Definition:Family of Surfaces/One-Parameter
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Definition
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.
Parameter
The value $c$ is the parameter of $F$.
Examples
Concentric Spheres
The equation:
- $x^2 + y^2 + z^2 = r^2$
is a one-parameter family of concentric spheres whose centers are at the origin of a Cartesian $3$-space and whose radii are the values of the parameter $r$.
Also see
- Results about one-parameter families of surfaces can be found here.