The Writhing Number of a Space Curve

Fuller, F. B.

doi:10.1073/pnas.68.4.815

Cited by 696 publications

(397 citation statements)

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“…The writhe number was introduced by Fuller 6 and it is used in curve theory to measure the extent to which a curve twists and coils around itself. In biomedical engineering, the 2D writhe number is used to study the shape and topology of DNA 18 or to characterize the shape of curves on 3D surfaces, such as the curves of sulci and gyri on the cortical surface.…”

Section: The Writhe Of Surfacesmentioning

confidence: 99%

“…To this end, we introduce a series of original shape descriptors based on the writhe number. 6 Similar to recent morphological studies, 9,12,13,16,17 the rupture status prediction is performed on a data set of aneurysms, which were labeled as being ruptured or not at the time of detection by the treating neurosurgeons. In the proposed methodology, aneurysms are isolated from the segmented vasculature represented as a 3D triangular mesh.…”

Section: Introductionmentioning

confidence: 99%

“…Writhe numbers are computed for vertices along the surface. Known in curve theory since its introduction by Fuller,6 the writhe number was recently extended to 3D surfaces. 11 There, 11 the writhe number of surfaces was used as a binary parameter to distinguish between healthy vessels and possible aneurysmal regions.…”

Section: Introductionmentioning

confidence: 99%

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3D Shape Analysis of Intracranial Aneurysms Using the Writhe Number as a Discriminant for Rupture

et al. 2011

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Abstract-Intracranial aneurysms are polymorphic focal arterial dilations, which harbor a variable risk of rupture leading to high morbidity and mortality. Increased detection of incidental aneurysms by non-invasive imaging has created a need for rupture risk stratification tools, in addition to simple aneurysm size, to guide optimal treatment strategy. To this end, shape analysis has emerged as a possible differentiator of rupture likelihood. A novel set of morphological parameters based on the writhe number are introduced here to describe aneurysms and discriminate rupture status. Classification in 117 saccular aneurysms (52 ruptured and 65 unruptured) is based on statistical analysis of writhe number distribution on the aneurysm surface. Aneurysms are analyzed both in isolation and including a portion of their parent vessel. Sidewall and bifurcation aneurysm subtypes were found to be best described by disjoint sets of shape parameters, yielding a morphological dichotomy between the two aneurysm classes. Writhe number analysis results in 86.7% accuracy on sidewall (SW) aneurysms and 71.2% accuracy on bifurcation (BF) aneurysms. This represents a 12% accuracy increase for both subtypes compared to the performance of seven established 2D and 3D indexes. The results support the utility of writhe number aneurysm shape analysis, with potential clinical value in rupture risk stratification.

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Section: The Writhe Of Surfacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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3D Shape Analysis of Intracranial Aneurysms Using the Writhe Number as a Discriminant for Rupture

et al. 2011

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“…It was introduced in [Cȃlugȃreanu, 1959-61] and named in [Fuller, 1971], and has proved its importance for molecular biologists in the study of knotted duplex DNA, and of the enzymes which affect it. Clearly, writhing number for knots is the analogue of helicity for vector fields.…”

Section: Helicity and Writhing Numbermentioning

confidence: 99%

Influence of Geometry and Topology on Helicity

Cantarella¹,

DeTurck²,

Gluck³

et al. 2013

Magnetic Helicity in Space and Laboratory Plasmas

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The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics and plasma physics. In this report we show how the relation between energy and helicity of a vector field is influenced by the geometry and topology of the domain on which it is defined. In particular, we will see that the standard model for the magnetic field in the Crab Nebula (equivalently, the spheromak field of plasma physics) is the unique energyminimizing divergence-free vector field of given nonzero helicity, defined on and tangent to the boundary of a round ball, and that the essential features of this energy-minimizing field persist even as the domain changes topological type. We will also see that when volume-preserving deformation of domain is permitted, the spheromak field is not the absolute energy-minimizing field with given helicity; instead, the round ball on which it is defined can be dimpled in at the poles and expanded out at the equator to further decrease the field energy while preserving helicity. Our numerical computations suggest that this volume-preserving, helicity-preserving, energy-decreasing deformation of domain and field converges to a singular domain, in which the north and south poles have been pressed together at the center, along with a corresponding singular field. Two fundamental problemsWe organize this report by focusing on two fundamental problems:1. Minimize energy among all divergence-free vector fields of given nonzero helicity, defined on and tangent to the boundary of a given domain.2. Find the above minimum over all domains of given volume.Such energy-minimizing vector fields provide models for stable force-free magnetic fields in gaseous nebulae and laboratory plasmas, while the search for them seems to bring out some of the deepest and most useful mathematics connected with helicity. Helicity and writhing numberThe helicity H(V ) of a smooth (meaning C ∞ ) vector field V on the domain Ω in 3-space, defined by the

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“…We envision travell~ng wave forms in DNA to be right-handed superhelices, containing approximately ten base·pairs per turn and having a·pitch of about 34 A. These structures balance the unwinding in the· helix with, right-handed superhelical writhe to keep the linkage 'invariant (Fuller, 1971), a feature that creates minimal perturbation in DNA structure. At .small amplitudes of oscillation, DNA behaves as an elastic body that accumulates strain energy in the sugar-phosphate chains through small departures of torsional ~ngles from their equilibrium values (compar~ Figures·lA and B).…”

mentioning

confidence: 99%