Set of curves from a function with variable parameter(s)
The Apollonian circles, two orthogonal families of circles.
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.
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when an additive constant of integration is introduced, it will usually be manipulated algebraically until it no longer represents a simple linear transformation.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 geometry, a familyofcurves is a set ofcurves, each of which is given by a function or parametrization in which one or more of the parameters is variable...
geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a familyofcurves; the dimension of the linear system corresponds...
psychology, ecology, etc. Rational curves are subdivided according to the degree of the polynomial. Line Plane curvesof degree 2 are known as conics or...
superposition of two perpendicular oscillations in x and y directions of different angular frequency (a and b). The resulting familyofcurves was investigated...
Given a familyofcurves, assumed to be differentiable, an isocline for that family is formed by the set of points at which some member of the family attains...
were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli...
mathematical study of changes in the qualitative or topological structure of a given familyofcurves, such as the integral curvesof a familyof vector fields...
Real Women Have Curves is a 2002 American comedy-drama film directed by Patricia Cardoso, based on the play of the same name by Josefina López, who co-authored...
is a gallery ofcurves used in mathematics, by Wikipedia page. See also list ofcurves. Line Circle Ellipse Parabola Hyperbola Cubic curve Cubic polynomial...
A dragon curve is any member of a familyof self-similar fractal curves, which can be approximated by recursive methods such as Lindenmayer systems. The...
trade in forms of debt, such as loans and bonds, use yield curves to determine their value. Shifts in the shape and slope of the yield curve are thought...
the curve. The evolute of an involute is the original curve. It is generalized by the roulette familyofcurves. That is, the involutes of a curve are...
of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves....
individual firms' supply curves are added horizontally). Economists distinguish between short-run and long-run supply curve. Short run refers to a time...
a family of elliptic curves degenerating to a rational curve with a cusp. One of the most important properties of stable curves is the fact that they...
short descriptions of redirect targets Differentiable curve – Study ofcurves from a differential point of view Differential geometry of surfaces Geodesic...
Budget share Engel curves describe how the proportion of household income spent on a good varies with income. Alternatively, Engel curves can also describe...
enough to include all non-singular cubic curves; see § Elliptic curves over a general field below.) An elliptic curve is an abelian variety – that is, it has...
curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy...
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