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.

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.

AFFINE MAURER–CARTAN INVARIANTS AND THEIR APPLICATIONS IN SELF-AFFINE FRACTALS

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850057 ◽  
Author(s):  
YUN YANG ◽  
YANHUA YU
Keyword(s):  
Optimization Methods ◽  
Moving Frame ◽  
Topological Properties ◽  
Affine Invariants ◽  
Planar Curve ◽  
Fixed Area ◽  
Moving Frame Method ◽  
Frame Method ◽  
Cartan Invariants ◽  
Equivalent Conditions

In this paper, we define the notion of affine curvatures on a discrete planar curve. By the moving frame method, they are in fact the discrete Maurer–Cartan invariants. It shows that two curves with the same curvature sequences are affinely equivalent. Conditions for the curves with some obvious geometric properties are obtained and examples with constant curvatures are considered. On the other hand, by using the affine invariants and optimization methods, it becomes possible to collect the IFSs of some self-affine fractals with desired geometrical or topological properties inside a fixed area. In order to estimate their Hausdorff dimensions, GPUs can be used to accelerate parallel computing tasks. Furthermore, the method could be used to a much broader class.

scholarly journals New Properties of Complex Functions with Mean Value Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Yuzhen Bai ◽  
Lei Wu
Keyword(s):  
Analytic Functions ◽  
Mean Value ◽  
Complex Functions ◽  
Real Functions ◽  
Equivalent Conditions

We apply mollifiers to study the properties of real functions which satisfy mean value conditions and present new equivalent conditions for complex analytic functions. New properties of complex functions with mean value conditions are given.

Rational Approximation of Real Functions

1988 ◽  
Author(s):  
P. P. Petrushev ◽  
Vasil Atanasov Popov
Keyword(s):  
Rational Approximation ◽  
Real Functions

The visual perception of arc length on a real surface

1997 ◽  
Author(s):  
J. Farley Norman ◽  
Joseph S. Lappin ◽  
Hideko F. Norman
Keyword(s):  
Visual Perception ◽  
Real Surface ◽  
Arc Length
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