• Honam Mathematical Journal
• /
• v.35 no.1
• /
• pp.17-27
• /
• 2013

We survey and study several volume formulas of a Euclidean tetrahedron.

• Journal of the Korean Mathematical Society
• /
• v.50 no.6
• /
• pp.1223-1256
• /
• 2013

We give a geometric proof of the analyticity of the volume of a tetrahedron in extended hyperbolic space, when vertices of the tetrahedron move continuously from inside to outside of a hyperbolic space keeping every face of the tetrahedron intersecting the hyperbolic space. Then we find a geometric and analytic interpretation of a truncated orthoscheme and Lambert cube in the hyperbolic space from the viewpoint of a tetrahedron in the extended hyperbolic space.

• Journal of the Korean Mathematical Society
• /
• v.49 no.4
• /
• pp.765-778
• /
• 2012

We construct a symmetric finite volume element (SFVE) scheme for a self-adjoint elliptic problem on tetrahedron grids and prove that our new scheme has optimal convergent order for the solution and has superconvergent order for the flux when grids are quasi-uniform and regular. The symmetry of our scheme is helpful to solve efficiently the corresponding discrete system. Numerical experiments are carried out to confirm the theoretical results.

• The Pure and Applied Mathematics
• /
• v.4 no.1
• /
• pp.1-6
• /
• 1997

We will construct the generalized law of cosines in a tetrahedron, in a natural way, which gives three dimensional Pythagoras' theorem and enables us to calculate the volume of an arbitrary tetrahedron.

• Proceedings of the KOSOMBE Conference
• /
• v.1998 no.11
• /
• pp.273-274
• /
• 1998

We construct a volume whose boundary is a tetrahedron with triangular faces intersecting the cutting planes along the given contours. This volume is obtained by calculating the Delaunay triangulation slice by slice, mapping 2D to 3D as tetrahedron. Also, eliminate extra-voronoi skeleton and non-solid tetrahedron. In this paper, we propose new method to eliminate non-solid tetrahedron based on the modified extra-voronoi skeleton path. This method enable us to do a compact tetrahedrisation and to reconstruct complex shapes.

• School Mathematics
• /
• v.8 no.1
• /
• pp.27-43
• /
• 2006

The objective of this paper is to design teaching units to foster secondary pre-service teachers' mathematising abilities. In these teaching units we focus on generalizing area of a 2-dimensional triangle and volume of a 3-dimensional tetrahedron to n-volume of n-simplex In this process of generalizing, principle of the permanence of equivalent forms and Cavalieri's principle are applied. To find n-volume of n-simplex, we define n-orthogonal triangular prism, and inquire into n-volume of it. And we find n-volume of n-simplex by using vectors and determinants. Through these teaching units, secondary pre-service teachers can understand and inquire into n-simplex which is generalized from a triangle and a tetrahedron, and n-volume of n-simplex which is generalized from area of a triangle and volume of a tetrahedron. They can also promote natural connection between school mathematics and academic mathematics.

• Journal of Positioning, Navigation, and Timing
• /
• v.9 no.2
• /
• pp.79-88
• /
• 2020

We determined Global Positioning System (GPS) satellite visibilities in signal-blockage simulations and then analyzed Position Dilution of Precision (PDOP) fluctuations obtained from those simulated satellite geometries. PDOP values under harsh signal-blockage simulation conditions become very high compared to those calculated with real observations. Especially when the number of observed satellites is four, which is the minimum requirement for GPS positioning, PDOP values instantaneously reached several hundreds or even several tens of thousands. It was also found that the volume of the tetrahedron composed with four satellites decreases significantly. When the correlation of the tetrahedron volume and PDOP was analyzed, we reached the following conclusions: PDOP values less than 4 can be acquired when the volume is larger than 103.2 × 10¹⁹ ㎥, and PDOP values increase beyond 50 when the volume is less than 6.0 × 10¹⁹ ㎥.

• Korean Journal of Computational Design and Engineering
• /
• v.22 no.1
• /
• pp.59-69
• /
• 2017

Internal capacity of a refrigerator is an important indicator for design and purchasing criteria. The components facing the internal space may have holes or gaps between parts. In traditional way, design engineers manually remodeled the parts to fill the holes and the gaps for enclosed boundary of the internal space. Then they calculated internal volume by subtracting the assembly of parts from its enclosing volume. However, filling holes and gaps is not an automated process requiring a plenty of labor and time. In this research, we have developed a voxel-based method to estimate the internal volume of a refrigerator automatically. It starts transforming all components facing the interior space into voxels and fills all holes and gaps automatically by the developed hole-filling algorithm to form a completely closed boundary of the assembly. Then, it identifies the boundary voxels that are facing to the internal voxels with any part of the component. After getting the intersection points between the boundary voxels and the surfaces of components, it generates the boundary surface of triangular facets with the intersection points. Finally, it estimates the internal volume by adding volume of each tetrahedron composed of a triangle of boundary surface and an arbitrary point.

• The Transactions of the Korean Institute of Electrical Engineers
• /
• v.37 no.12
• /
• pp.847-853
• /
• 1988

A method of three-demensional finite element mesh generation is presented in this paper. This method is based on the Delaunay's triangulation whose dual is Voronoi's diagram. A set of points is given on the boundary surface of the concerning domain and the initial tetrahedra are generated by the given set of points. Then, the quality of every tetrahedron is investigated and the interior points are generated near the tetrahedra which are inferior in quality and the tetrahedra of good quality can be controlled by the density of the initial boundary points. Regions with different material constant can be refined in tetrahedra respectively. To confirm the effectiveness of this algorithm,the total volume of tetrahedra was compared to the true volume and this mesh generator was applied to a three-dimensional electostatic problem.

• Journal of Institute of Control, Robotics and Systems
• /
• v.4 no.2
• /
• pp.220-225
• /
• 1998

In GPS (Global Positioning System) positioning, various measures can be used to select satellites or to evaluate the positioning results. Among these, GDOP (Geometric Dilution of Precision) and RGDOP (Relative GDOP) are the most frequently used. Although these measures are frequently used, the relationship between them is not clearly known. Moreover, the condition number is used as a traditional measure of numerical stability in solving linear equations. Sometimes, the volume of a tetrahedon made by the line of sight vector is used for simplicity. All of these measures share some common properties as well as differences. The relationships between these measures are analyzed in this paper.