Conic Arcs Research Articles

CURDIS: A template for incremental curve discretization algorithms and its application to conics

We introduce CURDIS, a template for algorithms to discretize arcs of regular curves by incrementally producing a list of support pixels covering the arc. In this template, algorithms proceed by finding the tangent quadrant at each point of the arc and determining which side the curve exits the pixel according to a tailored criterion. These two elements can be adapted for any type of curve, leading to algorithms dedicated to the shape of specific curves. While the calculation of the tangent quadrant for various curves, such as lines, conics, or cubics, is simple, it is more complex to analyze how pixels are traversed by the curve. In the case of conic arcs, we found a criterion for determining the pixel exit side. This leads us to present a new algorithm, called CURDIS-C, specific to the discretization of conics, for which we provide all the details. Surprisingly, the criterion for conics requires between one and three sign tests and four additions per pixel, making the algorithm efficient for resource-constrained systems and feasible for fixed-point or integer arithmetic implementations. Our algorithm also perfectly handles the pathological cases in which the conic intersects a pixel twice or changes quadrants multiple times within this pixel, achieving this generality at the cost of potentially computing up to two square roots per arc. We illustrate the use of CURDIS for the discretization of different curves, such as ellipses, hyperbolas, and parabolas, even when they degenerate into lines or corners.

Study on design of conical arc side-edge milling cutter and cutting performance under ultrasonic-assisted condition

Study on design of conical arc side-edge milling cutter and cutting performance under ultrasonic-assisted condition

Conic transfer arcs for Kepler's problem

A fundamental problem in spacecraft mission design is to find free-flight paths from one place to another that satisfy various design criteria. We explore the geometry of free-flight paths between departure and arrival points for Kepler's problem. Newton showed that these paths are conic arcs. We find the parameters for all conic paths between a departure and an arrival point as a function of one key variable called the inside angle. Once the paths are written in terms of this single parameter, then it is straightforward to find the path that takes a specified travel time (the Lambert problem) or to perform other optimizations such as minimizing the fuel costs.

Mass Points, Spaces of Spheres, ”hyperbolae” and Dandelin and Quételet Theorems

The use of mass points eases the definition of a branch of a hyperbola in the Euclidean plane based on a Rational Quadratic Bézier Curve. In the space of spheres, any circular cone, circular cylinder, torus, pencil of spheres or Dupin cyclide is represented by a rational quadratic Bézier curve that is conic arc seen as circle arc. The limit points of the Poncelet pencil or the singular points of the Dupin cyclide can be determined using the asymptotes of this circle. This article shows that the use of mass points simplifies the modelling of these pencils or these Dupin cyclides in the space of spheres. The determination of the Dandelin spheres ends this work as an application.

On polynomially integrable Birkhoff billiards on surfaces of constant curvature

The polynomial version of the Birkhoff Conjecture on integrable billiards on complete simply connected surfaces of constant curvature (plane, sphere, hyperbolic plane) was first stated, studied and solved in a particular case by Sergei Bolotin in 1990–1992. Here we present a complete solution of the polynomial version of the Birkhoff Conjecture. Namely, we show that every polynomially integrable real bounded planar billiard with C^2 -smooth connected boundary is an ellipse . We extend this result to billiards with piecewise smooth and not necessarily convex boundary on an arbitrary two-dimensional simply connected complete surface of constant curvature: plane, sphere, Lobachevsky–Poincaré (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in \mathbb R^3 equipped with an appropriate quadratic form. Namely, we show that a billiard is polynomially integrable if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments . We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given in two papers by Bialy and Mironov (in Euclidean and non-Euclidean cases respectively). Their geometric construction reduced the Polynomial Birkhoff Conjecture to a purely algebro-geometric problem to show that an irreducible algebraic curve in \mathbb {CP}^2 with certain properties is a conic. They have shown that its singular and inflection points lie in the complex light conic of the above-mentioned quadratic form. In the present paper we solve the above algebro-geometric problem completely.

Osculating conic biarcs

A circular biarc can be defined by using two points K and L with their (oriented) tangents gK and gL as input. It is well-known that one can determine a one parametric set of circular arc pairs k,ℓ such that k starts at K with tangent gK, ℓ ends at L with tangent gL and k and ℓ meet with a common tangent in an intermediate point P. In this paper we investigate a similar construction where we replace the circle biarcs by pairs of conic arcs. It turns out that in this case we can prescribe a conic k0 with a point K on it, another conic ℓ0 with a point L on it, and moreover an intermediate point P to obtain a unique pair k,ℓ of conics such that k osculates k0 in K, ℓ osculates ℓ0 in L and k and ℓ osculate each other in P. This also confirms a result of H. Pottmann from 1991. We use our method to solve an interpolation task of Hermite type whose input consists of a series of points with their curvature circles and another series of intermediate points. The output is a GC2 spline curve with conic arc segments.

Total curvature variation fairing for medial axis regularization

We present a new fairing method for planar curves, which is particularly well suited for the regularization of the medial axis of a planar domain. It is based on the concept of total variation regularization. The original boundary (given as a closed B-spline curve or several such curves for multiply connected domains) is approximated by another curve that possesses a smaller number of curvature extrema. Consequently, the modified curve leads to a smaller number of branches of the medial axis. In order to compute the medial axis, we use the state-of-the-art algorithm from [1] which is based on arc spline approximation and a domain decomposition approach. We improve this algorithm by using a different decomposition strategy that allows to reduce the number of base cases from 13 to only 5. Moreover, the algorithm reduces the number of conic arcs in the output by approx. 50%.

Construction Of Complete ( K,N )-Arcs In PG ( 2,8) FOR M < N

In this work, we construct complete (k,n)-arcs and we find some of them are maximum for some n, 2 < n < 8. if n = 2, where every arc which constructed by the equation of the conic called conic arc, and from it we constructs complete arcs and we prove it’s maximum by taking the union of two ,three and six conics , respectively. and then we show By adding the points of index zero for the (k6,6)-arc, (k7,7)-arc, respectively, we get a maximum complete (k8,8)-arc.Â

Iterative construction of Dupin cyclide characteristic circles using non-stationary Iterated Function Systems (IFS)

A Dupin cyclide can be defined, in two different ways, as the envelope of an one-parameter family of oriented spheres. Each family of spheres can be seen as a conic in the space of spheres. In this paper, we propose an algorithm to compute a characteristic circle of a Dupin cyclide from a point and the tangent at this point in the space of spheres. Then, we propose iterative algorithms (in the space of spheres) to compute (in 3D space) some characteristic circles of a Dupin cyclide which blends two particular canal surfaces. As a singular point of a Dupin cyclide is a point at infinity in the space of spheres, we use the massic points defined by Fiorot. As we subdivide conic arcs, these algorithms are better than the previous algorithms developed by Garnier and Gentil.

Analytical results for solar sail optimal missions with modulated radial thrust

The aim of this paper is to analyze the optimal trajectories of a spacecraft subjected to a modulated radial thrust, whose magnitude is inversely proportional to the square of the distance from the primary body. This case is representative of a Sun-facing solar sail with a passive attitude control system. In this study the sailcraft is assumed to perform a finite number of reorientation maneuvers to set the propelling acceleration to zero and generate suitable coasting arcs along the trajectory. Accordingly, the resulting generalized orbit is a sequence of either propelled or ballistic conic arcs, whose main characteristics (in terms of semimajor axis, eccentricity, and perihelion radius) can be calculated in closed form. As a result, the sailcraft optimal performance can be studied using an analytical approach. In particular, some compact relationships are drawn and discussed that allow one to find the optimal sailcraft characteristics required to reach a prescribed final orbit.

Phase Transitions and Structures of 5-{[(4’-Pentyloxy-4-biphenylyl)carbonyl]oxy}-1-pentyne

Phase transitions and structures of the monomer of side-chain liquid crystalline polyacetylene containing biphenyl mesogen,namely,5-{[(4'-pentyloxy-4-biphenylyl)carbonyl]oxy}-1-pentyne (A3EO5) were investigated by differential scanning calorimetry (DSC),one dimensional (1D) wide angle X-ray diffraction (WAXD),and polarized light microscopy equipped with hot-stage (PLM-hot stage).DSC and 1D-WAXD results show that A3EO5 forms an enantiotropic liquid crystal and displays four transitions during the cooling and subsequent heating processes.Upon cooling from isotropic phase to room temperature,the sample forms smectic A phase and then leads to smectic B phase with pseudo long-range orientation within the layer structure,further lowering the temperature,A3EO5 forms smectic E phase with orthorhombic packing within the smectic layer and enters crystal phase.PLM results show that the sample forms spherulitic texture,focal conic texture and concentric arcs texture during the cooling process.

Effect of REM addition of wire on CO2 gas shielded arc phenomenon

Carbon dioxide (CO2) gas shielded arc welding is the main arc welding method, but it generates a large amount of spatter during welding. The root cause of spatter lies in the fact that the droplet undergoes repeated irregular shaking. To solve this problem, spatter generation modes were clarified and the effects of polarity and rare earth metal (REM) addition of the wire on CO2 gas shielded arc welding were investigated. As a result, when welding is performed with an electrode negative (DCEN) polarity using REM added wire, it was found that a conical arc plasma is formed, and the droplet which is transferred from the wire tip to the molten pool is fine and continuous, in what is termed ‘spray transfer’. Thus, spatter generation was reduced to 10% of amount of the conventional CO2 gas shielded arc welding (from 0.058 to 0.005g/s).

An experimental study of point location in planar arrangements in CGAL

We study the performance in practice of various point-location algorithms implemented in CGAL (the Computational Geometry Algorithms Library), including a newly devised landmarks algorithm. Among the other algorithms studied are: a naïve approach, a “walk along a line” strategy, and a trapezoidal decomposition-based search structure. The current implementation addresses general arrangements of planar curves, including arrangements of nonlinear segments (e.g., conic arcs) and allows for degenerate input (for example, more than two curves intersecting in a single point or overlapping curves). The algorithms use exact geometric computation and thus result in the correct point location. In our landmarks algorithm (a.k.a. jump & walk), special points, “landmarks,” are chosen in a preprocessing stage, their place in the arrangement is found, and they are inserted into a data structure that enables efficient nearest-neighbor search. Given a query point, the nearest landmark is located and a “walk” strategy is applied from the landmark to the query point. We report on various experiments with arrangements composed of line segments or conic arcs. The results indicate that compared to the other algorithms tested, the landmarks approach is the most efficient, when the overall (amortized) cost of a query is taken into account, combining both preprocessing and query time. The simplicity of the algorithm enables an almost straightforward implementation and rather easy maintenance. The generic programming implementation allows versatility both in the selected type of landmarks and in the choice of the nearest-neighbor search structure. The end result is an efficient point-location algorithm that bypasses the alternative CGAL implementations in most practical aspects.

Application of high order expansions of two-point boundary value problems to astrodynamics

Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.

炭酸ガスアーク現象に及ぼすワイヤへのREM添加の影響

CO2gas shielded arc welding is major arc welding method, but generates large amount of spatter during welding. The root cause of spatter lies in the fact that the droplet undergoes repeated irregular shaking. To solve this problem, spatter generation modes were clarified and the effects of polarity and rare earth metal (REM) addition of the wire on CO2 gas shielded arc welding were investigated. As a result, when welding is performed with an electrode negative (DCEN) polarity using REM added wire, it was found that a conical arc plasma is formed, and the droplet which is transferred from the wire tip to the molten pool is fine and continuous, in what is termed "spray transfer". Thus, spatter generation was reduced to 10% of amount of the conventional CO2 gas shielded arc welding (from 0.058g/s to 0.005g/s).

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c1ɛ−1/4 + O(1), if the spline consists of parabolic arcs, and c2ɛ−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c1 and c2 are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline.

SU‐FF‐T‐436: Using a 3D Diode Array System to Verify 3D Dose Distribution

Purpose: We have been used a 3D diode array to verify 3D dose distributions from IMRT plans, and SRS plans using conical arcs or dynamical arcs. Some of the results are reported here. We are also going to study the possibility of using this diode array for routine QA of the treatment machine and the treatment planning system. Method and Materials: IMRT plans are developed on Varian Eclipse, and delivered on Varian Trilogy treatment unit by using sliding window technique. The SRS plans are developed on a BrainLab IPlan Dose, and delivered on the Trilogy. The 3D diode array is the Delta4 system from Scandidos. Delta4 has 1069 type Si diodes on two planes in a cylindrical PMMA phantom. In the treatment planning systems, a patient plan is copied to the phantom, and recalculated. The plan and dose distribution are transferred to the Delta4 system by using DICOM RT. In the Delta4 software, the planned dose and measured dose are compared by using 2D isodose display, profile comparisons, percentage dose deviation, and DTA and Gamma index. Results: We have used this method to verify 61 IMRT plans. The average percent diodes with deviation of 5% or less is 84.5%; the average distance to agreement (DTA) of 3mm or less is 86.7%. The average Gamma Index (5% and 3mm) of 1.0 or less is 91.8%. We have also used this system to verify the isocenter dose of BrainLab SRS plans, each using a cone from 7.5 mm to 30 mm with 180 degree arc. The average absolute difference between measurement and plan isocenter dose is 1.8%. Conclusion: The 3D diode system is a good tool to verify 3D dose distribution.

Exact and approximate construction of offset polygons

The Minkowski sum of two sets A , B ∈ R 2 , denoted A ⊕ B , is defined as { a + b ∣ a ∈ A , b ∈ B } . We describe an efficient and robust implementation of the construction of the Minkowski sum of a polygon in R 2 with a disc, an operation known as offsetting the polygon. Our software package includes a procedure for computing the exact offset of a straight-edge polygon, based on the arrangement of conic arcs computed using exact algebraic number-types. We also present a conservative approximation algorithm for offset computation that uses only rational arithmetic and decreases the running times by an order of magnitude in some cases, while having a guarantee on the quality of the result. The package will be included in the next public release of the Computational Geometry Algorithms Library, Cgal Version 3.3. It also integrates well with other Cgal packages; in particular, it is possible to perform regularized Boolean set-operations on the polygons the offset procedures generate.

Multiprimitive segmentation based on meaningful breakpoints for fitting digital planar curves with line segments and conic arcs

This paper presents a multiprimitive segmentation method with line segments and conic arcs based on the types of breakpoints. In this method, a joint tuning procedure is proposed to merge consecutive segments and adjust their locations to achieve more accurate and stable conic arcs. No threshold is required in the multiprimitive segmentation by using the proposed scheme. And, the types of breakpoints among line segments and conic arcs are defined and they are useful and meaningful for pattern recognition and shape analysis. Besides, the computational complexity of the proposed method is O( n log n) which is lower than most other conic fitting methods. Further, the concept of types of breakpoints can be easily extended to other primitives.

Phased array ultrasonic NDT system for tubes and pipes

One or more phased array ultrasonic probes in the shape of conical section arcs, or an entire conical section ring, emit beams at a fixed incident angle with respect to the outer surface of a tube or pipe under inspection. At any given moment, these beams strike the tube at a single small entry zone from number of different directions. After being refracted at the inspection angle corresponding to the fixed angle of incidence, defects having longitudinal, transverse and oblique orientations are detected in the three-dimensional volume of the tube wall. The number of entry zones required for complete coverage of the outer surface of the tube is a function of the size of the tube, the speed at which it passes through the inspection station and the rate of rotation of the tube.