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{{Unreferenced|date=December 2009}}

[[File:Apollonian circles.svg|thumb|The [[Apollonian circles]], two orthogonal families of circles.]]
A '''family of curves''' is a [[Set (mathematics)|set]] of [[curve]]s, each of which is given by a [[function (mathematics)|function]] or [[parametric equation|parametrization]] in which one or more of the parameters is variable. In general, the parameter(s) influence the shape of the curve in a way that is more complicated than a simple [[linear transformation]]. Sets of curves given by an implicit relation may also represent families of curves.

Families of curves appear frequently in solutions of [[differential equation]]s; when an additive [[constant of integration]] is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation. 

Families of curves may also arise in other areas. For example, all non-degenerate [[conic sections]] can be represented using a single polar equation with one parameter, the eccentricity of the curve: 

:<math>r(\theta) = {e \over 1+e \cos \theta}</math>

as the value of e changes, the appearance of the curve varies in a relatively complicated way.

== Applications ==
Families of curves may arise in various topics in geometry, including the [[envelope (mathematics)|envelope]] of a set of curves and the [[caustic (optics)|caustic]] of a given curve.

== Generalizations ==
In [[algebraic geometry]], an algebraic generalization is given by the notion of a [[linear system of divisors]].

{{DEFAULTSORT:Family Of Curves}}
[[Category:Algebraic geometry]]