrid Manufacturing System
Nowadays, part repair technology is gaining more interest from military and industries
due to the benefit of cost reducing as well as time and energy saving. Traditionally, part
repair is done in the repair department using welding process. The limitations of the
traditional welding process are becoming more and more noticeable when the accuracy
and reliability are required. Part repair process has been developed utilizing a hybrid
manufacturing system, in which the laser aided deposition and CNC cutting processes are
integrated. Part repair software is developed in order to facilitate the users. The system
and the software elevate the repair process to the next level, in which the accuracy,
reliability, and efficiency can be achieved. The concept of repair process is presented in
this paper. Verification and experimental results are also discussed.Mechanical Engineerin
Holography of the BTZ Black Hole, Inside and Out
We propose a 1+1 dimensional CFT dual structure for quantum gravity and
matter on the extended 2+1 dimensional BTZ black hole, realized as a quotient
of the Poincare patch of AdS. The quotient spacetime includes regions
beyond the singularity, "whiskers", containing timelike and lightlike closed
curves, which at first sight seem unphysical. The spacetime includes the usual
AdS-asymptotic boundaries outside the horizons as well as boundary components
inside the whiskers. We show that local boundary correlators with some
endpoints in the whisker regions: (i) are a protected class of amplitudes,
dominated by effective field theory even when the associated Witten diagrams
appear to traverse the singularity, (ii) describe well-defined
diffeomorphism-invariant quantum gravity amplitudes in BTZ, (iii) sharply probe
some of the physics inside the horizon but outside the singularity, and (iv)
are equivalent to correlators of specific non-local CFT operators in the
standard thermofield entangled state of two CFTs. In this sense, the whisker
regions can be considered as purely auxiliary spacetimes in which these useful
non-local CFT correlators can be rendered as local boundary correlators, and
their diagnostic value more readily understood. Our results follow by first
performing a novel reanalysis of the Rindler view of standard AdS/CFT duality
on the Poincare patch of AdS, followed by exploiting the simple quotient
structure of BTZ which turns the Rindler horizon into the BTZ black hole
horizon. While most of our checks are within gravitational effective field
theory, we arrive at a fully non-perturbative CFT proposal to probe the
UV-sensitive approach to the singularity.Comment: 52 pages, 15 figures. v2: Clarifications made throughout paper.
Derivation of (new) Section 8 corrected. Results and conclusions unchanged.
References adde
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