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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Different Finsler metrics and their geodesics resulting from selection H([alpha], [beta]) and E(x, y) arbitrary functions are discussed in the next section for family of curves that has two-parameter Weibull distribution.
Family of curves is divided into several groups according to [[gamma].sub.[DELTA]], [epsilon] or [[gamma].sub.[DELTA]], [[gamma].sub.c].
The modulus of a family of curves. For a family of curves [GAMMA] in the plane, we use the notation M([GAMMA]) for its modulus [17].
Formula 3 predicts a barrel-shaped family of curves, and statistically, this implies that the interaction is concentrated in the linear-by-quadratic interaction components (Anderson & Farkas, 1975).
For compressors with inlet guide vanes, compressor map is represented by a new family of curves that do not depend on suction conditions either.
The family of curves corresponding to different area ratios evidently collapsed to a single straight line by plotting the quantity [C.sub.b]/[([A.sub.b]/[A.sub.c]).sup.n] as a function of ([Q.sub.b]/[Q.sub.c]).
We shall see that the solution to the two questions stated above is essentially based on the idea of considering the values of a function along various curves from a given family of curves.
Let us remind that all secantoptics of a curve C for a fixed [beta] [member of] [0, [pi]), fixed [gamma] [member of] [0, [pi] - [beta]) and various [alpha] [member of] ([beta] + [gamma], [pi]) form two parameters family of curves [F.sub.[beta], [gamma]] ([alpha], t) (see [8]).
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.

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