Definition:Family of Surfaces
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Definition
A family of surfaces is a set of surfaces which are described with a common equation, in such a way that all such surfaces can be generated by varying one or more parameters.
Parameter
The parameters of a family of surfaces $\FF$ is a set of real numbers which, when varied, generate all the elements of $\FF$.
Examples
Classification
One-Parameter Family
Consider the implicit function $\map f {x, y, z, c} = 0$ in the Cartesian $3$-space where $c$ is a constant.
For each value of $c$, we have that $\map f {x, y, z, c} = 0$ defines a relation between $x$, $y$ and $z$ which can be graphed in cartesian $3$-space.
Thus, each value of $c$ defines a particular surface.
The complete set of all these surfaces for each value of $c$ is called a one-parameter family of surfaces.
Also see
- Results about families of surfaces can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): family: 2.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): family: 2.