12,138 research outputs found
Minimization of length and curvature on planar curves
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional ∫ √1+K 2 ds, depending both on length and curvature K. We fix starting and ending points as well as initial and final directions. For this functional, we find non-existence of minimizers on various functional spaces in which the problem is naturally formulated. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the "time-reparameterized" functional ∫γ(t)√1+Kγ2 dt for all boundary conditions if initial and final directions are considered regardless to orientation. ©2009 IEEE
Existence of planar curves minimizing length and curvature
In this paper we consider the problem of reconstructing a curve that is
partially hidden or corrupted by minimizing the functional , depending both on length and curvature . We fix
starting and ending points as well as initial and final directions.
For this functional we discuss the problem of existence of minimizers on
various functional spaces. We find non-existence of minimizers in cases in
which initial and final directions are considered with orientation. In this
case, minimizing sequences of trajectories can converge to curves with angles.
We instead prove existence of minimizers for the "time-reparameterized"
functional \int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt for all boundary
conditions if initial and final directions are considered regardless to
orientation. In this case, minimizers can present cusps (at most two) but not
angles
Optimization in Differentiable Manifolds in Order to Determine the Method of Construction of Prehistoric Wall-Paintings
In this paper a general methodology is introduced for the determination of
potential prototype curves used for the drawing of prehistoric wall-paintings.
The approach includes a) preprocessing of the wall-paintings contours to
properly partition them, according to their curvature, b) choice of prototype
curves families, c) analysis and optimization in 4-manifold for a first
estimation of the form of these prototypes, d) clustering of the contour parts
and the prototypes, to determine a minimal number of potential guides, e)
further optimization in 4-manifold, applied to each cluster separately, in
order to determine the exact functional form of the potential guides, together
with the corresponding drawn contour parts. The introduced methodology
simultaneously deals with two problems: a) the arbitrariness in data-points
orientation and b) the determination of one proper form for a prototype curve
that optimally fits the corresponding contour data. Arbitrariness in
orientation has been dealt with a novel curvature based error, while the proper
forms of curve prototypes have been exhaustively determined by embedding
curvature deformations of the prototypes into 4-manifolds. Application of this
methodology to celebrated wall-paintings excavated at Tyrins, Greece and the
Greek island of Thera, manifests it is highly probable that these
wall-paintings had been drawn by means of geometric guides that correspond to
linear spirals and hyperbolae. These geometric forms fit the drawings' lines
with an exceptionally low average error, less than 0.39mm. Hence, the approach
suggests the existence of accurate realizations of complicated geometric
entities, more than 1000 years before their axiomatic formulation in Classical
Ages
Some minimization problems for planar networks of elastic curve
In this note we announce some results that will appear in [6] (joint work
with also Matteo Novaga) on the minimization of the functional
, where is a network of
three curves with fixed equal angles at the two junctions. The informal
description of the results is accompanied by a partial review of the theory of
elasticae and a diffuse discussion about the onset of interesting variants of
the original problem passing from curves to networks. The considered energy
functional is given by the elastic energy and a term that penalize the
total length of the network. We will show that penalizing the length is
tantamount to fix it. The paper is concluded with the explicit computation of
the penalized elastic energy of the 'Figure Eight', namely the unique closed
elastica with self--intersections.Comment: 24 pages, 7 figure
On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves
For a smooth closed embedded planar curve , we consider the
minimization problem of the Willmore energy among immersed surfaces of a given
genus having the curve as boundary, without any
prescription on the conormal. By general lower bound estimates, in case
is a circle we prove that such problem is equivalent if restricted to
embedded surfaces, we prove that do not exist minimizers, and the infimum
equals , where is the energy of
the closed minimizing surface of genus . We also prove that the
same result also holds if is a straight line for the suitable
analogously defined minimization problem on asymptotically flat surfaces.\\
Then we study the case in which is compact, and the
competitors are restricted to a suitable class of varifolds
including embedded surfaces. We prove that under suitable assumptions
minimizers exists in this class of generalized surfaces
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