Family of curves

In geometry, a family of curves is a set of curves, each of which is given by a function or 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.

The Apollonian circles, two orthogonal families of circles.

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when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.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:

${\displaystyle r(\theta )={l \over 1+e\cos \theta }}$

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 of a set of curves and the caustic of a given curve.

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.

Generalizations

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

External links

• Weisstein, Eric W. "Family of Curves". MathWorld.