SOLID—TORUS

The placement information of the SOLID TORUS consists of the local origin and the local z-axis. 

ENTITY SWEEP = CLASS( LINEAR SWEEP, ROTATIONAL SWEEP )  

The entity SWEEP consists of a class containing the entities LINEAR SWEEP and ROTATIONAL SWEEP. 

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The first stage in this geometrical construction is as follows. Consider a solid-torus Hi G where Hi = x S, is a two-dimensional disk and is a circle. Let us embed a similar solid-torus II2 into Hi so that it intersects every disk up = constant in Hi, where (p is an angular variable, in two disjoint disks so that II2 makes two revolutions along Si without selfintersections as illustrated in Fig. 7.3.1. It is also assumed that II2 is about twice as long as that of Hi, and four times thinner. At the second stage, we embed a torus II3 into II2 in the same way as above, so that there are now four intersections of II3 with each disk ip = constant, two inside each previous pair of intersections. 

Fig. 12.2.3. The structure of intersection of the unstable manifold of a saddle-node periodic orbit L with a solid-torus-like cross-section Sq in the case m = 2. A trace of the intersection is a doubly-twisted curve 1. Consequently, it has at least two intersections with each level

The SOLID TORUS (see Figure 22 on page 125) is a construct primitive. It is created by revolving a circular disc lying in the xz-plane about the z-axis. The circular disc is assumed to be disjoint and coplanar to the z-axis. The shape description consists of radius large i.e. the distance from the axis to the center of the defining disc and of "radius small i.e. 

Platonic solids have been studied since antiquity and in a multiplicity of artistic and scientific contexts. More generally, polyhedral maps are ubiquitous in chemistry and crystallography. Their properties have been studied since Kepler. In the present book we are going to study classes of maps on the sphere or the torus and make a catalog of properties that would be helpful and useful to mathematicians and researchers in natural sciences. 

Figure 9.14 Nested invariant tori and a surface of section. Each dotted circle corresponds to a quasiperiodic trajectory at different E. When the radius of the circle collapses to zero, the trajectory is periodic. The dotted trajectory shown looping about the outermost invariant torus, will eventually fill the surface of that torus and generate a solid circle on the surface of section (from Child, 1991).

Analogously, the second theorem of Pappus states that the volume of the solid generated by the revolution of a figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. For a torus, the area of the cross section equals nr. Therefore, the volume of a torus is given by... 

A solid primitive is prepared for combination with other solid primitives or a more complex solid model under construction. It is created in its final position or repositioned after creation somewhere in the model space. Values of its dimensions are set and the solid primitive is ready for one of the element combination operations. The shapes of primitives are predefined for the modeling system or defined by engineers at application of the modeling system. Users apply one of the available solid generation rules starting from contours, sections, and curves as input entities. Primitives with predefined shape are called canonical. They are the cuboid, wedge, cylinder, cone, sphere, and torus (Figure 4-11). Inclusion of shapes other than canonical as predefined shapes is rare because application oriented shape definitions are better to define as form features. 

A. and Mukasyan, A. (2007) Solid Flame Combustion, 2nd edn. Torus Press,... 

Consequently the higher the curvature of a surface, the shorter the collapse time. If a bubble is not close to an interface, the opposite parts Amin move towards the interior faster than parts Amax/ leading to the formation of a torus. If the bubble is relatively close to a solid boundary, only the part Amin far from the wall can move, and a single jet forms and invades the bubble. A similar situation has been... 

FIGURE 9.16. Time-lapse photographs of a liquid marble descending a slope at a velocity of 1 m/s. The camera is tilted to the same inclination as the solid and the movement takes place toward the left (the bar indicates a distance of 1 cm). Because of the centrifugal force, the sphere deforms into a peanut (a) or a torus (b). (From P. Aussillous and D. Quere, in Nature, 4 1, p. 924 (2001). (c) 2001 McMillan Magazines, Ltd. Reproduced by permission.)... 

For bulk three-dimensional solids, g is equivalent to the number of cuts required to transform a solid structure into a structure topologically equivalent to a sphere (for instance, g = 0 for a polygonal sphere such as or C70, and g = 1 for a torus). Suppose, further, that the object is formed of polygons having different (0 number of sides. The total number of faces (F) is then... 

Rapidform XOR3 Yes (Solid + Surface) Plane, Cylinder, Cone, Sphere, Torus, Box Yes (Excellent) Yes (Fair)... 

Figure 12. Examples of allowable B REP topologies These are topologies of solid model representations for a cylinder and a torus.

In the case of a PLANAR SURFACE, ORIENTATION is. T. if the normal points from solid to void. In cylinders, cones, spheres and torus surfaces, ORIENTATION is. T. if the solid is on the inside, i.e. on the side of the axis, center and center of the minor circle respectively. 

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