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Ruled surface generated by two Bézier curves as directrices (red, green) 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. Formally, a ruled surface is a surface in is described by a parametric representation of the form
In geometry a conoid (from Greek κωνος 'cone' and -ειδης 'similar') is a ruled surface, whose rulings (lines) fulfill the additional conditions: (1) All rulings are parallel to a plane, the directrix plane. (2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix ...
Simple examples. A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
A hyperboloid of one sheet is a doubly ruled surface: it can be generated by either of two families of straight lines. Four images of hyperboloid towers. The Shukhov Tower in Polibino , the world's first hyperboloid structure , a water tower by Vladimir Shukhov at the All-Russian Exposition in Nizhny Novgorod , Russia
They are divided into minimal surfaces, ruled surfaces, non-orientable surfaces, quadrics, pseudospherical surfaces, algebraic surfaces, and other types of surfaces. Minimal surfaces [ edit ]
Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally. Essential types of generalized helicoids are ruled generalized helicoids. Their profile curves are lines and the surfaces are ruled surfaces. circular generalized helicoids. Their profile curves are circles.
Labs surface, a certain septic with 99 nodes; Endrass surface, a certain surface of degree 8 with 168 nodes; Sarti surface, a certain surface of degree 12 with 600 nodes; Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue ...
The helicoid is also a ruled surface (and a right conoid), meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it. Indeed, Catalan proved in 1842 that the helicoid and the plane were the only ruled minimal surfaces. [1] [2]