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Ruled surface

This article defines a property that makes sense for a surface embedded in \R^3, viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.
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Definition

A ruled surface is a surface in Euclidean space \R^3 with the property that for any point on the surface, there is a line through that point lying on the surface.

Because, by definition, a surface is connected, an alternative and equivalent definition of ruled surface is as a surface that can be swept by moving a line in space.

Parametric description

A ruled surface can be described by a parametric description of the form:

\! \mathbf{x}(u,v) = \mathbf{b}(u) + v \delta(u)

where \mathbf{b} and \delta are functions that take scalar inputs and give three-dimensional vector outputs. Here, the parameter u controls which line we are on, and the parameter v describes the location of the point on the line. In other words, for every fixed value of u, we get a fixed line described with a single parameter v. The surface is the union of these lines.

We use the following terminology:

  • The line for each fixed value of u is termed a ruling for the surface.
  • The function \mathbf{b} is termed the ruled surface directrix or the base curve. For any u, \mathbf{b}(u) describes the position of the line.
  • The value \delta(u) describes a direction vector along the line, and the function \delta is termed a director curve.

Examples

Ruled surface Equational/implicit description Functions \mathbf{b} and \delta in a possible parametric description Comment
Euclidean plane z = 0 (the xy-plane) \mathbf{b}(u) is the vector with coordinates (u,0,0) and \delta(u) is the vector (0,1,0). The Euclidean plane is in fact a doubly ruled surface and also a minimal surface.
right circular cylinder (infinite version) x^2 + y^2 = 1 (the right circular cylinder with base circle the unit circle in the xy-plane and axis along the z-axis). \mathbf{b}(u) is the vector with coordinates (\cos u, \sin u,0) and \delta(u) is the vector (0,0,1).
circular hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1 This is in fact a doubly ruled surface
elliptic hyperboloid of one sheet \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1
hyperbolic paraboloid z = \frac{y^2}{b^2} - \frac{x^2}{a^2} This is in fact a doubly ruled surface
helicoid y = x \tan(z/c) it is the only ruled minimal surface other than the plane.

Relation with other properties