Generalized Solids of Revolution

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3. Generalized Conics and Thier Properties

Generalized conic as defined in formula 1 is simple closed curve. For non-negative weights of foci with Euclidean and Manhattan metrics, it is always convex (also for some other metrics, see [Cech02] for details). The simplest generalized conics for Euclidean metrics is circle. It has only one focus and shape is not affected with its weight. Two equally weighted foci are used to create an ellipse. Releasing restriction to equal weights, Oval of Descartes [Camp82] is created. More foci give more complex curve. Generalized conic is more curved near foci, parts of curve far from foci are similar to circle arcs.

Manhattan metrics causes angled shape of generalized conics. The simplest shape is a square. Two foci can create hexagon or octagon, depending on their relative position in coordinate system.

3.1 Reducing Number of Foci

Evaluating of generalized conics is computationally complex task. The complexity rises with number of foci. Sometimes loosing some precision in exchange of speed is acceptable tradeoff. Influence of focus to shape of generalized conic depends on several parameters. It is position of focus (relative to other foci and also distance to generalized conic is important) and its weight relative to other foci. When focus has zero weight, it does not have any influence on shape. Therefore removing such focus is all right and introduces no error.

Joining few foci with the same metrics results into simpler definition of generalized conic. Not to introduce much error, group of foci has to be chosen carefully. The main criterion is their relative proximity in contrast to distance from curve. When radius of group is significantly small in comparison with distance from curve, they can be joined without causing too much error. Resulting focus should be placed to mass center of foci group. This heuristics takes only weights and relative positions between foci in consideration. More precise heuristics (which also includes relative position of foci to curve) can be made, but computational costs are much higher.

3.2 Degenrated Generalized Conic

When intersection point has minimal value of all values in the slice, generalized conic degenerates from closed curve into different object. When using Euclidean metrics, only single point or straight line can occur. For straight line, all foci have to be collinear. Other configurations create only one point when generalized conic degenerates. In case when Manhattan metrics is used, also rectangular area can occur. Foci set have to be symmetrical by two lines parallel to both main coordinate system axes.

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