Family of Surfaces

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Family of Surfaces

a set of surfaces that are dependent in a continuous manner on one or more parameters. Analytically, a family of surfaces can be defined by the one equation

(1) F(x, y, z, C₁, C₂,…,C_n) = 0

or by the three equations

x = Φ(u, v, C₁C₂,…, C_n)

(2) y = ψ(u, v, C₁, C₂,…, C_n)

z = X(u, v, C₁, C₂,…, C_n)

If the parameters C_i are assigned particular numerical values, then equations (1) and (2) become the equations of one surface in the family of surfaces. The functions F, Φ, ψ, and X are usually required to have continuous partial derivatives with respect to all the arguments.

The concept of an envelope plays an important role in the study of one- and two-parameter families of surfaces. The envelope of a one-parameter family of planes is called a developable surface (seeRULED SURFACE).

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