scholarly journals Harmonic Deformation of Planar Curves

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Eleutherius Symeonidis
Keyword(s):  
Harmonic Function ◽  
Planar Curves ◽  
Planar Curve ◽  
Harmonic Deformation ◽  
Specific Potential

We establish a principle of deformation of an arbitrary planar curve, so that the integral of a harmonic function over this curve does not change. The equations of deformation can be derived from a specific “potential.” Several applications are presented.

scholarly journals Affine Subspaces of Curvature Functions from Closed Planar Curves

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Arc Length ◽  
Planar Curves ◽  
Affine Subspaces ◽  
Planar Curve ◽  
Real Functions ◽  
Discrete Counterpart ◽  
Equivalent Conditions

AbstractGiven a pair of real functions (k, f), we study the conditions they must satisfy for $$k+\lambda f$$ k + λ f to be the curvature in the arc-length of a closed planar curve for all real $$\lambda $$ λ . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

scholarly journals A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

2007 ◽  
Vol 49 (2) ◽  
pp. 367-375 ◽  
Author(s):  
DZMITRY BADZIAHIN ◽  
JASON LEVESLEY
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
General Class ◽  
Simultaneous Approximation ◽  
Convergence Result ◽  
Planar Curves ◽  
Planar Curve ◽  
Induced Measure

AbstractLet $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

RECOGNITION OF AFFINE TRANSFORMED PLANAR CURVES BY EXTREMAL GEOMETRIC PROPERTIES

1993 ◽  
Vol 03 (02) ◽  
pp. 183-202 ◽  
Author(s):  
CRAIG GOTSMAN ◽  
MICHAEL WERMAN
Keyword(s):  
Digital Image ◽  
Affine Transformation ◽  
Time Complexity ◽  
Affine Invariant ◽  
Geometric Properties ◽  
Planar Curves ◽  
Geometric Methods ◽  
Planar Curve ◽  
Image Pixels

An algorithm for the recognition of a digital image of a planar curve which has undergone an affine transformation is presented. The algorithm is based on affine-invariant extremal geometric properties of curves, utilizes existing computational-geometric methods, and is relatively insensitive to noise. Its time complexity is linear in the number of image pixels on the curve. Extensions of our algorithm to deal with some cases of occlusion of the image curve and recognition under perspective transformations are also described. These algorithms are almost linear in the number of image pixels on the curve.

scholarly journals On weighted inhomogeneous Diophantine approximation on planar curves

2012 ◽  
Vol 154 (2) ◽  
pp. 225-241 ◽  
Author(s):  
MUMTAZ HUSSAIN ◽  
TATIANA YUSUPOVA
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Planar Curves ◽  
Planar Curve ◽  
Invent Math

AbstractThis paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdorff dimension results for the sets of simultaneously well-approximable points on planar curves, established in Badziahin and Levesley (Glasg. Math. J., 49(2):367–375, 2007), Beresnevich et al. (Ann. of Math. (2), 166(2):367–426, 2007), Beresnevich and Velani (Math. Ann., 337(4):769–796, 2007) and Vaughan and Velani (Invent. Math., 166(1):103–124, 2006).

An Algorithm for Representing Planar Curves in B-Splines

2014 ◽  
Vol 596 ◽  
pp. 149-153
Author(s):  
Zi Zhi Lin ◽  
Si Hui Shu
Keyword(s):  
Least Square ◽  
Second Step ◽  
B Spline ◽  
Planar Curves ◽  
B Splines ◽  
Planar Curve ◽  
Sample Data ◽  
Interpolation Curve ◽  
Data Points ◽  
The Given

An algorithm for representing planar curves in B-splines is presented in this paper. The representing problem is different from the approximation to data points; planar curve provided more information than data points. To make full use of the information, we propose a three-step representing approach: 1.Sample data points along with their tangent vectors from the planar curve according to the given accuracy. 2. Fit the sampled points by Bezier segments using local interpolation; compose these segments to an interpolation curve. 3. Approximate the interpolation curve using the best least approximation to get the final B-spline curve. Tangent information is used in the second step to construct the interpolation curve. In the third step, the system is always positive because of using the best least square approximation, so we can get more freedoms to approximate the interpolation curve. Finally, some examples of this algorithm demonstrate its usefulness and quality.

Conclusion

2018 ◽  
Author(s):  
Rachel Crossland
Keyword(s):  
Literature And Science ◽  
Evaluative Criteria ◽  
Specific Potential ◽  
The Creation ◽  
The Moment ◽  
Further Development

The conclusion returns to some of the ideas raised in the Introduction, specifically Gillian Beer’s suggestion that literature and science ‘share the moment’s discourse’. It argues for the relevance of this model to different periods and disciplines, while also suggesting some specific potential areas for further development in relation to the present study, including generalist periodicals. It also considers some of the evaluative criteria that have previously been suggested for studies in the field of literature and science, and raises some questions as to the direction in which that field of research should now move. The study concludes finally by suggesting that literature and science, as well as a range of other disciplines, some of which are included here, do more than share the moment’s discourse—they share in the creation, development, and modification of that discourse because they share the moment itself.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

scholarly journals Clothoid fitting and geometric Hermite subdivision

2021 ◽  
Vol 47 (4) ◽  
Author(s):  
Ulrich Reif ◽  
Andreas Weinmann
Keyword(s):  
Building Block ◽  
Numerical Results ◽  
Normal Vector ◽  
Subdivision Schemes ◽  
Planar Curves ◽  
Nonlinear Subdivision ◽  
New Strategy ◽  
Vector Information ◽  
Hermite Subdivision

AbstractWe consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

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