Toroidal Surfaces

The axis, centre and radii are essential for the geometric definition 

The reference point defines the parameterisation. The reference point must not lie on the central axis and should ideally be approximately at the centre of one of the minor circular sections of the torus. 

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In a toroidal traction drive, toroidal input and output disks face one another, separated by a number of rollers that contact the toroid surfaces. The rollers are mounted so that they can he tilted to vaiy the radius from the centerline of the disks where they contact the toroids, and therefore determine the input/output speed ratio of the rotating disks. A substantial axial force must be applied to the disks to prevent the rollers from slipping on the disk surfaces. To avoid excessive losses when the torque transmitted is low, this force needs to he modulated in proportion to the torque transmitted. 

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

If the quotient o>/a>0 is irrational, the path across the toroidal surface will return to a different point on the completion of each cycle. Eventually the trajectory will pass over every point on the surface of the torus without ever forming a closed loop. This is quasi-periodicity , and an example is shown in Fig. 13.11. The corresponding concentration histories do not necessarily give complex waveforms, as can be seen from the figure. However, the period of the oscillations is neither simply that of the natural cycle nor just that of the forcing term, but involves both. 

Early experience also showed that the induced plasma current in a tokamak generates a magnetic field that loops die minor axis nf Ihe torus. The field lines form helices along the toroidal surface the plasma must cross the lines to escape. It does so through the cumulative action of many random displacements caused by interparticle collisions, tin effect diffusing across the field lines and out of the system). Thermal energy is transported by much the same process. 

Parabolic focal conics are a special case of generic focal conic defects, which are composed of layers curved to form toroidal surfaces called Dupin cyclides (see Fig. 10-31). Each such structure contains a pair of disclination lines—one an ellipse and the other... 

T. Smith, G. E. Schneider, and M. M. Yovanovich, Numerical Study of Conduction Heat Transfer from Toroidal Surfaces Into an Infinite Domain, AIAA-92-2941, AIAA 27th Thermophysics Conference, Nashville, TN, 1992. 

Figure 15. A Canesian system applied to Figure 14 in a way that simplifies representation of the toroidal surface. The unit of length is the distance between the centers of two hexagons that share a common edge. The choices of a 120° (rather than 60°) angle between axes, and of a left-to-right vector as positive are adopted as standard conventions. In this example the coordinate positions of repeating A" hexagons at (a.c) = (9,0), and (b,d) = (4,1), abbreviated to "(9-4-1) suffice to define the parallelogram, and hence the toroidal surface.

Of the infinite hierarchy of finite closed surfaces beyond those homeomorphic to a sphere, a candidate for the next most important case is that of the toroidal surface S with xiS) = 0. Typical topohedral embeddings on a torus may be obtained from a rectangular fragment cut from the graphite lattice, say as in Figure 14. First the... 

Figure 11 Two different ways of embedding the toroidal graph into a toroidal surface...

Where a more intense spot focus of fairly monochromatic X-radiation is required than is available in either a Franks or a mirror-monochromator camera, an Elliott toroid camera can be used [4]. This camera uses a gold-coated mirror distributed on a toroidal surface, as shown in Fig. 11, to bring an anulus of X-radiation to a focus. 

Nevertheless, if the manipulator encumbrance increases the workspace volume will also increase. But sometimes the workspace volume may vanish even if L doesn t, since the workspace ring volume can degenerate into a toroidal surface for particular values of the link parameters, as it has been stressed for the workspace ring geometry of three-revolute chains, [11]. Furthermore, the minimum of the workspace volume can be the null value, but it is not related to the minimum encumbrance since it depends on the size of the degenerated toroidal surface. In this last case the two-revolute manipulator will be the optimal solution since its toroidal workspace can satisfy the same number of conditions for the workspace boundary surface as the case of three-revolute manipulators, [12], and probably it also occurs in the case of n-revolute manipulators. 

Fig.2. Ring workspace generation as an envelope of toroidal surfaces.

Particularly, one representative toroidal surface of the envelope is drawn and the cross-section contour of the ring workspace boundary is represented together with a cross-section contour of an enveloping torus. 

CYLINDRICAL SURFACE, CONICAL SURFACE, TOROIDAL SURFACE, PLANAR SURFACE, SURFACE OF REVOLUTION(DIM), SURFACE OF TRANSLATION(DIM), B SPLINE SURFACE ) ... 

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