Three Dimensional Acquisition and NURBS based Geometric Modelling of Natural Objects

2009 ◽  
Author(s):  
E. Dimas ◽  
E. Psarakis ◽  
D. Briassoulis ◽  
G. Moustakides
Keyword(s):  
Three Dimensional ◽  
Geometric Modelling ◽  
Natural Objects

scholarly journals FOUR-DIMENSIONAL BALL IN A GRAPHIC REPRESENTATION

2021 ◽  
pp. 37-46
Author(s):  
Helena Bidnichenko
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Geometric Modelling ◽  
Vector Model ◽  
Original Position ◽  
Two Dimensional ◽  
Axis Of Rotation ◽  
Dimensional Ball ◽  
Three Dimensional Space ◽  
Horizontal Vector

The paper presents a method for geometric modelling of a four-dimensional ball. For this, the regularities of the change in the shape of the projections of simple geometric images of two-dimensional and three-dimensional spaces during rotation are considered. Rotations of a segment and a circle around an axis are considered; it is shown that during rotation the shape of their projections changes from the maximum value to the degenerate projection. It was found that the set of points of the degenerate projection belongs to the axis of rotation, and each n-dimensional geometric image during rotation forms a body of a higher dimension, that is, one that belongs to (n + 1) -dimensional space. Identified regularities are extended to the four-dimensional space in which the ball is placed. It is shown that the axis of rotation of the ball will be a degenerate projection in the form of a circle, and the ball, when rotating, changes its size from a volumetric object to a flat circle, then increases again, but in the other direction (that is, it turns out), and then in reverse order to its original position. This rotation is more like a deformation, and such a ball of four-dimensional space is a hypersphere. For geometric modelling of the hypersphere and the possibility of its projection image, the article uses the vector model proposed by P.V. Filippov. The coordinate system 0xyzt is defined. The algebraic equation of the hypersphere is given by analogy with the three-dimensional space along certain coordinates of the center a, b, c, d. A variant of hypersection at t = 0 is considered, which confirms by equations obtaining a two-dimensional ball of three-dimensional space, a point (a ball of zero radius), which coincides with the center of the ball, or an imaginary ball. For the variant t = d, the equation of a two-dimensional ball is obtained, in which the radius is equal to R and the coordinates of all points along the 0t axis are equal to d. The variant of hypersection t = k turned out to be interesting, in which the equation of a two-dimensional sphere was obtained, in which the coordinates of all points along the 0t axis are equal to k, and the radius is . Horizontal vector projections of hypersection are constructed for different values of k. It is concluded that the set of horizontal vector projections of hypersections at t = k defines an ellipse.  

Challenges and Opportunities in Geometric Modelling of Complex Bio-Inspired 3D Objects Designed for Additive Manufacturing

2021 ◽  
pp. 1-27
Author(s):  
Nikita Letov ◽  
Pavan Tejaswi Velivela ◽  
Siyuan Sun ◽  
Yaoyao Fiona Zhao
Keyword(s):  
Additive Manufacturing ◽  
Computer Aided Design ◽  
Three Dimensional ◽  
Geometric Modelling ◽  
Survey Paper ◽  
Engineering Software ◽  
Modelling Method ◽  
Geometric Objects ◽  
Computer Aided ◽  
Challenges And Opportunities

Abstract Ever since its introduction over five decades ago, geometric solid modelling has been crucial for engineering design purposes and is used in engineering software packages such as computer-aided design (CAD), computer-aided manufacturing (CAM), computer-aided engineering (CAE), etc. Solid models produced by CAD software have been used to transfer geometric information from designers to manufacturers. Since the emergence of additive manufacturing (AM), a CAD file can also be directly uploaded to a three-dimensional (3D) printer and used for production. AM techniques allow manufacturing of complex geometric objects such as bio-inspired structures and lattice structures. These structures are shapes inspired by nature and periodical geometric shapes consisting of struts interconnecting in nodes. Both structures have unique properties such as significantly reduced weight. However, geometric modelling of such structures has significant challenges due to the inability of current techniques to handle their geometric complexity. This calls for a novel modelling method that would allow engineers to design complex geometric objects. This survey paper reviews geometric modelling methods of complex structures to support bio-inspired design created for AM which includes discussing reasoning behind bio-inspired design, limitations of current modelling approaches applied to bio-inspired structures, challenges encountered with geometric modelling and opportunities that these challenges reveal. Based on the review, a need for a novel geometric modelling method for bio-inspired geometries produced by AM is identified. A framework for such bio-inspired geometric modelling method is proposed as a part of this work.

scholarly journals FROM LIDAR DATA TOWARDS HBIM FOR STRUCTURAL EVALUATION

2019 ◽  
Vol XLII-2/W15 ◽  
pp. 125-132 ◽  
Author(s):  
R. Argiolas ◽  
A. Cazzani ◽  
E. Reccia ◽  
V. Bagnolo
Keyword(s):  
Three Dimensional ◽  
Point Clouds ◽  
3D Models ◽  
Geometric Modelling ◽  
Ongoing Research ◽  
Structural Evaluation ◽  
3D Point Clouds ◽  
First Results ◽  
Historical Architecture ◽  
Masonry Vaults

<p><strong>Abstract.</strong> In HBIM processes, the extraction of geometric components from 3D point clouds data can sometimes be a complex process. The so-called <q>Scan to BIM</q> process has been widely utilized: deriving 3D models from point clouds often a local modelling of geometric components is necessary. This leads in most cases to use external modelling tools or complex local modelling processes. In both cases, we often get a model that cannot be reused for other items belonging to the same category, contravening the BIM philosophy. Vaulted systems are a typical example of complex elements that we can find in historical architecture. The paper presents the first results of an ongoing research on geometric modelling and structural evaluation of masonry ribbed vaults. An algorithm is developed to generate a NURBS surface of masonry vaults that, starting from the data extrapolated from the point cloud, allows to obtain an HBIM family. The research aims to overcome the inability to reference to standardised objects in local modelling of historical architecture elements. Directed to a standardization in the geometric modelling process of 3D laser scan data, the developed workflow is a possible alternative to commonly used workflows. Particular attention is focused on a case study of stellar vaults, a special class of masonry ribbed vaults whose three-dimensional geometry features a star-shaped projection on the horizontal plane. The work is carried out to verify that this family can be used for the structural analysis of stellar masonry vaults.</p>

scholarly journals Manufacturability of Fractal Geometry

2004 ◽  
Vol 471-472 ◽  
pp. 722-726 ◽  
Author(s):  
Y.C. Yeung ◽  
Kai Ming Yu
Keyword(s):  
Fractal Geometry ◽  
State Of The Art ◽  
Three Dimensional ◽  
Rough Terrain ◽  
Seafloor Topography ◽  
Research Papers ◽  
Natural Objects ◽  
Manufacturing Techniques

Nowadays more and more aesthetic product developments, assemblage and decoration designs are taking aesthetically appealing forms of natural objects such as rough terrain, ripples on lakes, coastline and seafloor topography. They are mathematical definable via fractal geometry theory and emerge to attract a lot of attention. However, not many methods for manufacturing of fractal objects have been reported in the literature and no previous research papers concern the manufacturability of fractal geometry. The paper will, thus, give a tentative classification and nomenclature of fractal geometry. Then, a state-of-the-art overview of manufacturing techniques is presented. By bridging the gap between fractal geometry and manufacturing, those processes that are promising to manufacture the three dimensional (3D) fractal objects will be highlighted. Afterward, a brief overview of limitation of those processes will be discussed.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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