9,413 research outputs found
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
On the degree of the polynomial defining a planar algebraic curves of constant width
In this paper, we consider a family of closed planar algebraic curves
which are given in parametrization form via a trigonometric
polynomial . When is the boundary of a compact convex set, the
polynomial represents the support function of this set. Our aim is to
examine properties of the degree of the defining polynomial of this family of
curves in terms of the degree of . Thanks to the theory of elimination, we
compute the total degree and the partial degrees of this polynomial, and we
solve in addition a question raised by Rabinowitz in \cite{Rabi} on the lowest
degree polynomial whose graph is a non-circular curve of constant width.
Computations of partial degrees of the defining polynomial of algebraic
surfaces of constant width are also provided in the same way.Comment: 13 page
A Cubic Surface of Revolution
We develop a direct and elementary (calculus-free) exposition of the famous
cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second
elementary proof that the surface is of revolution.Comment: We12 pages. We have added a second elementary proof that the surface
is of revolution. We have added a two proofs of a special case of the
Salmon-Cayley theorem that every cubic surface has twenty-seven lines, a
section on the singularities of cubic surfaces and material on the general
problem of rational points on a cubic surfac
Stress Interference in Axisymmetric Torsion of a Transeversely Isotropic Body
An unbounded transversely isotropic body of revolution containing two spheroidal
cavities is subjected to torsion about its axis of elastic symmetry, which coincides
with its axis of revolution. At large' distances from the cavities the elastic field
approaches the Saint-Venant solution for the torsion of a circular cylinder. The
elasticity solution is obtained in series form and numerical results presented for
the case of two spherical cavities. Of primary interest is the degree of stress
interference between the. two perturbations, as a function of the spacing of the
cavities and the values of the elastic constants.Air Force Office of Scientific Research Grant No. AFOSR 82-004
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