Construction and properties of non-polynomial spline-curves

2014 ◽  
Author(s):  
Arne Lakså
Keyword(s):  
Polynomial Spline ◽  
Spline Curves

scholarly journals Spline Curves Formation Given Extreme Derivatives

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 47
Author(s):  
Konstantin Panchuk ◽  
Tatyana Myasoedova ◽  
Evgeniy Lyubchinov
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Model Development ◽  
Polynomial Spline ◽  
Systematic Research ◽  
Theoretical Problem ◽  
Spline Curve ◽  
Spline Curves ◽  
Derivatives Of ◽  
Bézier Spline

This paper is dedicated to development of mathematical models for polynomial spline curve formation given extreme vector derivatives. This theoretical problem is raised in the view of a wide variety of theoretical and practical problems considering motion of physical objects along certain trajectories with predetermined laws of variation of speed, acceleration, jerk, etc. The analysis of the existing body of work on computational geometry performed by the authors did not reveal any systematic research in mathematical model development dedicated to solution of similar tasks. The established purpose of the research is therefore to develop mathematical models of formation of spline curves based on polynomials of various orders modeling the determined trajectories. The paper presents mathematical models of spline curve formation given extreme derivatives of the initial orders. The paper considers construction of Hermite and Bézier spline curves of various orders consisting of various segments. The acquired mathematical models are generalized for the cases of vector derivatives of higher orders. The presented models are of systematic nature and are universal, i.e., they can be applied in formation of any polynomial spline curves given extreme vector derivatives. The paper provides a number of examples validating the presented models.

scholarly journals Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3084
Author(s):  
Andrea Raffo ◽  
Silvia Biasotti
Keyword(s):  
Digital Images ◽  
Spline Approximation ◽  
Detection Algorithm ◽  
Polynomial Spline ◽  
Automatic Method ◽  
Data Perturbation ◽  
Spline Curves ◽  
Curve Approximation ◽  
Local Approximations ◽  
Edge Points

The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a preprocessing step exploiting an edge detection algorithm; a splitting procedure to break the just-obtained set of edge points into smaller subsets; and a final step involving the use of a local curve approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), chosen for its robustness to data perturbation. The proposed method builds a sequence of polynomial spline curves, connected C0 in correspondence of cusps, G1 otherwise. To curb underfitting and overfitting, the computation of local approximations exploits the supervised learning paradigm. The effectiveness of the method is shown with simulation on real images from various application domains.

scholarly journals Chaotic Path Planning for 3D Area Coverage Using a Pseudo-Random Bit Generator from a 1D Chaotic Map

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1821
Author(s):  
Lazaros Moysis ◽  
Karthikeyan Rajagopal ◽  
Aleksandra V. Tutueva ◽  
Christos Volos ◽  
Beteley Teka ◽  
...  
Keyword(s):  
Path Planning ◽  
Chaotic Map ◽  
Dynamical Behavior ◽  
Area Coverage ◽  
Phase Portraits ◽  
Planning Strategy ◽  
Spline Curves ◽  
Key Space ◽  
Aerial Vehicle ◽  
Discrete Motion

This work proposes a one-dimensional chaotic map with a simple structure and three parameters. The phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams are first plotted to study the dynamical behavior of the map. It is seen that the map exhibits areas of constant chaos with respect to all parameters. This map is then applied to the problem of pseudo-random bit generation using a simple technique to generate four bits per iteration. It is shown that the algorithm passes all statistical NIST and ENT tests, as well as shows low correlation and an acceptable key space. The generated bitstream is applied to the problem of chaotic path planning, for an autonomous robot or generally an unmanned aerial vehicle (UAV) exploring a given 3D area. The aim is to ensure efficient area coverage, while also maintaining an unpredictable motion. Numerical simulations were performed to evaluate the performance of the path planning strategy, and it is shown that the coverage percentage converges exponentially to 100% as the number of iterations increases. The discrete motion is also adapted to a smooth one through the use of B-Spline curves.

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