Family of Curves


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family of curves

[‚fam·lē əv ′kərvz]
(mathematics)
A set of curves whose equations can be obtained by varying a finite number of parameters in a particular general equation.

Family of Curves

 

a set of curves that depend in a continuous manner on one or more parameters. In the plane, for example, a family of curves can be specified by an equation of the form

(*) F(x, y, C1, C2,…, Cn = 0

where C1,C2,…, Cn are parameters. If the parameters are assigned particular numerical values, then equation (*) defines a single curve. A family of curves on a surface is defined in a similar way. In this case, the Cartesian coordinates x and y in equation (*) are replaced by the intrinsic coordinates u and v on the surface.

It is usually assumed that F is a continuous function with respect to the set of its arguments and has a continuous partial derivative with respect to each argument. The concept of an envelope plays an important role in the study of one-parameter families in the plane or on an arbitrary surface.

References in periodicals archive ?
Family of curves is divided into several groups according to [[gamma].
We also remark that the case of symmetric hexagons can be dealt with by the Schwarz-Christoffel transformation and we relate its exterior modulus to a symmetry property of the modulus of a family of curves.
In particular, when their judgments are plotted in a factorial graph, the curves display the predicted barrel-shaped family of curves (see Figure 2).
The family of curves corresponding to different area ratios successfully collapsed to a single curve by plotting [C.
Let [GAMMA](a) be a family of curves (parametrized curves), passing through a.
Now consider the family of curves that represent the effect of changing system pressures on pump performance.
It should be noted that Equation 4 does not apply to the family of curves for disks, but only for the more spherical particles as noted above.
Neill developed a family of curves for estimating critical velocities for sediments for varying flow depths with grain sizes ranging from 0.
Zariski begins by describing that equisingularity and the moduli space, along with the Puiseux expansion, then covers equisingularity invariants, parametrizations, the topology of moduli space, including an explicit determination of the rare cases when the space is compact, and gives examples applications of deformation theory to the moduli problem, including the determination of the generic component for a particular family of curves.
Three methods were used to develop a family of curves representing stable channel dimensions under a variety of sediment supply conditions: "method one," which, over a hydrologic period, yields the net accumulation of sediment within a design reach; "method two," which can be used to estimate amounts of down-cutting or deposition expected for short- and long-term evaluations; and the "SAM method," which partitions channel roughness between the channel bed and banks, and uses resistance and sediment transport formulae to compute stable channel dimension curves.
The analysis of the parameters associated with a family of curves can give insight into the behavior of the family and can be used within the context of a mathematical model to determine a policy for the implementation of the model in a real-world setting.
1) Forster and Sober's solution to the curve-fitting problem provides us with a way to choose one curve from among a set of curves where each curve is from a different family of curves ([1994], pp.

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