On the Line Congruences

Taş, Ferhat; Gürsoy, Osman

doi:10.36890/iejg.545123

Cited by 5 publications

(6 citation statements)

References 10 publications

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“…Varieties of lines with two degrees of freedom, such as the line reflections produced by two rotating mirrors, are called line congruences [27]. In our case we coin the name two-mirror congruence.…”

Section: Line Reflections By Rotating Mirrorsmentioning

confidence: 99%

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On the Angular Control of Rotating Lasers by Means of Line Calculus on Hyperboloids

Penne

Boi

Vanlanduit

2023

Sensors

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We propose a new paradigm for modelling and calibrating laser scanners with rotation symmetry, as is the case for lidars or for galvanometric laser systems with one or two rotating mirrors. Instead of bothering about the intrinsic parameters of a physical model, we use the geometric properties of the device to model it as a specific configuration of lines, which can be recovered by a line-data-driven procedure. Compared to universal data-driven methods that train general line models, our algebraic-geometric approach only requires a few measurements. We elaborate the case of a galvanometric laser scanner with two mirrors, that we model as a grid of hyperboloids represented by a grid of 3×3 lines. This provides a new type of look-up table, containing not more than nine elements, lines rather than points, where we replace the approximating interpolation with exact affine combinations of lines. The proposed method is validated in a realistic virtual setting. As a collateral contribution, we present a robust algorithm for fitting ruled surfaces of revolution on noisy line measurements.

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Section: Line Reflections By Rotating Mirrorsmentioning

confidence: 99%

“…The centres of the different hyperboloids H(L(α), B), being the points p(α) on B with minimal distance to L(α), are not equal (except in degenerate cases). Therefore, the congruence of laser lines emitted by a 2M-GLS does not constitute a linear line congruence [17,27,37,38].…”

Section: Warningmentioning

confidence: 99%

On the Angular Control of Rotating Lasers by Means of Line Calculus on Hyperboloids

Penne

Boi

Vanlanduit

2023

Sensors

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“…Ruled surfaces are a class of mathematical surfaces that can be generated by moving lines, also known as generators [2][3][4]. In CAD systems, such surfaces can be generated by moving the cross-section along the rail curve [5].…”

Section: Introductionmentioning

confidence: 99%

Developable Ruled Surfaces Generated by the Curvature Axis of a Curve

Taş,

Ziatdinov

2023

Axioms

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Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of a lot of discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple method of generating ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, as these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.

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“…The ruled surfaces are used in robotics, architecture, mechanics, and other areas. These surfaces are studied in many spaces such as Euclidean, Lorenzian, Galilean, and pseudo‐Galilean spaces 4–8,10,50–64 …”

Section: Introductionmentioning

confidence: 99%

The new characterization of ruled surfaces corresponding dual Bézier curves

İncesu

2021

Math Methods in App Sciences

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Let be a Bézier curve in D3, that is, the control points of are the vectors in D3 = {u = (u1, u2, u3) : ui ∈ D,i = 1,2,3} which is the set of dual vectors whose each coordinate components are dual numbers defined as a+εa* : a,a* ∈ R,ε ≠ 0; ε2 = 0. So the spherical projection of is a spherical curve denoted by on the unit sphere in D3 and every point of corresponds to a directed line in real space R3 by Study transformation. In this study, the ruled surface X(t, ν) corresponding to this projection curve of dual Bézier curve is stated in terms of the parametric equation of the real and dual part of a given dual Bézier curve . Also, in this study, some fundamental characteristics such as the striction curve, the Gaussian curvature, and the distribution parameter of the ruled surface X(t, ν) corresponding to this projection curve of the dual Bézier curve are investigated. These concepts at any point are stated in terms of the control points of the given dual Bézier curve .

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On the Line Congruences

Cited by 5 publications

References 10 publications

On the Angular Control of Rotating Lasers by Means of Line Calculus on Hyperboloids

On the Angular Control of Rotating Lasers by Means of Line Calculus on Hyperboloids

Developable Ruled Surfaces Generated by the Curvature Axis of a Curve

The new characterization of ruled surfaces corresponding dual Bézier curves

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