- Source: Conical spiral
In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor projection is a plane spiral. If the floor projection is a logarithmic spiral, it is called conchospiral (from conch).
Parametric representation
In the
x
{\displaystyle x}
-
y
{\displaystyle y}
-plane a spiral with parametric representation
x
=
r
(
φ
)
cos
φ
,
y
=
r
(
φ
)
sin
φ
{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }
a third coordinate
z
(
φ
)
{\displaystyle z(\varphi )}
can be added such that the space curve lies on the cone with equation
m
2
(
x
2
+
y
2
)
=
(
z
−
z
0
)
2
,
m
>
0
{\displaystyle \;m^{2}(x^{2}+y^{2})=(z-z_{0})^{2}\ ,\ m>0\;}
:
x
=
r
(
φ
)
cos
φ
,
y
=
r
(
φ
)
sin
φ
,
z
=
z
0
+
m
r
(
φ
)
.
{\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}
Such curves are called conical spirals. They were known to Pappos.
Parameter
m
{\displaystyle m}
is the slope of the cone's lines with respect to the
x
{\displaystyle x}
-
y
{\displaystyle y}
-plane.
A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone.
= Examples
=1) Starting with an archimedean spiral
r
(
φ
)
=
a
φ
{\displaystyle \;r(\varphi )=a\varphi \;}
gives the conical spiral (see diagram)
x
=
a
φ
cos
φ
,
y
=
a
φ
sin
φ
,
z
=
z
0
+
m
a
φ
,
φ
≥
0
.
{\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}
In this case the conical spiral can be seen as the intersection curve of the cone with a helicoid.
2) The second diagram shows a conical spiral with a Fermat's spiral
r
(
φ
)
=
±
a
φ
{\displaystyle \;r(\varphi )=\pm a{\sqrt {\varphi }}\;}
as floor plan.
3) The third example has a logarithmic spiral
r
(
φ
)
=
a
e
k
φ
{\displaystyle \;r(\varphi )=ae^{k\varphi }\;}
as floor plan. Its special feature is its constant slope (see below).
Introducing the abbreviation
K
=
e
k
{\displaystyle K=e^{k}}
gives the description:
r
(
φ
)
=
a
K
φ
{\displaystyle r(\varphi )=aK^{\varphi }}
.
4) Example 4 is based on a hyperbolic spiral
r
(
φ
)
=
a
/
φ
{\displaystyle \;r(\varphi )=a/\varphi \;}
. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for
φ
→
0
{\displaystyle \varphi \to 0}
.
Properties
The following investigation deals with conical spirals of the form
r
=
a
φ
n
{\displaystyle r=a\varphi ^{n}}
and
r
=
a
e
k
φ
{\displaystyle r=ae^{k\varphi }}
, respectively.
= Slope
=The slope at a point of a conical spiral is the slope of this point's tangent with respect to the
x
{\displaystyle x}
-
y
{\displaystyle y}
-plane. The corresponding angle is its slope angle (see diagram):
tan
β
=
z
′
(
x
′
)
2
+
(
y
′
)
2
=
m
r
′
(
r
′
)
2
+
r
2
.
{\displaystyle \tan \beta ={\frac {z'}{\sqrt {(x')^{2}+(y')^{2}}}}={\frac {mr'}{\sqrt {(r')^{2}+r^{2}}}}\ .}
A spiral with
r
=
a
φ
n
{\displaystyle r=a\varphi ^{n}}
gives:
tan
β
=
m
n
n
2
+
φ
2
.
{\displaystyle \tan \beta ={\frac {mn}{\sqrt {n^{2}+\varphi ^{2}}}}\ .}
For an archimedean spiral,
n
=
1
{\displaystyle n=1}
, and hence its slope is
tan
β
=
m
1
+
φ
2
.
{\displaystyle \ \tan \beta ={\tfrac {m}{\sqrt {1+\varphi ^{2}}}}\ .}
For a logarithmic spiral with
r
=
a
e
k
φ
{\displaystyle r=ae^{k\varphi }}
the slope is
tan
β
=
m
k
1
+
k
2
{\displaystyle \ \tan \beta ={\tfrac {mk}{\sqrt {1+k^{2}}}}\ }
(
constant!
{\displaystyle \color {red}{\text{ constant!}}}
).
Because of this property a conchospiral is called an equiangular conical spiral.
= Arclength
=The length of an arc of a conical spiral can be determined by
L
=
∫
φ
1
φ
2
(
x
′
)
2
+
(
y
′
)
2
+
(
z
′
)
2
d
φ
=
∫
φ
1
φ
2
(
1
+
m
2
)
(
r
′
)
2
+
r
2
d
φ
.
{\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(x')^{2}+(y')^{2}+(z')^{2}}}\,\mathrm {d} \varphi =\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {(1+m^{2})(r')^{2}+r^{2}}}\,\mathrm {d} \varphi \ .}
For an archimedean spiral the integral can be solved with help of a table of integrals, analogously to the planar case:
L
=
a
2
[
φ
(
1
+
m
2
)
+
φ
2
+
(
1
+
m
2
)
ln
(
φ
+
(
1
+
m
2
)
+
φ
2
)
]
φ
1
φ
2
.
{\displaystyle L={\frac {a}{2}}\left[\varphi {\sqrt {(1+m^{2})+\varphi ^{2}}}+(1+m^{2})\ln \left(\varphi +{\sqrt {(1+m^{2})+\varphi ^{2}}}\right)\right]_{\varphi _{1}}^{\varphi _{2}}\ .}
For a logarithmic spiral the integral can be solved easily:
L
=
(
1
+
m
2
)
k
2
+
1
k
(
r
(
φ
2
)
−
r
(
φ
1
)
)
.
{\displaystyle L={\frac {\sqrt {(1+m^{2})k^{2}+1}}{k}}(r{\big (}\varphi _{2})-r(\varphi _{1}){\big )}\ .}
In other cases elliptical integrals occur.
= Development
=For the development of a conical spiral the distance
ρ
(
φ
)
{\displaystyle \rho (\varphi )}
of a curve point
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
to the cone's apex
(
0
,
0
,
z
0
)
{\displaystyle (0,0,z_{0})}
and the relation between the angle
φ
{\displaystyle \varphi }
and the corresponding angle
ψ
{\displaystyle \psi }
of the development have to be determined:
ρ
=
x
2
+
y
2
+
(
z
−
z
0
)
2
=
1
+
m
2
r
,
{\displaystyle \rho ={\sqrt {x^{2}+y^{2}+(z-z_{0})^{2}}}={\sqrt {1+m^{2}}}\;r\ ,}
φ
=
1
+
m
2
ψ
.
{\displaystyle \varphi ={\sqrt {1+m^{2}}}\psi \ .}
Hence the polar representation of the developed conical spiral is:
ρ
(
ψ
)
=
1
+
m
2
r
(
1
+
m
2
ψ
)
{\displaystyle \rho (\psi )={\sqrt {1+m^{2}}}\;r({\sqrt {1+m^{2}}}\psi )}
In case of
r
=
a
φ
n
{\displaystyle r=a\varphi ^{n}}
the polar representation of the developed curve is
ρ
=
a
1
+
m
2
n
+
1
ψ
n
,
{\displaystyle \rho =a{\sqrt {1+m^{2}}}^{\,n+1}\psi ^{n},}
which describes a spiral of the same type.
If the floor plan of a conical spiral is an archimedean spiral than its development is an archimedean spiral.
In case of a hyperbolic spiral (
n
=
−
1
{\displaystyle n=-1}
) the development is congruent to the floor plan spiral.
In case of a logarithmic spiral
r
=
a
e
k
φ
{\displaystyle r=ae^{k\varphi }}
the development is a logarithmic spiral:
ρ
=
a
1
+
m
2
e
k
1
+
m
2
ψ
.
{\displaystyle \rho =a{\sqrt {1+m^{2}}}\;e^{k{\sqrt {1+m^{2}}}\psi }\ .}
= Tangent trace
=The collection of intersection points of the tangents of a conical spiral with the
x
{\displaystyle x}
-
y
{\displaystyle y}
-plane (plane through the cone's apex) is called its tangent trace.
For the conical spiral
(
r
cos
φ
,
r
sin
φ
,
m
r
)
{\displaystyle (r\cos \varphi ,r\sin \varphi ,mr)}
the tangent vector is
(
r
′
cos
φ
−
r
sin
φ
,
r
′
sin
φ
+
r
cos
φ
,
m
r
′
)
T
{\displaystyle (r'\cos \varphi -r\sin \varphi ,r'\sin \varphi +r\cos \varphi ,mr')^{T}}
and the tangent:
x
(
t
)
=
r
cos
φ
+
t
(
r
′
cos
φ
−
r
sin
φ
)
,
{\displaystyle x(t)=r\cos \varphi +t(r'\cos \varphi -r\sin \varphi )\ ,}
y
(
t
)
=
r
sin
φ
+
t
(
r
′
sin
φ
+
r
cos
φ
)
,
{\displaystyle y(t)=r\sin \varphi +t(r'\sin \varphi +r\cos \varphi )\ ,}
z
(
t
)
=
m
r
+
t
m
r
′
.
{\displaystyle z(t)=mr+tmr'\ .}
The intersection point with the
x
{\displaystyle x}
-
y
{\displaystyle y}
-plane has parameter
t
=
−
r
/
r
′
{\displaystyle t=-r/r'}
and the intersection point is
(
r
2
r
′
sin
φ
,
−
r
2
r
′
cos
φ
,
0
)
.
{\displaystyle \left({\frac {r^{2}}{r'}}\sin \varphi ,-{\frac {r^{2}}{r'}}\cos \varphi ,0\right)\ .}
r
=
a
φ
n
{\displaystyle r=a\varphi ^{n}}
gives
r
2
r
′
=
a
n
φ
n
+
1
{\displaystyle \ {\tfrac {r^{2}}{r'}}={\tfrac {a}{n}}\varphi ^{n+1}\ }
and the tangent trace is a spiral. In the case
n
=
−
1
{\displaystyle n=-1}
(hyperbolic spiral) the tangent trace degenerates to a circle with radius
a
{\displaystyle a}
(see diagram). For
r
=
a
e
k
φ
{\displaystyle r=ae^{k\varphi }}
one has
r
2
r
′
=
r
k
{\displaystyle \ {\tfrac {r^{2}}{r'}}={\tfrac {r}{k}}\ }
and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.
References
External links
Jamnitzer-Galerie: 3D-Spiralen. Archived 2021-07-02 at the Wayback Machine.
Weisstein, Eric W. "Conical Spiral". MathWorld.
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