Curves and Surfaces in Three-Dimensional Euclidean Space

2015 ◽  
pp. 69-96
Author(s):  
Mikhail Itskov
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Curves And Surfaces

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

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