Family of curves

In general, the parameter(s) influence the shape of the curve in a way that is more complicated than a simple linear transformation.

Families of curves appear frequently in solutions of differential equations; when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.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: as the value of e changes, the appearance of the curve varies in a relatively complicated way.

In machine learning, neural networks are families of curves with parameters chosen by an optimization algorithm e.g. to minimize the value of a loss function on a given training dataset.