Kähler structures on spaces of framed curves

Needham, Tom

doi:10.1007/s10455-018-9595-3

Cited by 9 publications

(16 citation statements)

References 31 publications

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“…The following theorem relates the complicated metric g S on framed path space to this simple L 2 -metric. This result is stated in [30], but not proved. We include a proof in the appendix (Section 7.1).…”

Section: The Classical Hopf Mapmentioning

confidence: 85%

“…We then utilize quaternionic algebra to give a coordinate transformation which flattens a particular choice of framed curve elastic metric. This construction was described in previous articles [31,30], where the focus was on symplectic geometry and theoretical applications. The exposition provided here is focused on algorithms and applications to shape analysis.…”

Section: Main Contributions Andmentioning

confidence: 99%

“…It can be deduced from the fact that the fundamental group of SOp3q is Z{2Z that the manifold p S c has two connected components. The component that a framed pγ, V q curve belongs to is determined by the topological linking number of γ with a small pushoff along the V -direction, measured modulo-2 (a more detailed explanation is given in [30,32]). S c by scaling, rotation and reparameterization, respectively.…”

Section: Shape Analysis Of Closed Framed Curvesmentioning

confidence: 99%

“…[30]). The frame-Hopf map induces an isometryGr2 pLCq \ Gr2 pACq Ñ p S c {pR`ˆSOp3qˆS 1 qwith respect to the induced L 2 metric and g S .…”

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confidence: 99%

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Shape Analysis of Framed Space Curves

Needham

2019

J Math Imaging Vis

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In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. Averages of collections of plane and space curves are computed via a novel algorithm utilizing flag means. arXiv:1807.03477v1 [math.DG] 10 Jul 2018 Theorem 1. 1 ([39]). The space ImmpS 1 , R 2 q{ttransl., rot., scal.u, endowed with the elastic metric g a,a is locally isometric to the Grassmann manifold Gr 2 pC 8 pS 1 , Rqq of two-dimensional planes in the vector space C 8 pS 1 , Rq, endowed with its canonical L 2 metric.

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Section: The Classical Hopf Mapmentioning

confidence: 85%

Section: Main Contributions Andmentioning

confidence: 99%

Section: Shape Analysis Of Closed Framed Curvesmentioning

confidence: 99%

“…[30]). The frame-Hopf map induces an isometryGr2 pLCq \ Gr2 pACq Ñ p S c {pR`ˆSOp3qˆS 1 qwith respect to the induced L 2 metric and g S .…”

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confidence: 99%

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Shape Analysis of Framed Space Curves

Needham

2019

J Math Imaging Vis

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“…The construction for open framed curves is classical and widely-used. For closed framed curves, we will use recent work of the author which shows that the space of periodic framed loops corresponds to an infinite-dimensional Grassmann manifold [24]. We show that, in either case, the generalized energy functional E transforms into the Riemannian energy functional on quaternionic path space (Corollary 2.4) and we use this to give a simple expression for the gradient of E (Corollary 2.5).…”

Section: 3mentioning

confidence: 99%

Knot types of generalized Kirchhoff rods

Needham

2019

J. Knot Theory Ramifications

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Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimensional Grassmann manifold. The critical framed curves have surprisingly simple parameterizations, but they still exhibit interesting topological features. In particular, we show that for each critical energy level there is a one-parameter family of framed curves whose base curves pass through exactly two torus knot types, echoing a similar result of Ivey and Singer for classical Kirchhoff energy. In contrast to the classical theory, the generalized functional has knotted critical points which are not torus knots.

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