geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a family of curves is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

. Sets of curves given by an implicit relation may also represent families of curves. Families of curves appear frequently in solutions of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...

s; when an additive

constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...

is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.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 In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...

can be represented using a single

polar equation In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...

with one parameter, the eccentricity 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 An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...

of a set of curves and the

caustic Caustic most commonly refers to: * Causticity, a property of various corrosive substances ** Sodium hydroxide, sometimes called ''caustic soda'' ** Potassium hydroxide, sometimes called ''caustic potash'' ** Calcium oxide, sometimes called ''caust ...

of a given curve.

Generalizations

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, an algebraic generalization is given by the notion of a

linear system of divisors In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the fo ...

External links

* {{DEFAULTSORT:Family Of Curves Algebraic geometry