Parametric Spiral And Its Application As Transition Curve

Lengkung Bezier merupakan suatu perwakilan lengkungan yang paling popular digunakan di dalam applikasi Rekabentuk Berbantukan Komputer (RBK) dan Rekabentuk Geometrik Berbantukan Komputer (RGBK). The Bezier curve representation is frequently utilized in computer-aided design (CAD) and computer-aided geometric design (CAGD) applications. The curve is defined geometrically, which means that the parameters have geometric meaning; they are just points in three-dimensional space

Employing Pythagorean Hodograph curves for artistic patterns

In this paper we present a novel design element creator tool for the digital artist. The purpose of our tool is to support the creation of vines, swirls, swooshes and floral components. To create visually pleasing and gentle curves we employ Pythagorean Hodograph quintic spiral curves to join a hierarchy of control circles defined by the user. The control circles are joined by spiral segments with at least G2 continuity, ensuring smooth and seamless transitions. The control circles give the user a fast and intuitive way to define the desired curve. The resulting curves can be exported as cubic Bezier curves for further use in vector graphics applications

G2 Bézier-Like Cubic as the S-Transition and C-Spiral Curves and Its Application in Designing a Spur Gear Tooth

Many of the researchers or designers are mostly used the involute curves (known as one of the approximation curves) to design the profile of gears. This study intends to develop the transition (S transition and C spiral) curves using Bézier–like cubic curve function with G2 (curvature) continuity as the degree of smoothness. Method of designing the transition curves is adapted by using circle to circle templates. While the transition curves are completely finished, it will be applied in redesigning the profile of spur gear. The mathematical proofs and simple models are also shown

Fairing arc spline and designing by using cubic bézier spiral segments

This paper considers how to smooth three kinds of G 1 biarc models, the C-, S-, and J-shaped transitions, by replacing their parts with spiral segments using a single cubic Bézier curve. Arc spline is smoothed to G 2continuity. Use of a single curve rather than two has the benefit because designers and implementers have fewer entities to be concerned. Arc spline is planar, tangent continuous, piecewise curves made of circular arcs and straight line segments. It is important in manufacturing industries because of its use in the cutting paths for numerically controlled cutting machinery. Main contribution of this paper is to minimize the number of curvature extrema in cubic transition curves for further use in industrial applications such as non-holonomic robot path planning, highways or railways, and spur gear tooth designing

INTEGRATING SPUR GEAR TEETH DESIGN AND ITS ANALYSIS WITH G2 PARAMETRIC BÉZIER-LIKE CUBIC TRANSITION AND SPIRAL CURVES

An involute curve (or known as an approximated curve) is mostly used in designing the gear teeth (profile) especially in spur gear. Conversely, this study has the intention to redesign the spur gear teeth using the transition (S transition and C spiral) curves (also known as the exact curves) with curvature continuity (G2) as the degree of smoothness. Method of designing the transition curves is adapted from the circle to circle templates. The applicability of the new teeth model with the chosen material, Stainless Steel Grade 304 is determined using Linear Static Analysis, Fatigue Analysis and Design Efficiency (DE). Several concepts and the related examples are shown throughout this study

An experimental study of the buckling of complete spherical shells

Buckling of complete spherical shells to examine Tsien energy hypothesi

Geometry and tool motion planning for curvature adapted CNC machining

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and oscu- lating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software

A descriptive and evaluative bibliography of mathematics filmstrips.

Submitted by A.W. Clark and R.W. Allen for the degree of Master of Arts and by C.H. Gardner and R.F. Sweeney for the degree of Master of Education. Thesis (Ed.M.)--Boston UniversityThe purpose of this paper is to present in one volume (1) a bibliography of all mathematics filmstrips from those suitable for the first grade to those suitable for use in senior high school and college, (2) an accurate description of each filmstrip, and (3) unbiased evaluations of each filmstrip by qualified teachers invited to take part in the project. Concomitant problems. The foregoing three parts were the heart of the problem and the portion nearly completely solved. There were, however, concomitant problems which have been partially solved by this work. The first of these concerns the limited use of filmstrips by mathematics teachers. Undoubtedly many do not believe in using filmstrips in mathematics classes. Others have never given serious thought about the advisability of using filmstrips. In later sections of this chapter and throughout this work evidence is cited to support the contention that filmstrips should have serious consideration, and that they are useful in mathematics classes. The second concomitant problem concerns the revision of current filmstrips and production of new ones. The filmstrip producers were supplied, upon their request, with summaries of the evaluations. Summaries were supplied only at the producer's request; for unless they were interested enough to request the summaries, they probably would not be interested in changing or improving their filmstrips. Summary. The problem, then, had three major parts: listing , describing, and evaluating mathematics filmstrips, and two concomitant parts: arousing the mathematics teacher's interest in filmstrips, and encouraging producers to make better productions and necessary revisions in current productions. [TRUNCATED

Identification of Transition Curves in Vehicular Roads and Railways

In the paper attention is focused on the necessity to systematize the procedure for determining the shape of transition curves used in vehicular roads and railway routes. There has been presented a universal method of identifying curvature in transition curves by using differential equations. Curvature equations for such known forms of transitioncurves as clothoid, quartic parabola, the Bloss curve, cosinusoid and sinusoid, have been worked out and by the use these equations it was possible to determine some appropriate Cartesian coordinates. In addition some approximate solutions obtained in consequence of making certain simplifying assumptions orientated mainly towards railway routes, have been provided. Notice has been taken of limitations occurring in the application of smooth transition curves in railway systems, which can be caused by very small values of the horizontal ordinates in the initial region. This problem has provided an inspiration for finding a new family of the so-called parametric transition curves, being more advantageous not only over the clothoid but also over cubic parabola as far as dynamics is concerned

Post-Quantum Cryptography: Riemann Primitives and Chrysalis

The Chrysalis project is a proposed method for post-quantum cryptography using the Riemann sphere. To this end, Riemann primitives are introduced in addition to a novel implementation of this new method. Chrysalis itself is the first cryptographic scheme to rely on Holomorphic Learning with Errors, which is a complex form of Learning with Errors relying on the Gauss Circle Problem within the Riemann sphere. The principle security reduction proposed by this novel cryptographic scheme applies complex analysis of a Riemannian manifold along with tangent bundles relative to a disjoint union of subsets based upon a maximal element. A surjective function allows the mapping of multivariate integrals onto subspaces. The proposed NP-Hard problem for security reduction is the non-commutative Grothendieck problem. The reduction of this problem is achieved by applying bilinear matrices in terms of the holomorphic vector bundle such that coordinate systems are intersected via surjective functions between each holomorphic expression. The result is an arbitrarily selected set of points within constraints of bilinear matrix inequalities approximate to the non-commutative problem. This is achieved by applying the quadratic form of bilinear matrices to a linear matrix inequality.Comment: Originally available on ResearchGate and now archive