Inertia tensor of a triangle in barycentric coordinates

2021 ◽  
Vol 79 (7) ◽  
pp. 589-599
Author(s):  
U-Rae Kim ◽  
Dong-Won Jung ◽  
Chaehyun Yu ◽  
Wooyong Han ◽  
Jungil Lee
Keyword(s):  
Inertia Tensor ◽  
Barycentric Coordinates

scholarly journals Criteria for copositive matrices using simplices and barycentric coordinates

1995 ◽  
Vol 220 ◽  
pp. 9-30 ◽  
Author(s):  
Lars-Erik Andersson ◽  
Gengzhe Chang ◽  
Tommy Elfving
Keyword(s):  
Barycentric Coordinates ◽  
Copositive Matrices

Random nested tetrahedra

1998 ◽  
Vol 30 (03) ◽  
pp. 619-627 ◽  
Author(s):  
Gérard Letac ◽  
Marco Scarsini
Keyword(s):  
Random Walk ◽  
Convex Hull ◽  
Affine Space ◽  
Barycentric Coordinates ◽  
Geometrical Application

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, α n of E. Choose n independent points β1, β2, …, β n randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, α n ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

scholarly journals Nets with collisions (unstable nets) and crystal chemistry

2013 ◽  
Vol 69 (6) ◽  
pp. 535-542 ◽  
Author(s):  
Olaf Delgado-Friedrichs ◽  
Stephen T. Hyde ◽  
Shin-Won Mun ◽  
Michael O'Keeffe ◽  
Davide M. Proserpio
Keyword(s):  
Crystal Structures ◽  
Crystal Chemistry ◽  
Barycentric Coordinates ◽  
Real Crystal ◽  
Crystallographic Symmetry

Nets in which different vertices have identical barycentric coordinates (i.e.have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termedunforced.

The Evaluation of Moments of Regions With Piecewise-Linearly Approximated Boundaries via Line Integration

1989 ◽  
Author(s):  
J. Angeles ◽  
M. J. Al-Daccak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Inertia Tensor ◽  
Divergence Theorem ◽  
Dimensional Euclidean Space ◽  
Repeated Application ◽  
The Subject ◽  
First Moment

Abstract The subject of this paper is the computation of the first three moments of bounded regions imbedded in the three-dimensional Euclidean space. The method adopted here is based upon a repeated application of Gauss’s Divergence Theorem to reduce the computation of the said moments — volume, vector first moment and inertia tensor — to line integration. Explicit, readily implementable formulae are developed to evaluate the said moments for arbitrary solids, given their piecewise-linearly approximated boundary. An example is included that illustrates the applicability of the formulae.

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