- Source: Conoid
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 plane. Hence all rulings are perpendicular to the axis.
Because of (1) any conoid is a Catalan surface and can be represented parametrically by
x
(
u
,
v
)
=
c
(
u
)
+
v
r
(
u
)
{\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)\ }
Any curve x(u0,v) with fixed parameter u = u0 is a ruling, c(u) describes the directrix and the vectors r(u) are all parallel to the directrix plane. The planarity of the vectors r(u) can be represented by
det
(
r
,
r
˙
,
r
¨
)
=
0
{\displaystyle \det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0}
.
If the directrix is a circle, the conoid is called a circular conoid.
The term conoid was already used by Archimedes in his treatise On Conoids and Spheroides.
Examples
= Right circular conoid
=The parametric representation
x
(
u
,
v
)
=
(
cos
u
,
sin
u
,
0
)
+
v
(
0
,
−
sin
u
,
z
0
)
,
0
≤
u
<
2
π
,
v
∈
R
{\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,0)+v(0,-\sin u,z_{0})\ ,\ 0\leq u<2\pi ,v\in \mathbb {R} }
describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line
(
x
,
0
,
z
0
)
x
∈
R
.
{\displaystyle (x,0,z_{0})\ x\in \mathbb {R} \ .}
Special features:
The intersection with a horizontal plane is an ellipse.
(
1
−
x
2
)
(
z
−
z
0
)
2
−
y
2
z
0
2
=
0
{\displaystyle (1-x^{2})(z-z_{0})^{2}-y^{2}z_{0}^{2}=0}
is an implicit representation. Hence the right circular conoid is a surface of degree 4.
Kepler's rule gives for a right circular conoid with radius
r
{\displaystyle r}
and height
h
{\displaystyle h}
the exact volume:
V
=
π
2
r
2
h
{\displaystyle V={\tfrac {\pi }{2}}r^{2}h}
.
The implicit representation is fulfilled by the points of the line
(
x
,
0
,
z
0
)
{\displaystyle (x,0,z_{0})}
, too. For these points there exist no tangent planes. Such points are called singular.
= Parabolic conoid
=The parametric representation
x
(
u
,
v
)
=
(
1
,
u
,
−
u
2
)
+
v
(
−
1
,
0
,
u
2
)
{\displaystyle \mathbf {x} (u,v)=\left(1,u,-u^{2}\right)+v\left(-1,0,u^{2}\right)}
=
(
1
−
v
,
u
,
−
(
1
−
v
)
u
2
)
,
u
,
v
∈
R
,
{\displaystyle =\left(1-v,u,-(1-v)u^{2}\right)\ ,u,v\in \mathbb {R} \ ,}
describes a parabolic conoid with the equation
z
=
−
x
y
2
{\displaystyle z=-xy^{2}}
. The conoid has a parabola as directrix, the y-axis as axis and a plane parallel to the x-z-plane as directrix plane. It is used by architects as roof surface (s. below).
The parabolic conoid has no singular points.
= Further examples
=hyperbolic paraboloid
Plücker conoid
Whitney Umbrella
helicoid
Applications
= Mathematics
=There are a lot of conoids with singular points, which are investigated in algebraic geometry.
= Architecture
=Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid (s. parabolic conoid).
External links
mathworld: Plücker conoid
"Conoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
References
A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, FL:CRC Press, 2006. [1] (ISBN 978-1-58488-448-4)
Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE [2] (ISBN 978-0-8176-4074-3)
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Conoid
- Clavicle
- Conoid ligament
- Plücker's conoid
- Right conoid
- On Conoids and Spheroids
- List of surfaces
- Fan vault
- Apicomplexan life cycle
- Whitney umbrella
