Accurate Drawings of Three Dimensional Figures

1944 ◽  
Vol 37 (8) ◽  
pp. 350-353
Author(s):  
Margaret Joseph
Keyword(s):  
Three Dimensional ◽  
Regular Tetrahedron ◽  
Solid Geometry

Every teacher of solid geometry has no doubt experienced having a student fail to prove an original exercise just because his drawing of the figure required was so much out of proportion that he failed to establish the proper relationship between lines, angles, triangles, or planes necessary in the proof. If, for example, a student would draw a square for the base of an oblique prism or a circle for the base of a cone, instead of the parallelogram and ellipse as they appear in perspective, then the whole figure is distorted. Just recently I had a student come to me for help in deriving the formula for the altitude of a regular tetrahedron in terms of its edge e. As soon as I saw the rough sketch he was using, where the faces were scalene triangles, I knew why he had two unknowns in the equation that he was trying to solve. The altitude did not meet the base of the figure at its center. I made a better drawing for him and he immediately recognized his error and proceeded with the solution without further help from me.

scholarly journals Numerical Study of the Zero Velocity Surface and Transfer Trajectory of a Circular Restricted Five-Body Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
R. F. Wang ◽  
F. B. Gao
Keyword(s):  
Body Problem ◽  
Numerical Study ◽  
Dimensional Space ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Regular Tetrahedron ◽  
Transfer Trajectory ◽  
Zero Velocity ◽  
Zero Velocity Surfaces ◽  
Zero Velocity Surface

We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedron configuration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as a small celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfer trajectory of the fifth particle skimming over four primaries then is designed.

Equipment workspace analysis in infrastructure projects

2007 ◽  
Vol 34 (10) ◽  
pp. 1247-1256 ◽  
Author(s):  
Amin Hammad ◽  
Cheng Zhang ◽  
Mohamed Al-Hussein ◽  
Germain Cardinal
Keyword(s):  
Three Dimensional ◽  
Construction Equipment ◽  
Prototype System ◽  
Constructive Solid Geometry ◽  
Infrastructure Projects ◽  
Bridge Construction ◽  
Solid Geometry ◽  
Workspace Analysis ◽  
Computational Aspects

Workspace conflicts are one of the important problems that can delay construction activities, reduce productivity, or cause accidents that threaten the safety of workers. Workspace planning is particularly important in the case of large infrastructure projects, such as bridge construction, and rehabilitation projects where equipment with complex workspace requirement is required. This paper aims to extend the previous research on workspace representation and analysis in the case of large infrastructure projects focusing on the following issues: (i) specific representation of equipment workspaces using composite shapes, (ii) semiautomatic generation and analysis of workspaces, and (iii) development of a prototype system that can generate workspaces and detect spatiotemporal conflicts in a three-dimensional environment. The computational aspects of the proposed approach are discussed and demonstrated through the development of a prototype system, which has been applied in a case study of the deck replacement of Jacques Cartier Bridge in Montréal.Key words: workspace analysis, spatiotemporal conflicts, construction equipment, infrastructure projects, bridge redecking, constructive solid geometry.

scholarly journals Design and analysis of three-dimensional printing of a porous titanium scaffold

2021 ◽  
Vol 22 (1) ◽  
Author(s):  
Jiajie Yang ◽  
Yaqiang Li ◽  
Xiaojian Shi ◽  
Meihua Shen ◽  
Kaibing Shi ◽  
...  
Keyword(s):  
Compressive Strength ◽  
3D Printing ◽  
Pore Size ◽  
Three Dimensional ◽  
Structural Models ◽  
Porous Titanium ◽  
Knee Joints ◽  
Regular Tetrahedron ◽  
Diamond Structure ◽  
Pore Morphology

Abstract Objective Mechanic strength, pore morphology and size are key factors for the three-dimensional (3D) printing of porous titanium scaffolds, therefore, developing optimal structure for the 3D printed titanium scaffold to fill bone defects in knee joints is instructive and important. Methods Structural models of titanium scaffolds with fifteen different pore unit were designed with 3D printing computer software; five different scaffold shapes were designed: imitation diamond-60°, imitation diamond-90°, imitation diamond-120°, regular tetrahedron and regular hexahedron. Each structural shape was evaluated with three pore sizes (400, 600 and 800 μm), and fifteen types of cylindrical models (size: 20 mm; height: 20 mm). Autodesk Inventor software was used to determine the strength and safety of the models by simulating simple strength acting on the knee joints. We analyzed the data and found suitable models for the design of 3D printing of porous titanium scaffolds. Results Fifteen different types of pore unit structural models were evaluated under positive pressure and lateral pressure; the compressive strength reduced when the pore size increased. Under torsional pressure, the strengths of the imitation diamond structure were similar when the pore size increased, and the strengths of the regular tetrahedron and regular hexahedron structures reduced when the pore size increased. In each case, the compressive strength of the regular hexahedron structure was highest, that of the regular tetrahedron was second highest, and that of the imitation diamond structure was relatively low. Fifteen types of cylindrical models under a set force were evaluated, and the sequence of comprehensive compressive strength, from strong to weak was: regular hexahedron > regular tetrahedron > imitation diamond-120° > imitation diamond-90° > imitation diamond-60°. The compressive strength of cylinder models was higher when the pore size was smaller. Conclusion The pore size and pore morphology were important factors influencing the compressive strength. The strength of each structure reduced when the pore size (400, 600 and 800 μm) increased. The models of regular hexahedron, regular tetrahedron and imitation diamond-120°appeared to meet the conditions of large pore sizes and high compressive strength.

Automated FE Analysis for Heat Sink of LED Modules

2013 ◽  
Vol 302 ◽  
pp. 765-771 ◽  
Author(s):  
Joon Seong Lee ◽  
Dong Keun Park ◽  
Youn Jong Choi
Keyword(s):  
Heat Sink ◽  
Geometric Model ◽  
Three Dimensional ◽  
Fe Analysis ◽  
Present System ◽  
Analysis Model ◽  
Solid Geometry ◽  
Analysis System ◽  
Definition Of ◽  
Interactive Operations

This paper describes an automatic finite element (FE) mesh generation for FE analysis of LED modules. It is consisting of element generation, bubble packing and solid geometry modeler. This automated FE analysis system including bubble packing method consists of three sub-processes: (a) definition of geometric model, i.e. analysis model, (b) generation of bubbles, and (c) generation of elements. One of commercial solid modelers is employed for three-dimensional solid structures. Bubble is generated if its distance from existing bubble points is similar to the bubble spacing function at the point. The Delaunay method is introduced as a basic tool for element generation. The developed system allows designers to evaluate detailed physical behaviors of structures through some simple interactive operations to their geometry models. To demonstrate practical performances of the present system, the system was used to an analysis of heat sink. Practical performances of the present system are demonstrated through several examples for heat sink of LED modules.

The Art of Teaching: A Helpful Technique in Teaching Solid Geometry

1940 ◽  
Vol 33 (1) ◽  
pp. 39-40
Author(s):  
James V. Bernardo
Keyword(s):  
Three Dimensional ◽  
The Real ◽  
Solid Geometry ◽  
Average Student ◽  
Art Of Teaching

It has been my experience; as it has been undoubtedly that of many who teach solid geometry, to find that the three-dimensional concepts are not easily conceived by the average student. He does not comprehend fully the meaning of the drawings of “solid” figures in one plane. To develop an aptitude for drawing and for interpreting figures is the real job for the teacher who is attempting to expound the propositions of the sixth, seventh, and eighth “Books” of Euclid.

Looking at Solid Geometry Through Perspective

1941 ◽  
Vol 34 (4) ◽  
pp. 147-150
Author(s):  
Ethel Spearman
Keyword(s):  
Secondary Education ◽  
Three Dimensional ◽  
Joint Commission ◽  
Pictorial Representation ◽  
Solid Geometry ◽  
The Joint Commission

“Much attention should be given to the visualization of spatial figures and relations, to the representation of three-dimensional figures on paper, and to the solution of problems in mensuration,” says the Joint Commission among its suggestions on solid geometry in its report on “The Place of Mathematics in Secondary Education.”* These ideas are incorporated in the notion of perspective. Perspective may be regarded as a practical means for securing a rigorous reciprocal metrical relationship between the shapes of objects as definitely located in space and their pictorial representation. It may be regarded as the rationalization of sight. This Commission further points out that “the problems in mensuration offer opportunity for correlation of solid geometry with arithmetic, algebra, and trigonometry. Part of the importance of the geometry of the sphere comes from the perspective that it makes possible.”

Three-Dimensional Probabilistically Shaped CAP Modulation Based on Constellation Design Using Regular Tetrahedron Cells

2020 ◽  
Vol 38 (7) ◽  
pp. 1728-1734 ◽  
Author(s):  
Jianxin Ren ◽  
Bo Liu ◽  
Xiangyu Wu ◽  
Lijia Zhang ◽  
Yaya Mao ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Regular Tetrahedron ◽  
Constellation Design

On a 3-Dimensional Isoperimetric Problem

1970 ◽  
Vol 13 (4) ◽  
pp. 447-449 ◽  
Author(s):  
Magelone Kömhoff
Keyword(s):  
Surface Area ◽  
Convex Polyhedron ◽  
Three Dimensional ◽  
Isoperimetric Problem ◽  
Total Surface Area ◽  
Edge Length ◽  
Regular Tetrahedron ◽  
3 Dimensional ◽  
Image Position

Let L(P) denote the total edge length and A(P) the total surface area of a three-dimensional convex polyhedron P. In [5] it was shown that if P belongs to the set of all polyhedra with triangular faces then for all with equality if and only if is a regular tetrahedron.It is not difficult to establish the inequality

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