13,398 research outputs found
Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
We discuss how the shape of a special Cosserat rod can be represented as a
path in the special Euclidean algebra. By shape we mean all those geometric
features that are invariant under isometries of the three-dimensional ambient
space. The representation of the shape as a path in the special Euclidean
algebra is intrinsic to the description of the mechanical properties of a rod,
since it is given directly in terms of the strain fields that stimulate the
elastic response of special Cosserat rods. Moreover, such a representation
leads naturally to discretization schemes that avoid the need for the expensive
reconstruction of the strains from the discretized placement and for
interpolation procedures which introduce some arbitrariness in popular
numerical schemes. Given the shape of a rod and the positioning of one of its
cross sections, the full placement in the ambient space can be uniquely
reconstructed and described by means of a base curve endowed with a material
frame. By viewing a geometric curve as a rod with degenerate point-like cross
sections, we highlight the essential difference between rods and framed curves,
and clarify why the family of relatively parallel adapted frames is not
suitable for describing the mechanics of rods but is the appropriate tool for
dealing with the geometry of curves.Comment: Revised version; 25 pages; 7 figure
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From on-line sketching to 2D and 3D geometry: A fuzzy knowledge based system
The paper describes the development of a fuzzy knowledge based prototype system for conceptual design. This real time system is designed to infer userâs sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles, arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tools, but also accepts userâs interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry
Computation of protein geometry and its applications: Packing and function prediction
This chapter discusses geometric models of biomolecules and geometric
constructs, including the union of ball model, the weigthed Voronoi diagram,
the weighted Delaunay triangulation, and the alpha shapes. These geometric
constructs enable fast and analytical computaton of shapes of biomoleculres
(including features such as voids and pockets) and metric properties (such as
area and volume). The algorithms of Delaunay triangulation, computation of
voids and pockets, as well volume/area computation are also described. In
addition, applications in packing analysis of protein structures and protein
function prediction are also discussed.Comment: 32 pages, 9 figure
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
Approximation, generalisation and deconstruction of planar curves
CISRG discussion paper ; 1
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