In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give ...the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.
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The present paper is focused on time-like circular surfaces and singularities in Minkowski 3-space. The timelike circular surface with a constant radius could be swept out by moving a Lorentzian ...circle with its center while following a non-lightlike curve called the spine curve. In the present study, we have parameterized timelike circular surfaces and examined their geometric properties, such as singularities and striction curves, corresponding with those of ruled surfaces. After that, a different kind of timelike circular surface was determined and named the timelike roller coaster surface. Meanwhile, we support the results of this work with some examples.
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The approach of the paper is on spacelike circular surfaces in the Minkowski 3-space. A spacelike circular surface is a one-parameter family of Lorentzian circles with a fixed radius regarding a ...non-null curve, which acts as the spine curve, and it has symmetrical properties. In the study, we have parametrized spacelike circular surfaces and have provided their geometric and singularity properties such as Gaussian and mean curvatures, comparing them with those of ruled surfaces and the classification of singularities. Furthermore, the conditions for spacelike roller coaster surfaces to be flat or minimal surfaces are obtained. Meanwhile, we support the results of the approach with some examples.
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E. Study map is one of the most basic and powerful mathematical tools to study lines in line geometry, it has symmetry property. In this paper, based on the E. Study map, clear expressions were ...developed for the differential properties of one-parameter Lorentzian dual spherical movements that are coordinate systems independent. This eliminates the requirement of demanding coordinates transformations necessary in the determination of the canonical systems. With the proposed technique, new proofs for Euler–Savary, and Disteli’s formulae were derived.
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In this paper, we give the parametric equation of the Bishop frame for a timelike sweeping surface with a unit speed timelike curve in Minkowski 3-space. We introduce a new geometric invariant to ...explain the geometric properties and local singularities of this timelike surface. We derive the sufficient and necessary conditions for this timelike surface to be a timelike developable ruled surface. Afterwards, we take advantage of singularity theory to give the classification of singularities of this timelike developable surface. Furthermore, we give some representative examples to show the applications of the theoretical results.
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The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the ...sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.
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In this paper, explicit expressions were improved for timelike ruled surfaces with the similarity of hyperbolic dual spherical movements. From this, the well known Hamilton and Mannhiem formulae of ...surfaces theory are attained at the hyperbolic line space. Then, by employing the E. Study map, a new timelike Plücker conoid is immediately founded and its geometrical properties are examined. In addition, via the height dual function, we specified the connection among the timelike ruled surface and the order of contact with its timelike Disteli-axis. Lastly, a classification for a timelike line to be a stationary timelike Disteli-axis is attained and explained in detail. Our findings contribute to a deeper realization of the cooperation between hyperbolic spatial movements and timelike ruled surfaces, with potential implementations in fields such as robotics and mechanical engineering.
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This work extends some classical results of Bertrand curves to timelike ruled and developable surfaces using the E. Study map. This provides support to define two timelike ruled surfaces which are ...offset in the sense of Bertrand. It is proved that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied among their dual invariants. In addition, some new results and theorems concerning the developability of the Bertrand offsets of timelike ruled surfaces are gained.
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In this paper, we obtain the necessary and sufficient conditions of a surface pencil pair interpolating a Bertrand pair as common asymptotic curves in Euclidean 3-space E3. Afterwards, the conclusion ...to the ruled surface pencil pair is also obtained. Meanwhile, the epitomes are stated to emphasize that the proposed methods are effective in product manufacturing by adjusting the shapes of the surface pencil pair.
A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, ...the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space G3. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method.
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