On the relationship between plane and solid geometry

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

The place and teaching of calculus in secondary schools

Thesis (M.A.)--Boston Universit

From on-line sketching to 2D and 3D geometry: A fuzzy knowledge based system

The paper describes the development of a fuzzy knowledge based prototype system for conceptual design. This real time system is designed to infer user’s sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles, arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tools, but also accepts user’s interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry

Accuracy and Mechanical Properties of Open-Cell Microstructures Fabricated by Selective Laser Sintering

This paper investigates the applicability of selective laser sintering (SLS) for the manufacture of scaffold geometries for bone tissue engineering applications. Porous scaffold geometries with open-cell structure and relative density of 10-60 v% were computationally designed and fabricated by selective laser sintering using polyamide powder. Strut and pore sizes ranging from 0.4 - 1 mm and 1.2 -2 mm are explored. The effect of process parameters on compressive properties and accuracy of scaffolds was examined and outline laser power and scan spacing were identified as significant factors. In general, the designed scaffold geometry was not accurately fabricated on the micron-scale. The smallest successfully fabricated strut and pore size was 0.4 mm and 1.2 mm, respectively. It was found that selective laser sintering has the potential to fabricate hard tissue engineering scaffolds. However the technology is not able to replicate exact geometries on the micron-scale but by accounting for errors resulting from the diameter of the laser and from the manufacturing induced geometrical deformations in different building directions, the exact dimensions of the manufactured scaffolds can be predicted and controlled indirectly, which corresponds favorably with its application in computer aided tissue engineering.Mechanical Engineerin

A finite element modelling methodology for the non-linear stiffness evaluation of adhesively bonded single lap-joints. Part 2, Novel shell mesh to minimise analysis time

A new modelling methodology is presented that enables the stiffness of adhesively bonded single lap-joints to be included in the finite element analysis of whole vehicle bodies. This work was driven by the need to significantly reduce computing resources for vehicle analysis. To achieve this goal the adhesive bond line and adherends are modelled by a relatively ‘small’ number of shell elements to replace the usual solid element mesh for a reliable analysis. Previous work in Part 1 has provided the necessary background information to develop and verify the new finite element analysis that reduces the solution runtime by a factor of 1000. Although a joint’s non-linear stiffness is reliably simulated to failure load, it is recognised by the authors that the coarse shell mesh cannot provide accurate peak stresses or peak strains for the successful application of a numerical failure criterion. Given that the new modelling methodology is very quick to apply to existing shell models of vehicle bodies, it is recommended for use by the stress analyst who requires, say at the preliminary design stage, whole vehicle stiffness performance in a significantly reduced timeframe

Structural effect on the static spin and charge correlations in La1.875_{1.875}Ba0.125−x_{0.125-x}Srx_{x}CuO4_{4}

We report the results of elastic neutron scattering measurements performed on 1/8-hole doped La$_{1.875}$Ba$_{0.125-x}$Sr$_{x}$CuO$_{4}$ single crystals with {\it x}=0.05, 0.06, 0.075 and 0.085. In the low-temperature less-orthorhombic (LTLO, {\it Pccn} symmetry) phase, the charge-density-wave (CDW) and spin-density-wave (SDW) wavevectors were found to tilt in a low-symmetric direction with one-dimensional anisotropy in the CuO$_{2}$ plane, while they were aligned along the high-symmetry axis in the low-temperature tetragonal (LTT, {\it P}4$_2$/{\it ncm} symmetry) phase. The coincident direction of two wavevectors suggests a close relation between CDW and SDW orders. The SDW wavevector systematically deviates from the Cu-O bond direction in the LTLO phase upon Sr substitution and the tilt angle in the LTLO phase is smaller than that in the low-temperature orthorhombic phase (LTO, {\it B}{\it mab} symmetry) with comparable in-plane orthorhombic distortion. These results demonstrate a correlation between the corrugated pattern of CuO$_{2}$ plane and the deviations.Comment: 6 pages, 7figure