Ruled surface
In geometry, a surface S in 3-dimensional Euclidean space is ruled (also called a scroll) if through every point of S, there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.
A ruled surface can be described as the set of points swept by a moving straight line.
For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle.
The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces.
The plane is the only surface which contains at least three distinct lines through each of its points (Fuchs & Tabachnikov 2007).
The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry.
In algebraic geometry, ruled surfaces are sometimes considered to be surfaces in affine or projective space over a field, but they are also sometimes considered as abstract algebraic surfaces without an embedding into affine or projective space, in which case "straight line" is understood to mean an affine or projective line.
A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines.
The directrix may collapse to a point (in case of a cone, see example below).
The ruled surface above may alternatively be described by with the second directrix
To go back to the first description starting with two non intersecting curves
However, the specific parametric representations of them also influence the shape of the ruled surface.
A right circular cylinder is given by the equation It can be parameterized as with A right circular cylinder is given by the equation It can be parameterized as with In this case one could have used the apex as the directrix, i.e. and as the line directions.
This shows that the directrix of a ruled surface may degenerate to a point.
The parametric representation has two horizontal circles as directrices.
For A hyperboloid of one sheet is a doubly ruled surface.
If the two directrices in (CD) are the lines one gets which is the hyperbolic paraboloid that interpolates the 4 points
For the example shown in the diagram: The hyperbolic paraboloid has the equation
The ruled surface with contains a Möbius strip.
[3] For the determination of the normal vector at a point one needs the partial derivatives of the representation
The linear dependency of three vectors can be checked using the determinant of these vectors: A smooth surface with zero Gaussian curvature is called developable into a plane, or just developable.
The determinant condition can be used to prove the following statement: The generators of any ruled surface coalesce with one family of its asymptotic lines.
For developable surfaces they also form one family of its lines of curvature.
[5] The determinant condition for developable surfaces is used to determine numerically developable connections between space curves (directrices).
[6] An impression of the usage of developable surfaces in Computer Aided Design (CAD) is given in Interactive design of developable surfaces.
[8] In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a straight line through any given point.
This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line through any point.
This is equivalent to saying that they are birational to the product of a curve and a projective line.
Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a fibration over a curve with fibers that are projective lines.
Doubly ruled surfaces are the inspiration for curved hyperboloid structures that can be built with a latticework of straight elements, namely: The RM-81 Agena rocket engine employed straight cooling channels that were laid out in a ruled surface to form the throat of the nozzle section.
