3,552 research outputs found
The Distance Geometry of Music
We demonstrate relationships between the classic Euclidean algorithm and many
other fields of study, particularly in the context of music and distance
geometry. Specifically, we show how the structure of the Euclidean algorithm
defines a family of rhythms which encompass over forty timelines
(\emph{ostinatos}) from traditional world music. We prove that these
\emph{Euclidean rhythms} have the mathematical property that their onset
patterns are distributed as evenly as possible: they maximize the sum of the
Euclidean distances between all pairs of onsets, viewing onsets as points on a
circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this
notion of \emph{evenness}. We also show that essentially all Euclidean rhythms
are \emph{deep}: each distinct distance between onsets occurs with a unique
multiplicity, and these multiplicies form an interval . Finally,
we characterize all deep rhythms, showing that they form a subclass of
generated rhythms, which in turn proves a useful property called shelling. All
of our results for musical rhythms apply equally well to musical scales. In
addition, many of the problems we explore are interesting in their own right as
distance geometry problems on the circle; some of the same problems were
explored by Erd\H{o}s in the plane.Comment: This is the full version of the paper: "The distance geometry of deep
rhythms and scales." 17th Canadian Conference on Computational Geometry (CCCG
'05), University of Windsor, Canada, 200
Distance Geometry for Kissing Spheres
A kissing sphere is a sphere that is tangent to a fixed reference ball. We
develop in this paper a distance geometry for kissing spheres, which turns out
to be a generalization of the classical Euclidean distance geometry.Comment: 11 pages, 2 picture
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Distance Geometry in Quasihypermetric Spaces. III
Let be a compact metric space and let denote the
space of all finite signed Borel measures on . Define by
and set , where ranges over the collection of signed
measures in of total mass 1. This paper, with two earlier
papers [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric
spaces. I and II], investigates the geometric constant and its
relationship to the metric properties of and the functional-analytic
properties of a certain subspace of when equipped with a
natural semi-inner product. Specifically, this paper explores links between the
properties of and metric embeddings of , and the properties of
when is a finite metric space.Comment: 20 pages. References [10] and [11] are arXiv:0809.0740v1 [math.MG]
and arXiv:0809.0744v1 [math.MG
Distance geometry in active structures
The final publication is available at link.springer.comDistance constraints are an emerging formulation that offers intuitive geometrical interpretation of otherwise complex problems. The formulation can be applied in problems such as position and singularity analysis and path planning of mechanisms and structures. This paper reviews the recent advances in distance geometry, providing a unified view of these apparently disparate problems. This survey reviews algebraic and numerical techniques, and is, to the best of our knowledge, the first attempt to summarize the different approaches relating to distance-based formulations.Peer ReviewedPostprint (author's final draft
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