Null testing toroidal surface and biconic surface with cylinder compensator

2013 ◽  
Vol 11 (s2) ◽  
pp. S22202-322206
Author(s):  
Zhuang Liu Zhuang Liu ◽  
Yan Gong Yan Gong
Keyword(s):  
Toroidal Surface

Workspace Cusp Locations, and Simplified Workspace Boundary Equations for General RRR Regional Structures of Manipulators

1996 ◽  
Author(s):  
Peregrine E. J. Riley ◽  
Louis E. Torfason
Keyword(s):  
Complex Geometry ◽  
Spatial Location ◽  
Polynomial Equation ◽  
Toroidal Surface ◽  
General Complex ◽  
Cusp Points ◽  
Joint Variable ◽  
Boundary Equations ◽  
Workspace Boundary

Abstract General, complex geometry forms of RRR regional structures are often avoided due to the presence of inner boundaries within the workspace which tend to complicate robot guidance. Despite the added complexity, certain RRR geometries may have useful applications as they contain large workspace regions where four alternate configurations may be used to reach a given spatial location. Cusp points often appear on the workspace boundaries of general RRR regional structures, and determining their precise location may be useful for both design and guidance purposes. A twelfth degree polynomial equation in the outer joint variable is derived which defines the location of non-trivial cusps in the workspace. A new closed form workspace boundary equation is derived in the outer joint variable and x coordinate of the toroidal surface generated by rotation of the two outer revolutes. If the outer joint variable is incremented, a quadratic in x is formed at each step which enables a very efficient determination of the workspace boundaries while also providing the coordinates of the boundary on the toroidal surface.

Corneal astigmatism axis standard based on toroidal surface design

2018 ◽  
Author(s):  
Jiyan Zhang ◽  
Wenli Liu ◽  
Baoyu Hong ◽  
Lei Yang ◽  
Xiang Ding ◽  
...  
Keyword(s):  
Corneal Astigmatism ◽  
Surface Design ◽  
Toroidal Surface

scholarly journals Green's function for the Laplace–Beltrami operator on a toroidal surface

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120479 ◽  
Author(s):  
Christopher C. Green ◽  
Jonathan S. Marshall
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Vortex Dynamics ◽  
Stereographic Projection ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Toroidal Surface ◽  
Torus Surface ◽  
Dilogarithm Function ◽  
Prime Function

Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

THE BOSONIC SIGMA MODEL ON A TOROIDAL SURFACE

2004 ◽  
Vol 19 (16) ◽  
pp. 2713-2720
Author(s):  
D. G. C. McKEON
Keyword(s):  
Riemannian Manifold ◽  
Sigma Model ◽  
Three Dimensional ◽  
The Other ◽  
Target Space ◽  
Dimensional Euclidean Space ◽  
Nonlinear Sigma Model ◽  
Two Dimensional ◽  
Toroidal Surface ◽  
Spherical Basis

The nonlinear sigma model with a two-dimensional basis space and an n-dimensional target space is considered. Two different basis spaces are considered; the first is an 0(2)×0(2) subspace of the 0(2,2) projective space related to the Minkowski basis space, and the other is a toroidal space embedded into three-dimensional Euclidean space, characterized by radii R and r. The target space is taken to be an arbitrarily curved Riemannian manifold. One-loop dependence on the renormalization induced scale μ is shown in the toroidal basis space to be the same as in a flat or spherical basis space.

scholarly journals KINEMATICAL FEATURES OF FORMATION OF A FLANGE WITH MINIMUM TENSION OF A TRANSIENT TOROIDAL SURFACE

2018 ◽  
pp. 132-136
Author(s):  
M. I. Sidorenko
Keyword(s):  
Free Surface ◽  
Cylindrical Part ◽  
Contact Friction ◽  
Internal Cavity ◽  
Toroidal Surface ◽  
Plastic Forming ◽  
Plastic Shaping ◽  
Deformation Force ◽  
Outer Plane

The technology of plastic forming of wide flanges in tube billets with the predicted length of the transitional toroidal section between the outer plane of the flange and the internal cavity of the pipe is proposed. The procedure for calculating the length of this section is given. In order to eliminate the toroidal portion in the flange formed during the flanging of the pipe, it is proposed to perform its plastic shaping by depositing the cylindrical part of the workpiece. Equations for calculating the extent of the free surface on the toroidal part of the workpiece when it is shaped, depending on the coefficient of contact friction and the presence of a radial support of the flange are obtained. The variant of forming in the flange the toroidal section in the stamp with the compensation cavity is proposed. Equations for calculating the deformation force and the extent of the free surface are given.

An Object-Space Based Machining Simulation in Milling: Part 2 — By Toroidal Surface

2009 ◽  
Author(s):  
Eyyup Aras
Keyword(s):  
Linear Equations ◽  
Nonlinear Function ◽  
Intersection Point ◽  
Machining Process ◽  
End Mill ◽  
Root Finding ◽  
Toroidal Surface ◽  
Non Linear ◽  
Cartesian Coordinate ◽  
Tool Motion

This is the second part of a two-part paper presenting an efficient parametric approach for updating the in-process workpiece represented by the Z-map. With the Z-map representation, the machining process can be simulated by intersecting z-axis aligned vectors with cutter swept envelopes. In this paper the vector-envelope intersections are formulized for the toroidal section of a fillet-end mill which may be oriented arbitrarily in space. For a given tool motion a toroidal surface generates more than one envelope. In NC machining because the torus is considered as one of the constituent parts of a fillet-end mill, only some parts of the torus envelopes, called contact envelopes, can intersect with Z-map vectors. In this paper an analysis is developed for separating the contact-envelopes from the non-contact ones. When a fillet-end mill has an orientation along the vertical z-axis of the Cartesian coordinate system, which happens in 2 1/2 and 3-axis machining, the number of intersections between a Z-map vector and the swept envelope of a toroidal section of the fillet-end mill is maximum one. For finding this single intersection point one of the numerical root finding methods, i.e. bisection, can be applied to the nonlinear function obtained from vector-envelope intersections. On the other hand when a fillet-end mill has an arbitrary orientation, the number of intersections can be more than one and therefore the numerical root finding methods cannot be applied directly. Therefore for addressing those multiple intersections, a system of non-linear equations in several variables, obtained by intersecting a Z-map vector with the envelope surface of the toroidal section of a fillet-end mill, is transformed into a single variable non-linear function. Then developing a nonlinear root finding analysis which guarantees the root(s) in the given interval, those intersections are obtained.

scholarly journals Arrays of dark optical traps on a toroidal surface

2019 ◽  
Vol 99 (1) ◽  
Author(s):  
Jakub Bělín ◽  
V. E. Lembessis ◽  
A. Lyras ◽  
O. Aldossary ◽  
Johannes Courtial
Keyword(s):  
Optical Traps ◽  
Toroidal Surface

Five Axis Swept Profiles of Torus Like Cutters via Separation of Inner and Outer Characteristic Curves

2018 ◽  
Author(s):  
Eyyup Aras
Keyword(s):  
Tool Path ◽  
Rotation Axis ◽  
Linear Interpolation ◽  
Unit Vector ◽  
Industrial Applications ◽  
Body Motion ◽  
Moving Frames ◽  
Closed Form Solutions ◽  
Toroidal Surface ◽  
Five Axis

A broadly applicable formulation for identifying the swept profiles (SWP) generated by subsets of a toroidal surface is presented. While the problem of locating the entire SWP of a torus has been extensively addressed in the literature, this rarely addressed problem is of significance to NC machining with non-standard shape of milling tools. A torus, generated by revolving a circle about an axis coplanar with the circle, is made up of inner and outer parts of a tube. The common use of the torus is in a fillet-end mill which contains only the fourth quadrant of a cross section of the tube. However, in the industrial applications the different regions of the torus geometry appear. Especially we can see this on the profile cutters, such as the corner-rounding and concave-radius end mills. Also to the best of our knowledge, the interior of the torus-tube is either neglected or represented by B-spline curves in literature. In case of common milling tool surfaces such as sphere, cylinder and frustum there exists only one SWP in any instance of a tool movement. But, in case of the toroidal surface there exist two sophisticated SWPs and we need to consider only one of them in tool swept envelope generation. Therefore, considering the complexity of five-axis tool motions there is a need not only to distinguish the front from the rear of the cutter but also the exterior from the interior of a tube. This paper presents a methodology and algorithms for analytically formulating the SWP of any sub-set of the torus in five-axis tool motions. By introducing the rigid body motion theory, two moving frames along with a fixed frame are defined. Arbitrary poses of a tool between tool path locations are interpolated by a spherical linear interpolation (slerp) whose effect is a rotation with uniform angular velocity around a fixed rotation axis. For the problem of NC simulation, by using the envelope theory the closed-form solutions of swept profiles are formulated as two-unit vector functions.

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