scholarly journals The μ-Basis of Improper Rational Parametric Surface and Its Application

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 640
Author(s):  
Sonia Pérez-Díaz ◽  
Li-Yong Shen
Keyword(s):  
Inversion Formula ◽  
Rational Surface ◽  
Parametric Surface ◽  
Rational Map ◽  
Geometric Properties ◽  
Theoretical Frame ◽  
Surface Representations ◽  
Greatest Common Divisors ◽  
Computational Processes ◽  
Algebraic Tool

The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

Parametric Surfaced-Based Geological Reservoir Representation: A Computer Graphics Tool for Improved Decision Making in Immersive Environments

2020 ◽  
Author(s):  
Seyyedmohammad Moulaeifard ◽  
Florian Wellmann
Keyword(s):  
Decision Making ◽  
Computer Graphics ◽  
Subdivision Surfaces ◽  
Control Points ◽  
Parametric Surface ◽  
Geological Modelling ◽  
Spline Surfaces ◽  
Geological Models ◽  
Surface Representations ◽  
Implicit Representations

<p>Uncertainties are an inherent part of geological interpretation and immersive rendering has the potential to play a key role in gaining better insights. However, most 3D geological models have a limited possibility of manual, fast and smooth modification in order to make better decisions and interpretations. Here we present examples of parametric surface representations which use control points as a possibility to bring interactivity to geological modelling in immersive frameworks.</p><p>In fact, using 2D surfaces of 3D solid objects is a typical representation of 3D models. Two of the major ways for surface representation in computer graphics are implicit representations and parametric surface representations. Parametric surface representations, unlike implicit representations, are based on control points. Manipulating these control points makes it easy and intuitive to modify geological models smoothly and fast, with a potential to more interactive decision-making.</p><p>We present two different examples of parametric surface approaches; Spline Surfaces and Subdivision Surfaces. Spline surfaces, e.g. Bezier or NURBS surfaces, are a popular and common standard for CAD (Computer-Aided Design). Also, these surfaces are on the basis of parametric- based curves and a set of weighted control points. Subdivision Surfaces define smooth surfaces after a series of refinement which can be controlled by control points. Subdivision surfaces are not only a popular method for making free form models but also a common tool in animation, computer games and entertainment industry.</p><p>Recently, research has been done based on using spline surfaces to model diverse geological structures and reservoirs. Similar to applications in computer graphics, using these methods in geological modelling can have specific considerations. Model refinement (e.g. adding new control points) and the requirement of many patches with geometrical constraints for the representation of complex geometries are some of the main difficulties of using spline surfaces. In this presentation, we will discuss several of these aspects and show two promising and controllable techniques for intuitive use of parametric surface-based representations in 3D geological and reservoir modelling.</p>

Three-Dimensional Reconstruction of Native Androctonus Australis Hemocyanin

1990 ◽  
Vol 48 (1) ◽  
pp. 452-453
Author(s):  
Nicolas Boisset ◽  
Jean-Christophe Taveau ◽  
Jean Lamy ◽  
Terence Wagenknecht ◽  
Michael Radermacher ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Solid Body ◽  
Body Surface ◽  
Three Dimensional ◽  
Staining Technique ◽  
Three Dimensional Reconstruction ◽  
Dimensional Reconstruction ◽  
View Reconstruction ◽  
Surface Representations ◽  
Top View

Hemocyanin, the respiratory pigment of the scorpion Androctonus australis is composed of 24 kidney shaped subunits. A model of architecture supported by many indirect arguments has been deduced from electron microscopy (EM) and immuno-EM. To ascertain, the disposition of the subunits within the oligomer, the 24mer was submitted to three-dimensional reconstruction by the method of single-exposure random-conical tilt series.A sample of native hemocyanin, prepared with the double layer negative staining technique, was observed by transmisson electron microscopy under low-dose conditions. Six 3D-reconstructions were carried out indenpendently from top, side and 45°views. The results are composed of solid-body surface representations, and slices extracted from the reconstruction volume.The main two characters of the molecule previously reported by Van Heel and Frank, were constantly found in the solid-body surface representations. These features are the presence of two different faces called flip and flop and a rocking of the molecule around an axis passing through diagonnally opposed hexamers. Furthermore, in the solid-body surface of the top view reconstruction, the positions and orientations of the bridges connecting the half molecules were found in excellent agreement with those predicted by the model.

Arguing for Design Thinking Interventions as a Form of Artistic Interventions

2016 ◽  
Vol 10 ◽  
pp. 40-47
Author(s):  
Ulla Johansson Sköldberg ◽  
Jill Woodilla
Keyword(s):  
Organizational Change ◽  
Design Thinking ◽  
Organizational Innovation ◽  
Frame Of Reference ◽  
Full Potential ◽  
Artistic Process ◽  
Theoretical Frame ◽  
Artistic Interventions ◽  
Innovation And Change ◽  
Empirical Material

Drawing on data from two projects where artists used their artistic competence as organizational change facilitators, we argue for a theoretical coupling of the discourse(s) of design thinking to research streams within art-and-management. The artistic dimension of design, the practice perspective and the artistic process should be considered if we are to understand the full potential of design thinking for companies. This paper describes two artistic intervention projects that highlight valuable ways artists can contribute to organizational innovation and change.  We begin with the theoretical frame of reference and a short methodological statement, followed by the empirical material.  In the analysis section we point to ways in which such interventions are similar to ones led by designers when we consider the designer’s process as individualized and contextualized.  Finally, we draw conclusions.

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

Conclusion

2018 ◽  
Author(s):  
Thibaut d'Hubert
Keyword(s):  
South Asia ◽  
Quasi Experimental ◽  
Key Issues ◽  
Per Se ◽  
Theoretical Frame ◽  
Opening Up

In the conclusion, I come back on key issues of my analysis of Ālāol’s poetics. Whereas performance and the absence of theoretical frame recorded in treatises on grammar or poetics are defining features of the vernacular tradition, we witness attempts to describe and systematize vernacular poetics in eastern South Asia. Sanskrit played a major role in this attempt at systematizing vernacular poetics to foster connoisseurship. The domain of reference of vernacular poets was not poetics per se or rhetoric, but lyrical arts and musicology. But efforts to describe vernacular poetics also display an awareness of the importance of heteroglossia and fluidity in vernacular aesthetics in contrast with Sanskrit. The opening up of the Sanskrit episteme constituted by vernacular poetics also made possible the recourse to literary models and quasi-experimental uses of vernacular poetic idioms. Old Maithili, Avadhi, and Persian were visible components of the making of vernacular poetics in Bengal.

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