Geometry, Algebra, and Algorithms

Local curvature of maximally nondegenerate Radon-like transforms

“…Using this formulation, it is possible to give a slightly more intrinsic definition of N R (Q) that agrees exactly with (6). To that end, let {ũ i } d1 i=1 , {ṽ i } k i=1 , and { wi } d i=1 be orthonormal bases of R d1 , R k , and R d , respectively.…”

“…Note that the convex hull is taken in [0, ∞) d1+k+d . This is equivalent to (6) simply because every orthonormal basis can be mapped to the standard basis by an orthogonal matrix, and likewise the action of any orthogonal matrix on the standard basis is to map it to an orthonormal basis. The relevance of N R (Q) is demonstrated by the following lemma.…”

“…. , p M (x, y), −z + F (x, y)) has determinant − det D y p(x, y) = 0 and consequently has rank M + 1 (see [6,9.6,Definition 7 and Theorem 8] for the relationship between Jacobians and the dimension of a variety). As a consequence, there must be a nontrivial polynomial…”

Joint-Space Characterization of a Medical Parallel Robot Based on a Dual Quaternion Representation of SE(3)

“…The parameterization singularity introduced by the tangent of half angle substitution (the rotation angle of π) may be avoided by defining the parameterization, such that the rotation angle of π is not in the operational workspace or it is equivalent to a known singularity of the mechanism. For an in-depth introduction into algebraic geometry (with great focus on polynomial ideals of affine and projective varieties), and computational methods to work with these polynomial ideals (e.g., Groebner bases) and their properties, the reader is referred to [23].…”