Given a generic rational curve C in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly C. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.
Let K be an algebraic number field of degree d and discriminant ∆ over Q. Let A be an associative algebra over K given by structure constants such that A ∼ = M n (K) holds for some positive integer n. Suppose that d, n and |∆| are bounded. Then an isomorphism A → M n (K) can be constructed by a polynomial time ff-algorithm. An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields.As a consequence, we obtain a polynomial time ff-algorithm to compute ismorphisms of central simple algebras of bounded degree over K.Theorem 2. Let A be a Q-subalgebra of M n (R) isomorphic to M n (Q) and let Λ be a maximal Z-order in A. Then there exists an element C ∈ Λ which has rank 1 as a matrix, and whose Frobenius norm C is less than n.Remark 3. When we apply the above theorem, the Frobenius norm · will be inherited from M n (R), with respect to an arbitrary embedding of A into M n (R). Recall that for a matrix X ∈ M n (R) we have X = T r(X T X).Proof. The isomorphism A ∼ = M n (Q) extends to an automorphism of M n (R). Therefore, by the Noether-Skolem Theorem, there exists a matrix P ∈ M n (R) such that A = P M n (Q)P −1 . Let Λ ′ denote the standard maximal order M n (Z) in M n (Q). The theory of maximal orders in central
In this paper we introduce a new technique, based on dual quaternions, for the analysis of closed linkages with revolute joints: the theory of bonds. The bond structure comprises a lot of information on closed revolute chains with a one-parametric mobility. We demonstrate the usefulness of bond theory by giving a new and transparent proof for the well-known classification of overconstrained 5R linkages.
For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings, possibly non-generic. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings.
Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.
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