scholarly journals Computing concave hull with closed curve smoothing: performance, concaveness measure and applications

2018 ◽  
Vol 136 ◽  
pp. 479-488 ◽  
Author(s):  
Darina Kalinina ◽  
Varvara Ingilevich ◽  
Anastasiia Lantseva ◽  
Sergei Ivanov
Keyword(s):  
Closed Curve ◽  
Curve Smoothing

The Numerical Evaluation of Curvilinear Integrals and Areas

1968 ◽  
Vol 72 (694) ◽  
pp. 887-888
Author(s):  
T. R. F. Nonweiler
Keyword(s):  
Digital Computer ◽  
Numerical Evaluation ◽  
Closed Curve ◽  
Pitching Moment ◽  
Pressure Measurements ◽  
Force Components

We envisage here that it is desired to evaluate (using a digital computer) the integral of a function f (x, y) with respect to x (say), along a curve γ in the xy-plane; the value of the function and of the co-ordinates of the curve being known only at discrete, and possibly irregularly spaced, points of γ. Such a problem frequently arises, especially where γ is a closed curve: as for instance in aerodynamics when it is desired to estimate the force components and pitching moment inferred from an experimentally determined set of pressures taken over an aerofoil; here the function values are related to the pressure measurements and γ is the aerofoil contour.

Optimal space filling for additive manufacturing

2016 ◽  
Vol 22 (4) ◽  
pp. 660-675 ◽  
Author(s):  
Sajan Kapil ◽  
Prathamesh Joshi ◽  
Hari Vithasth Yagani ◽  
Dhirendra Rana ◽  
Pravin Milind Kulkarni ◽  
...  
Keyword(s):  
Additive Manufacturing ◽  
Manufacturing Process ◽  
Layered Manufacturing ◽  
Closed Curve ◽  
Numerical Control ◽  
Surface Finishing ◽  
Content Type ◽  
Single Pass ◽  
Connected Area ◽  
Area Filling

Purpose In additive manufacturing (AM) process, the physical properties of the products made by fractal toolpaths are better as compared to those made by conventional toolpaths. Also, it is desirable to minimize the number of tool retractions. The purpose of this study is to describe three different methods to generate fractal-based computer numerical control (CNC) toolpath for area filling of a closed curve with minimum or zero tool retractions. Design/methodology/approach This work describes three different methods to generate fractal-based CNC toolpath for area filling of a closed curve with minimum or zero tool retractions. In the first method, a large fractal square is placed over the outer boundary and then rest of the unwanted curve is trimmed out. To reduce the number of retractions, ends of the trimmed toolpath are connected in such a way that overlapping within the existing toolpath is avoided. In the second method, the trimming of the fractal is similar to the first method but the ends of trimmed toolpath are connected such that the overlapping is found at the boundaries only. The toolpath in the third method is a combination of fractal and zigzag curves. This toolpath is capable of filling a given connected area in a single pass without any tool retraction and toolpath overlap within a tolerance value equal to stepover of the toolpath. Findings The generated toolpath has several applications in AM and constant Z-height surface finishing. Experiments have been performed to verify the toolpath by depositing material by hybrid layered manufacturing process. Research limitations/implications Third toolpath method is suitable for the hybrid layered manufacturing process only because the toolpath overlapping tolerance may not be enough for other AM processes. Originality/value Development of a CNC toolpath for AM specifically hybrid layered manufacturing which can completely fill any arbitrary connected area in single pass while maintaining a constant stepover.

Entanglement complexity of graphs in Z3

1992 ◽  
Vol 111 (1) ◽  
pp. 75-91 ◽  
Author(s):  
C. E. Soteros ◽  
D. W. Sumners ◽  
S. G. Whittington
Keyword(s):  
Crossing Number ◽  
Equivalence Classes ◽  
Closed Curve ◽  
Good Measure ◽  
Average Value ◽  
Cut Edge ◽  
Expected Complexity ◽  
Almost All

AbstractIn this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

scholarly journals On elliptic curves with a closed component passing through a hexagon

2019 ◽  
Vol 27 (2) ◽  
pp. 67-82
Author(s):  
Miroslav Kureš
Keyword(s):  
Elliptic Curves ◽  
Numerical Experiments ◽  
Algebraic Curves ◽  
Closed Curve ◽  
Special Shape ◽  
The Third ◽  
Convex Pentagon

AbstractIn general, there exists an ellipse passing through the vertices of a convex pentagon, but any ellipse passing through the vertices of a convex hexagon does not have to exist. Thus, attention is turned to algebraic curves of the third degree, namely to the closed component of certain elliptic curves. This closed curve will be called the spekboom curve. Results of numerical experiments and some hypotheses regarding hexagons of special shape connected with the existence of this curve passing through the vertices are presented and suggested. Some properties of the spekboom curve are described, too.

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