Computation of Spatial Displacements From Geometric Features

1993 ◽  
Vol 115 (1) ◽  
pp. 95-102 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
Minimum Number ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes, and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimum number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimum number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

Computation of Spatial Displacements From Geometric Features

1991 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

Abstract This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimal number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines (each pair being non-planar) are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimal number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

Computation of Spatial Displacements From Redundant Geometric Features

1992 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Symmetric Matrix ◽  
Automatic Generation ◽  
Geometric Features ◽  
Least Eigenvalue ◽  
Special Cases ◽  
Minimum Number ◽  
Feature Information ◽  
Spatial Displacements ◽  
Simple Features ◽  
Cubic Function

Abstract This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1992). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

Computation of Spatial Displacements from Redundant Geometric Features

1994 ◽  
Vol 116 (4) ◽  
pp. 1073-1080 ◽  
Author(s):  
Q. J. Ge ◽  
B. Ravani
Keyword(s):  
Symmetric Matrix ◽  
Automatic Generation ◽  
Geometric Features ◽  
Least Eigenvalue ◽  
Special Cases ◽  
Minimum Number ◽  
Feature Information ◽  
Spatial Displacements ◽  
Simple Features ◽  
Cubic Function

This paper follows a previous one on the computation of spatial displacements (Ravani and Ge, 1993). The first paper dealt with the problem of computing spatial displacements from a minimum number of simple features of points, lines, planes, and their combinations. The present paper deals with the same problem using a redundant set of the simple geometric features. The problem for redundant information is formulated as a least squares problem which includes all simple features. A Clifford algebra is used to unify the handling of various feature information. An algorithm for determining the best orientation is developed which involves finding the eigenvector associated with the least eigenvalue of a 4 × 4 symmetric matrix. The best translation is found to be a rational cubic function of the best orientation. Special cases are discussed which yield the best orientation in closed form. In addition, simple algorithms are provided for automatic generation of body-fixed coordinate frames from various feature information. The results have applications in robot and world model calibration for off-line programming and computer vision.

scholarly journals On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

2008 ◽  
Vol 17 (2) ◽  
pp. 203-224 ◽  
Author(s):  
ADRIAN DUMITRESCU ◽  
CSABA D. TÓTH
Keyword(s):  
Unit Volume ◽  
Time Algorithm ◽  
Combinatorial Problems ◽  
List Type ◽  
Type Number ◽  
Point Sets ◽  
Minimum Number ◽  
Three Space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined bynpoints in 3-space, and in general inddimensions.(i)The number of tetrahedra of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$3is at most$\frac{2}{3}n^3-O(n^2)$, and there are point sets for which this number is$\frac{3}{16}n^3-O(n^2)$. We also present anO(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every$k,d\in \mathbb{N}, 1\leq k \leq d$, the maximum number ofk-dimensional simplices of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$dis Θ(nk).(ii)The number of unit volume tetrahedra determined bynpoints in$\mathbb{R}$3isO(n7/2), and there are point sets for which this number is Ω(n3log logn).(iii)For every$d\in \mathbb{N}$, the minimum number of distinct volumes of all full-dimensional simplices determined bynpoints in$\mathbb{R}$d, not all on a hyperplane, is Θ(n).

Enumeration of Contact Geometries for Kinematic Registration Using Tactile Sensing Fixtures

2005 ◽  
Vol 128 (1) ◽  
pp. 34-45 ◽  
Author(s):  
Walter W. Nederbragt ◽  
Bahram Ravani
Keyword(s):  
Group Theory ◽  
Data Driven ◽  
Tactile Sensing ◽  
Geometric Features ◽  
Type Synthesis ◽  
Robot Calibration ◽  
Planar Surfaces ◽  
Spherical Surfaces ◽  
Kinematic Registration ◽  
Geometric Elements

This paper uses group theory for enumeration of contacts between geometric elements necessary for kinematic registration or part referencing in robotics. The results are applied to type synthesis of tactile sensing mechanical fixtures. Kinematic registration is an important step in robot calibration and in data driven automation. Although the scope of the paper is limited to geometric contacts involving points, lines, planar surfaces, cylindrical surfaces, and spherical surfaces, the techniques developed are general and can be applied to other geometric features and non-tactile sensing elements used in robotic calibration and part referencing.

scholarly journals Proposta curricular no âmbito educacional: relevâncias para construção de aprendizagens na escola / Curriculum proposal in the educational field: relevance for building learning at school

2021 ◽  
Vol 15 (58) ◽  
pp. 49-57
Author(s):  
Danubia De Jesus Souza ◽  
Pedro Fernando Dos Santos
Keyword(s):  
Intellectual Development ◽  
Theoretical Foundation ◽  
National Curriculum ◽  
School Context ◽  
Fundamental Importance ◽  
Field School ◽  
The World ◽  
The Common ◽  
Educational Field

Resumo: Esse artigo faz uma reflexão da proposta curricular no âmbito educacional, cujo principal objetivo é analisar o documento mediante sua importância para os fazeres no contexto escolar. A análise é feita a partir dos estudos proporcionados pela extensão na Residência Pedagógica-CAPES-FACHUS e desenvolvidos na escola-campo, Doutor Severino Alves de Sá, no município de Salgueiro-PE. Sua fundamentação teve como base a Lei 9.394/96(LDB), A Base Nacional Comum Curricular – BNCC, assim como Veiga 2002 e Castro 2019. A metodologia elencada no desenrolar do conteúdo é decorrente de uma pesquisa bibliográfica de estudo analítico sobre o objeto aqui citado. Os resultados obtidos vêm ao encontro de todo exposto apresentado na fundamentação teórica embasada na LDB e BNCC, assim como nas afirmações dos autores e todo o conteúdo estudado. Vale afirmar que a análise do conteúdo proporcionado pela escola é de bastante relevância na construção da Proposta Curricular disponibilizada é parte integrante de seu PPP, atendendo assim todas as expectativas moduladas no decorrer do estudo. Os dados encontrados dispõem de componentes curriculares associados a conteúdos que sempre estão em pauta na atualidade. Considerando-se, portanto, que a proposta pedagógica curricular é um documento de fundamental importância para os seguimentos de conteúdo escolar em prol da formação integral do aluno, do desenvolvimento intelectual e de sua consciência no mundo do saber e das transformações que por ele se dão. Palavras-Chave: Proposta curricular. Disciplinas. Contexto escolar. Abstract: This article reflects on the curriculum proposal in the educational field, whose main objective is to analyze the document according to its importance for the actions in the school context. The analysis is made from the studies provided by the extension in the Pedagogical Residency-CAPES-FACHUS and developed in the field-school, Doutor Severino Alves de Sá, in the municipality of Salgueiro-PE. Its theoretical foundation was based on Law 9,394/96 (LDB), The Common National Curriculum Base – BNCC, as well as Veiga 2002 and Castro 2019. cited here. The results obtained are in line with all the above presented in the theoretical foundation based on the LDB and BNCC, as well as the authors' statements and all the studied content. It is worth stating that the analysis of the content provided by the school is very relevant in the construction of the Curriculum Proposal made available and is an integral part of the school's PPP, thus meeting all the expectations found in the course of the study. The data found have curricular components associated with contents that are always on the agenda today. Considering, therefore, that the curricular pedagogical proposal is a document of fundamental importance for the segments of school content in favor of the integral formation of the student, intellectual development and their awareness in the world of knowledge and the transformations that take place through it. Keywords: Curriculum proposal. Disciplines. School context. 

Multifactorial Decomposition of Inequality: The Case of CAP

2016 ◽  
pp. 59
Author(s):  
Arsen Palestini ◽  
Giuseppe Pignataro
Keyword(s):  
Agricultural Policy ◽  
Theoretical Foundation ◽  
Transfer Scheme ◽  
Income Sources ◽  
Ex Post ◽  
Distribution Process ◽  
Shapley Values ◽  
Marginal Contributions ◽  
The Common ◽  
The Impact

The purpose of this article is to propose a theoretical foundation on the impact of a transfer scheme on income inequality in the redistribution process among participants in a related agreement. Our example involves the study of the Common Agricultural Policy implemented by EU Countries. First, we show that ex-post inequality (after the distribution process) may increase if either initial aggregate income or the amounts of fiscal contributions are sufficiently high. Second, we characterize a multifactorial methodology according to Palestini and Pignataro (2014) to gauge the impact of redistribution and the effects of different income sources to the inequality profile. Finally, we propose an exercise where a hypothetical policy is implemented and we apply the Banzhaf and Shapley values to determine the marginal contributions of each factor to overall inequality.

Modeling Spatial Displacements Using Clifford Algebra

2003 ◽  
Vol 126 (3) ◽  
pp. 420-424 ◽  
Author(s):  
Glen Mullineux
Keyword(s):  
Euclidean Space ◽  
Clifford Algebra ◽  
Geometric Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Real Numbers ◽  
B Spline ◽  
Spatial Displacements ◽  
Basis Vectors ◽  
Three Dimensional Space

This paper looks at the use of a Clifford (or geometric) algebra for handling both rotations and translations in Euclidean space. The algebra is constructed over the real numbers using four basis vectors. Three of these generate a subalgebra which models three-dimensional space; the fourth acts as a projective coordinate. Spatial displacements are represented by bivectors of a certain form. The application to the generation of smooth motions using Be´zier and B-spline techniques is illustrated.

Interpolation of Spatial Displacements Using the Clifford Algebra of E4

1999 ◽  
Vol 121 (1) ◽  
pp. 39-44 ◽  
Author(s):  
K. R. Etzel ◽  
J. M. McCarthy
Keyword(s):  
Clifford Algebra ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Coordinate Frame ◽  
Spatial Motion ◽  
Hypercomplex Numbers ◽  
Key Frames ◽  
Spatial Displacements ◽  
Three Dimensional Space

In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.

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