This article defines a property that makes sense for a surface embedded in , viz three-dimensional Euclidean space. The property is invariant under orthogonal transformations and scaling, i.e., under all similarity transformations.
View other such properties
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.
A ruled surface can be described by a parametric description of the form:
where and are functions that take scalar inputs and give three-dimensional vector outputs. Here, the parameter controls which line we are on, and the parameter describes the location of the point on the line. In other words, for every fixed value of , we get a fixed line described with a single parameter . The surface is the union of these lines.
We use the following terminology:
- The line for each fixed value of is termed a ruling for the surface.
- The function is termed the ruled surface directrix or the base curve. For any , describes the position of the line.
- The value describes a direction vector along the line, and the function is termed a director curve.
|Ruled surface||Equational/implicit description||Functions and in a possible parametric description||Comment|
|Euclidean plane||(the -plane)||is the vector with coordinates and is the vector .||The Euclidean plane is in fact a doubly ruled surface and also a minimal surface.|
|right circular cylinder (infinite version)||(the right circular cylinder with base circle the unit circle in the -plane and axis along the -axis).||is the vector with coordinates and is the vector .|
|circular hyperboloid of one sheet||This is in fact a doubly ruled surface|
|elliptic hyperboloid of one sheet|
|hyperbolic paraboloid||This is in fact a doubly ruled surface|
|helicoid||it is the only ruled minimal surface other than the plane.|