Ruled surface

${\displaystyle \ x^{2}+y^{2}=a^{2}\ }$:

${\displaystyle \mathbf {x} (u,v)=(a\cos u,a\sin u,v)^{T}}$

${\displaystyle ={\color {red}(a\cos u,a\sin u,0)^{T}}\;+\;v\;{\color {blue}(0,0,1)^{T}}}$

${\displaystyle =(1-v)\;{\color {red}(a\cos u,a\sin u,0)^{T}}\;+\;v\;{\color {green}(a\cos u,a\sin u,1)^{T}}\ .}$

with

${\displaystyle \mathbf {c} (u)=(a\cos u,a\sin u,0)^{T}\ ,\ \mathbf {r} (u)=(0,0,1)^{T}\ ,\ \mathbf {d} (u)=(a\cos u,a\sin u,1)^{T}\ .}$

${\displaystyle \ x^{2}+y^{2}=z^{2}\ }$:

${\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,1)^{T}\;+\;v\;(\cos u,\sin u,1)^{T}}$

${\displaystyle =(1-v)\;(\cos u,\sin u,1)^{T}\;+\;v\;(2\cos u,2\sin u,2)^{T}.}$

with ${\displaystyle \quad \mathbf {c} (u)=(\cos u,\sin u,1)^{T}\;=\;\mathbf {r} (u)\ ,\quad \mathbf {d} (u)=(2\cos u,2\sin u,2)^{T}\ .}$
In this case one could have used the apex as the directrix, i.e.: ${\displaystyle \ \mathbf {c} (u)=(0,0,0)^{T}\ }$ and ${\displaystyle \ \mathbf {r} (u)=(\cos u,\sin u,1)^{T}\ }$ as the line directions.

For any cone one can choose the apex as the directrix. This case shows: The directrix of a ruled surface may degenerate to a point.

helicoid

${\displaystyle \mathbf {x} (u,v)=\;(v\cos u,v\sin u,ku)^{T}\;}$

${\displaystyle =\;(0,0,ku)^{T}\;+\;v\;(\cos u,\sin u,0)^{T}\ }$

${\displaystyle =\;(1-v)\;(0,0,ku)^{T}\;+\;v\;(\cos u,\sin u,ku)^{T}\ .}$

The directrix ${\displaystyle \ \mathbf {c} (u)=(0,0,ku)^{T}\;}$ is the z-axis, the line directions are ${\displaystyle \;\mathbf {r} (u)=\ (\cos u,\sin u,0)^{T}\;}$ and the second directrix ${\displaystyle \ \mathbf {d} (u)=(\cos u,\sin u,ku)^{T}\ }$ is a helix.

The helicoid is a special case of the ruled generalized helicoids.

hyperboloid of one sheet for ${\displaystyle \varphi =63^{\circ }}$

The parametric representation

${\displaystyle \mathbf {x} (u,v)=(1-v)\;(\cos(u-\varphi ),\sin(u-\varphi ),-1)^{T}\;+\;v\;(\cos(u+\varphi ),\sin(u+\varphi ),1)^{T}}$

has two horizontal circles as directrices. The additional parameter ${\displaystyle \varphi }$ allows to vary the parametric representations of the circles. For

${\displaystyle \varphi =0\ }$ one gets the cylinder ${\displaystyle x^{2}+y^{2}=1}$, for

${\displaystyle \varphi =\pi /2\ }$ one gets the cone ${\displaystyle x^{2}+y^{2}=z^{2}}$ and for

${\displaystyle 0<\varphi <\pi /2\ }$ one gets a hyperboloid of one sheet with equation ${\displaystyle \ {\tfrac {x^{2}+y^{2}}{a^{2}}}-{\tfrac {z^{2}}{c^{2}}}=1\ }$ and the semi axes ${\displaystyle \ a=\cos \varphi \;,\;c=\cot \varphi }$.

A hyperboloid of one sheet is a doubly ruled surface.

Hyperbolic paraboloid

If the two directrices in (CD) are the lines

${\displaystyle \mathbf {c} (u)=(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2},\quad \mathbf {d} (u)=(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}}$

one gets

${\displaystyle \mathbf {x} (u,v)=(1-v){\big (}(1-u)\mathbf {a} _{1}+u\mathbf {a} _{2}{\big )}\ +\ v{\big (}(1-u)\mathbf {b} _{1}+u\mathbf {b} _{2}{\big )}\ }$,

which is the hyperbolic paraboloid that interpolates the 4 points ${\displaystyle \ \mathbf {a} _{1},\;\mathbf {a} _{2},\;\mathbf {b} _{1},\;\mathbf {b} _{2}\ }$ bilinearly.[1]

Obviously the ruled surface is a doubly ruled surface, because any point lies on two lines of the surface.

For the example shown in the diagram:

${\displaystyle \ \mathbf {a} _{1}=(0,0,0)^{T},\;\mathbf {a} _{2}=(1,0,0)^{T},\;\mathbf {b} _{1}=(0,1,0)^{T},\;\mathbf {b} _{2}=(1,1,1)^{T}\ }$.

The hyperbolic paraboloid has the equation ${\displaystyle z=xy}$.

Möbius strip

The ruled surface

${\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\;\mathbf {r} (u)}$

with

${\displaystyle \mathbf {c} (u)=(\cos 2u,\sin 2u,0)^{T}\ }$ (circle as directrix),

${\displaystyle \mathbf {r} (u)=(\cos u\cos 2u,\cos u\sin 2u,\sin u)^{T}\ ,\quad 0\leq u<\pi \ ,}$

contains a Möbius strip.

The diagram shows the Möbius strip for ${\displaystyle -0.3\leq v\leq 0.3}$.

A simple calculation shows ${\displaystyle \det(\mathbf {\dot {c}} (0)\;,\;\mathbf {\dot {r}} (0)\;,\;\mathbf {r} (0))\;\neq \;0\ }$ (see next section). Hence the given realization of a Möbius strip is not developable. But there exist developable Möbius strips.[2]

• Conoid
• Catalan surface
• Developable rollers (oloid, sphericon)