Conchoidal transform of two plane curves

The conchoid of a plane curve $C$ is constructed using a fixed circle $B$ in the affine plane. We generalize the classical definition so that we obtain a conchoid from any pair of curves $B$ and $C$ in the projective plane. We present two definitions, one purely algebraic through resultants and a more geometric one using an incidence correspondence in \PP^2 \times \PP^2. We prove, among other things, that the conchoid of a generic curve of fixed degree is irreducible, we determine its singularities and give a formula for its degree and genus. In the final section we return to the classical case: for any given curve $C$ we give a criterion for its conchoid to be irreducible and we give a procedure to determine when a curve is the conchoid of another.Comment: 18 pages Revised version: slight title change, improved exposition, fixed proof of Theorem 5.3 Accepted for publication in Appl. Algebra Eng., Commun. Comput

Motion Planning of Legged Robots

We study the problem of computing the free space F of a simple legged robot called the spider robot. The body of this robot is a single point and the legs are attached to the body. The robot is subject to two constraints: each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). Moreover, the robot can only put its feet on some regions, called the foothold regions. The free space F is the set of positions of the body of the robot such that there exists a set of accessible footholds for which the robot is stable. We present an efficient algorithm that computes F in O(n2 log n) time using O(n2 alpha(n)) space for n discrete point footholds where alpha(n) is an extremely slowly growing function (alpha(n) <= 3 for any practical value of n). We also present an algorithm for computing F when the foothold regions are pairwise disjoint polygons with n edges in total. This algorithm computes F in O(n2 alpha8(n) log n) time using O(n2 alpha8(n)) space (alpha8(n) is also an extremely slowly growing function). These results are close to optimal since Omega(n2) is a lower bound for the size of F.Comment: 29 pages, 22 figures, prelininar results presented at WAFR94 and IEEE Robotics & Automation 9

Conchoid surfaces of spheres

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction

Infinite Products of Large Random Matrices and Matrix-valued Diffusion

We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of hermitian and complex matrices, we observe an emergence of "topological phase transition" in the spectrum, after some critical diffusion time $\tau_{\rm crit}$ is reached. In the case of the particular product of two hermitian ensembles, we observe also an unusual localization-delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulae obtained in this work by numerical simulation.Comment: 39 pages, 12 figures; v2: references added; v3: version to appear in Nucl. Phys.

From spider robots to half disk robots

International audienceWe study the problem of computing the set F of accessible and stable placements of a spider robot. The body of this robot is a single point and the legs are line segments attached to the body. The robot can only put its feet on some regions, called the foothold regions. Moreover, the robot is subject to two constraints: Each leg has a maximal extension R (accessibility constraint) and the body of the robot must lie above the convex hull of its feet (stability constraint). We present an efficient algorithm to compute F. If the foothold regions are polygons with n edges in total, our algorithm computes F in O(n^2 log n) time and O(n^2 alpha(n)) space where alpha is the inverse of the Ackerman's function. Omega(n^2) is a lower bound for the size of F

Block design of a wheelset for railway transport

The paper presents an analysis of the features of geometric contacting of the rolling surfaces of a block design of a wheelset and a rail. The peculiarities of the contact slipping by the rolling surfaces of the contact stresses are considered

BASIC APPLICATIONS OF THE q-DERIVATIVE FOR A GENERAL SUBFAMILY OF ANALYTIC FUNCTIONS SUBORDINATE TO k-JACOBSTHAL NUMBERS

This research paper deals with some radius problems, the basic geometricproperties, general coecient and inclusion relations that are established for functionsin a general subfamily of analytic functions subordinate to k-Jacobsthal numbers