Family of curves

Parabolic cluster bundles in and tufts in ${\displaystyle f_{a}(x)=ax^{2}}$
${\displaystyle (0,0)}$ ${\displaystyle a}$

A family of curves , also a family of functions , a family of functions or a parameter function , is a set of different curves whose mapping rules differ in at least one parameter . Special cases are the tuft , a one-parameter group, and the bundle , a group with a point common to all functions.

definition

The family is a set of points on a curve, curves on a surface or surfaces in space , each of which is described by an equation or a system of equations with variable parameters.

According to another definition , a family of curves results from the graph of a function, in which a free parameter of the relevant function is varied in a parametric representation .

Dynamic geometry systems are particularly suitable for visualizing functional groups .

special cases

• If all the diagrams of the family of functions are straight lines , one speaks of a family of straight lines.
  • If the individual straight lines also run parallel, they are called a set of parallels .
  • If all the straight lines involved intersect at one point, it is a straight line bundle .
  • If all the straight lines involved both intersect at a point and lie in a plane , it is a straight line bundle .

• If all curves of the family are parabolas , one speaks of a parabola family.

Examples

• all curves of the family of curves belonging to the function run parallel to the axis (straight line). The parameter of this family of curves is . ${\displaystyle \!\ f_{p}(x)=p}$ ${\displaystyle x}$ ${\displaystyle p}$
• all curves of the family belonging to the function are parabolas through the coordinate origin (see illustration). The parameter is . ${\displaystyle f_{a}(x)=ax^{2},\,a\neq 0}$ ${\displaystyle a}$
• all curves of the family belonging to the relation are concentric circles . The parameter is here . ${\displaystyle \!\ r^{2}=x^{2}+y^{2}\ \Leftrightarrow {}\ r={\sqrt {x^{2}+y^{2}}}}$ ${\displaystyle r}$

literature

• Set of curves In: Schülerduden - Mathematics II . Bibliographisches Institut & FA Brockhaus, 2004, ISBN 3-411-04275-3 , pp. 241–242
• Mark Yes. Vygodskij: Higher mathematics at hand: definitions, theorems, examples . Springer 2013, ISBN 9783322901132 , p. 696

Web links

• Eric W. Weisstein : Family of Curves . In: MathWorld (English).
• Group of functions on .onlinemathe.de
• Family of curves on www.wh54.de