geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a family of curves is a set of

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s, each of which is given by a function or parametrization in which one or more of the

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s 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 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when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.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 Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

of the curve: :

r(\theta) =

as the value of 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 Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...

neural networks A neural network is a group of interconnected units called neurons that send signals to one another. Neurons can be either Cell (biology), biological cells or signal pathways. While individual neurons are simple, many of them together in a netwo ...

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

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

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

External links

* {{DEFAULTSORT:Family Of Curves Algebraic geometry