Morley’s theorem revisited: Origami construction and automated proof

Morley’s theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. The construction of Morley’s triangle by the straightedge and compass method is impossible because of the well-known impossibility result for angle trisection. However, by origami, the construction of an angle trisector is possible, and hence that of Morley’s triangle. In this paper we present a computational origami construction of Morley’s triangle and an automated correctness proof of the generalized Morley’s theorem.During the computational origami construction, geometrical constraints in symbolic representation are generated and accumulated. Those constraints are then transformed into algebraic forms, i.e. a set of polynomials, which in turn are used to prove the correctness of the construction. The automated proof is based on the Gröbner bases method. The timings of the experiments of the Gröbner bases computations for our proofs are given. They vary greatly depending on the origami construction methods, the algorithms for the Gröbner bases computation, and variable orderings.     

A method for the mechanical derivation of formulas in elementary geometry

A method for mechanical derivation of formulas and calculation of quantities in geometry is given with several examples to illustrate its applications.The work described in this article was supported by NSF grant DRC-8503498.     

On the algebraic formulation of certain geometry statements and mechanical geometry theorem proving

In this paper we analyze the algebraic formulations of certain geometry statements appearing in recent literature related to mechanical geometry theorem proving and give several examples to show that one of these formulations can cause serious problems. We clarify a formulation which is essentially due to W. T. Wu and, in our opinion, is the most satisfactory.This author was supported by NSF Grants DCR-8503498 and CCR-8702108.     

Towards the generation of 3D OpenStreetMap building models from single contributed photographs

3D city models are valuable and useful for many experts using this information for a wide array of environmental and sustainable analyses and services. To produce 3D city models, conventional production processes are typically conducted by authoritative mapping agencies, which mostly rely on aerial photogrammetry and laser scanning. Consequently, obtaining public domain 3D city models is challenging and limited, where the majority of open data is collected and mapped by participatory mapping driven communities. These are still limited to 2D data collection proficiencies due to the used mapping infrastructures and technological limitations. Thus, the 3rd (height) dimension is mostly missing from these maps and models, resulting in the fact that public domain 3D city models are still limited, and only scarcely used for environmental applications. Perhaps one of the most important features in 3D city models are the buildings, since they serve as a major geospatial element in many environmental applications. Our objective is to use a single contributed photograph and OpenStreetMap vector data to precisely calculate the photographed building height, and add this data to the OpenStreetMap map to enable the creation of open source Level of Detail 1 (LoD1) city models. To this end, we have developed a Newton's-based method in optimization to accurately calculate building heights from single contributed photographs taken by citizens using smartphones or tablets. An Android app, OpenStreetHeight, is developed to carry out the experiments. Based on the various medium-height buildings that were photographed using the app and processed using the developed algorithms we received accurate building height values. When compared to reliable reference field measurements, the average height mean absolute error was 30 cm. Combined with the OpenStreetMap footprint vector data, we were able to produce an average LoD1 volume mean absolute error of less than 5%, satisfying the CityGML standard quality. This research presents a framework for a semi-automatic crowdsourced user-generated content calculation of OpenStreetMap building heights, and the creation of reliable and accurate LoD1 building models, as a first step to enhance the already established 2D OpenStreetMap map infrastructure to the 3D domain. This enables expanding the use of OpenStreetMap as a comprehensive and detailed representation of our urban environment for various environmental and sustainable applications and analyses.     

Representation of a human head with bi-cubic B-splines technique based on the laser scanning technique in 3D surface anthropometry

Three-dimensional (3D) anthropometry based on the laser scanning technique not only provides one-dimensional measurements calculated in accordance with the landmarks which are pre-located on the human body surface manually, but also the 3D shape information between the landmarks. This new technique used in recent ergonomic research has brought new challenges to resolving the application problem that was generally avoided by anthropometric experts in their researches. The current research problem is concentrating on how to shift and develop one-dimensional measurements (1D landmarks) into three-dimensional measurements (3D land-surfaces). The main purpose of this paper is to test whether the function of B-splines can be used to fit 3D scanned human heads, and to for further study to develop a computer aided ergonomic design tool (CAED). The result shows that B-splines surfaces can effectively reconstruct 3D human heads based on the laser scanning technique.