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,…, C_n = 0

where C_1,C₂,…, C_n 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.