Godfried T. Toussaint

McGill University, Montréal, Quebec, Canada

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Publications (169)53.74 Total impact

  • Binay K.bhattacharya, Asishmukhopadhyay, Godfried T.toussaint
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    ABSTRACT: A simple polygon P is said to be weakly extrenally visible from a line segment L if it lies outside P and for every point p on the boundary of P there is a point q on L such that no point in the interior of lies inside P. In this paper, a linear time algorithm is proposed for computing a shortest line segment from which P is weakly externally visible. This is done by a suitable generalization of a linear time algorithm which solves the same problem for a convex polygon.
    International Journal of Computational Geometry & Applications 11/2011; 09(01). · 0.18 Impact Factor
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    ABSTRACT: Given a planar polygon (or chain) with a list of edges {e1, e2, e3, …, en-1, en}, we examine the effect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. We show that a star-shaped polygon can be convexified using O(n2) edge-swaps, while maintaining simplicity, and that this is tight in the worst case. We show that determining whether a signed polygon P can be transformed to one that has rotational or mirror symmetry with P, using transpositions, takes Θ(n log n) time. We prove that the problem of deciding whether transpositions can modify a polygon to fit inside a rectangle is weakly NP-complete. Finally we give an O(n log n) time algorithm to compute the maximum endpoint distance for an oriented chain.
    International Journal of Computational Geometry & Applications 11/2011; 21(01). · 0.18 Impact Factor
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    ABSTRACT: Given a planar polygon (or chain) with a list of edges fe 1 ; e 2 ; e 3 ; : : : ; e n 1 ; e n g, we examine the eect of several operations that permute this edge list, resulting in the formation of a new polygon. The main operations that we consider are: reversals which involve inverting the order of a sublist, transpositions which involve interchanging subchains (sublists), and edge-swaps which are a special case and involve interchanging two consecutive edges. Using these permuting operations, we explore the complexity of performing certain actions, such as convexifying a given polygon or obtaining its mirror image. When each edge of the given polygon has also been assigned a direction we say that the polygon is signed. In this case any edge involved in a reversal changes direction. The complexity of some problems varies depending on whether a polygon is signed or unsigned. An additional restriction in many cases is that polygons remain simple after every permutation.
    Int. J. Comput. Geometry Appl. 01/2011; 21:87-100.
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    Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, August 10-12, 2011; 01/2011
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    Csaba D. Tóth, Godfried T. Toussaint, Andrew Winslow
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    ABSTRACT: An open edge of a simple polygon is the set of points in the relative interior of an edge. We revisit several art gallery problems, previously considered for closed edge guards, using open edge guards. A guard edge of a polygon is an edge that sees every point inside the polygon. We show that every simple non-starshaped polygon admits at most one open guard edge, and give a simple new proof that it admits at most three closed guard edges. We also characterize open guard edges using a special type of kernel. Finally, we present lower bound constructions for simple polygons with n vertices that require ⌊n/3⌋ open edge guards, and conjecture that this bound is tight.
    Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, August 10-12, 2011; 01/2011
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    David Rappaport, Godfried T. Toussaint, Mustafa Mohamad
    Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, August 10-12, 2011; 01/2011
  • Yang Liu, Godfried T. Toussaint
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    ABSTRACT: The marble pavement of the Cathedral in the Tuscan city of Siena in Italy has been described as one of the marvels of the world. Over the centuries much has been written about its biblical and political characters, the stories depicted in its figurative mosaics, the artists responsible for creating the mosaics, the types of marble used and the history of their construction. The many frieze patterns framing the figurative mosaics are noteworthy examples of geometric design, and yet, they have been conspicuously overlooked in the literature concerning this pavement. Here, the geometric frieze patterns found on the pavement, walls and ceiling of the Siena Cathedral are analysed in terms of their underlying geometric structure, the optical effects, such as multi-stable perception, that they engender in the viewer and a typology of patterns of repetition.
    Journal of Mathematics and the Arts 01/2011; 5(3):115-127.
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    ABSTRACT: Abstract
    Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, Winnipeg, Manitoba, Canada, August 9-11, 2010; 01/2010
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    Godfried Toussaint
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    ABSTRACT: Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.
    Computational Geometry. 01/2010;
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    ABSTRACT: The Hexachordal Theorem may be interpreted in terms of scales, or rhythms, or as abstract mathematics. In terms of scales it claims that the complement of a chord that uses half the pitches of a scale is homometric to—i.e., has the same interval structure as—the original chord. In terms of onsets it claims that the complement of a rhythm with the same number of beats as rests is homometric to the original rhythm. We generalize the theorem in two directions: from points on a discrete circle (the mathematical model encompassing both scales and rhythms) to a continuous domain, and simultaneously from the discrete presence or absence of a pitch/onset to a continuous strength or weight of that pitch/onset. Athough this is a significant generalization of the Hexachordal Theorem, having all discrete versions as corollaries, our proof is arguably simpler than some that have appeared in the literature. We also establish the natural analog of what is sometimes known as Patterson’s second theorem: if two equal-weight rhythms are homometric, so are their complements.
    06/2009: pages 11-21;
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    ABSTRACT: Musical cyclic rhythms with a cycle length (timespan) of 8 or 16 pulses are called binary; those with 6 or 12 pulses are called ternary. The process of mapping a ternary rhythm of, say 12 pulses, to a rhythm of 16 pulses, such that musicologically salient properties are preserved is termed binarization. By analogy, the converse process of mapping a binary rhythm to a ternary rhythm is referred to as ternarization. New algorithms are proposed and investigated for the binarization and ternarization of musical rhythms with the goal of understanding the historical evolution of traditional rhythms through inter-cultural contacts. The algorithms also have applications to automated rhythmic pattern generation, and may be incorporated in composition software tools.
    05/2009;
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    Yang Liu, Godfried Toussaint
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    ABSTRACT: The traditional sona drawing art of Angola is compared from the structural point of view to similar geometric arts found in other parts of the world. A simple algorithm is proposed for constructing a sub-class of sona drawings, called perfect monolinear sona drawings, on a given set of points in the plane, that exhibit the topological structure of a tree. The application of the tree sona drawings obtained with this algorithm to the visual arts and Discipline Based Art Education is illustrated. 1



Introduction
 Many cultures all over the world have traditional visual art and design practices that unite them with a common thread: the use of geometric curves with certain geometric properties [19]. Such practices include sona drawings from Angola in Africa [9, 10], kolam drawings from South India [18, 33], Celtic knots [22, 26], Sand drawings from the South Pacific Islands [34], and Chinese knots [19]. A sona drawing consists of a single curve that starts at one point, meanders about a group of points previously drawn, and ends at the same place it started. Sometimes more than one curve is drawn, but the most challenging, and thus prized, drawings consist of one curve, and are called monolinear. Another desirable property of these drawings is that every region must contain exactly one point. Such a sona drawing will be called perfect. Fig. 1: Two symmetric sona drawings.
    01/2009;
  • Eric Thul, Godfried T. Toussaint
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    ABSTRACT: Eight measures of musical rhythm complexity are compared to each other using two families of real-world rhythms: subsaharan African timelines and North Indian talas. The complexity measures that are designed to measure syncopation, and that agree with human judgements, also agree with each other across the two families. Thus they may be considered as culturally robust measures, at least for these two families of rhythms. Furthermore, according to these measures the African timelines are more complex than the North Indian talas. On the other hand, the more mathematical measures of complexity are less culturally robust, and suggest that some North Indian talas are more complex, in this sense, than the African timelines.
    Canadian Conference on Computer Science & Software Engineering, C3S2E 2008, Montreal, Quebec, Canada, May 12-13, 2008, Proceedings; 01/2008
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    Joseph O'Rourke, Perouz Taslakian, Godfried T. Toussaint
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    ABSTRACT: Homometric rhythms (chords) are those with the same histogram or multiset of intervals (distances). The purpose of this note is threefold. First, to point out the potential importance of isospectral vertices in a pair of homometric rhythms. Second, to establish a method ("pumping") for generating an infinite sequence of homometric rhythms that include isospectral vertices. And finally, to introduce the notion of polyphonic homo- metric rhythms, which apparently have not been previ- ously explored.
    Proceedings of the 20th Annual Canadian Conference on Computational Geometry, Montreal, Canada, August 13-15, 2008; 01/2008
  • Eric Thul, Godfried T. Toussaint
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    ABSTRACT: Six measures of musical rhythm complexity were compared experimentally to human difficulty of performance (performance complexity) using two data sets of rhythms, via phylogenetic trees of the rank-correlation coefficient matrices obtained from rankings of the rhythms according to the complexity measures. The results suggest the hypothesis that measures of rhythmic syncopation that are based on a weighted metrical hierarchy, are better predictors of human performance difficulty than measures based on cognitive complexity, weighted distances from onsets to beats, or mathematical irregularity.
    Canadian Conference on Computer Science & Software Engineering, C3S2E 2008, Montreal, Quebec, Canada, May 12-13, 2008, Proceedings; 01/2008
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    Francisco Gomez Martin, Perouz Taslakian, Godfried T. Toussaint
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    ABSTRACT: In this paper we define four operations on musical rhythms that preserve a property called maximal evenness. The operations we define are shadow, complementation, concatenation, and alternation. The proofs of the theorems are omitted from this abstract.
    01/2008;
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    Eric Thul, Godfried T. Toussaint
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    ABSTRACT: Thirty two measures of rhythm complexity are compared using three widely different rhythm data sets. Twenty-two of these measures have been investigated in a limited con- text in the past, and ten new measures are explored here. Some of these measures are mathematically inspired, some were designed to measure syncopation, some were intended to predict various measures of human performance, some are based on constructs from music theory, such as Press- ing's cognitive complexity, and others are direct measures of different aspects of human performance, such as perceptual complexity, meter complexity, and performance complex- ity. In each data set the rhythms are ranked either accord- ing to increasing complexity using the judgements of human subjects, or using calculations with the computational mod- els. Spearman rank correlation coefficients are computed between all pairs of rhythm rankings. Then phylogenetic trees are used to visualize and cluster the correlation co- efficients. Among the many conclusions evident from the results, there are several observations common to all three data sets that are worthy of note. The syncopation measures form a tight cluster far from other clusters. The human per- formance measures fall in the same cluster as the syncopa- tion measures. The complexity measures based on statisti- cal properties of the inter-onset-interval histograms are poor predictors of syncopation or human performance complex- ity. Finally, this research suggests several open problems.
    ISMIR 2008, 9th International Conference on Music Information Retrieval, Drexel University, Philadelphia, PA, USA, September 14-18, 2008; 01/2008
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    ABSTRACT: 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 $1,2,...,k-1$. 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.
    06/2007;
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    Conference Paper: Vertex Pops and Popturns.
    Proceedings of the 19th Annual Canadian Conference on Computational Geometry, CCCG 2007, August 20-22, 2007, Carleton University, Ottawa, Canada; 01/2007
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    Justin Colannino, Godfried T. Toussaint
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    ABSTRACT: Recently, a number of interesting algorithmic problems have arisen from the emergence, in a number of countries, of kidney exchange schemes, whereby live donors are matched with recipients according to compatibility and other considerations. One such ...
    Inf. Process. Lett. 01/2007; 103:44.

Publication Stats

3k Citations
53.74 Total Impact Points

Institutions

  • 1974–2011
    • McGill University
      • • School of Computer Science
      • • Centre for Interdisciplinary Research in Music Media & Technology (CIRMMT)
      Montréal, Quebec, Canada
  • 2010
    • Harvard University
      • Department of Music
      Cambridge, Massachusetts, United States
  • 2005
    • Universidad de Sevilla
      • Applied Mathematics II
      Sevilla, Andalusia, Spain
  • 1997–2002
    • Simon Fraser University
      • School of Computing Science
      Burnaby, British Columbia, Canada
  • 1992
    • University of Kentucky
      • Department of Computer Science
      Lexington, KY, United States
  • 1990
    • Kyushu University
      Hukuoka, Fukuoka, Japan
  • 1979–1980
    • Concordia University Montreal
      Montréal, Quebec, Canada