Gebruiker:Martinvl/Fibonaccireeks: Verskil tussen weergawes
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==Spirale in drie dimensies==
[[File:Spiral-cone-arch-s.svg|thumb|upright=0.8|Conic spiral with Archimedean spiral as floor plan]]
=== Conical spirals ===
{{mainarticle|conical spiral}}
If in the <math>x</math>-<math>y</math>-plane a spiral with parametric representation
:<math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi</math>
is given, then there can be added a third coordinate <math>z(\varphi)</math>, such that the now space curve lies on the [[cone]] with equation <math>\;m(x^2+y^2)=(z-z_0)^2\ ,\ m>0\;</math>:
* <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color{red}{z=z_0 + mr(\varphi)} \ .</math>
Spirals based on this procedure are called '''conical spirals'''.
;Example
Starting with an ''archimedean spiral'' <math>\;r(\varphi)=a\varphi\;</math> one gets the conical spiral (see diagram)
:<math>x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .</math>
[[File:Kugel-spirale-1-2.svg|thumb|upright=1.2|Spherical spiral with <math>c=8</math>]]
=== Spherical spirals ===
If one represents a sphere of radius <math>r</math> by:
: <math>
\begin{array}{cll}
x &=& r \cdot \sin \theta \cdot \cos \varphi \\
y &=& r \cdot \sin \theta \cdot \sin \varphi \\
z &=& r \cdot \cos \theta
\end{array}
</math>
and sets the linear dependency <math>\; \varphi=c\theta , \; c> 2\; ,</math> for the angle coordinates, one gets a [[spherical curve]] called '''spherical spiral'''<ref>Kuno Fladt: ''Analytische Geometrie spezieller Flächen und Raumkurven'', Springer-Verlag, 2013, {{ISBN|3322853659}}, 9783322853653, S. 132</ref> with the parametric representation (with <math>c</math> equal to twice the number of turns)
* <math>
\begin{array}{cll}
x &=& r \cdot \sin \theta \cdot \cos{\color{red} c\theta} \\
y &=& r \cdot \sin \theta \cdot \sin {\color{red}c\theta} \\
z &=& r \cdot \cos \theta\qquad \qquad 0\le\theta\le \pi \ .
\end{array}
</math>
Spherical spirals were known to Pappus, too.
Remark: a [[rhumb line]] is ''not'' a spherical spiral in this sense.
<gallery>
KUGSPI-5 Archimedische Kugelspirale.gif|Spherical spiral
KUGSPI-9_Loxodrome.gif|Loxodrome
</gallery>
A [[rhumb line]] (also known as a loxodrome or "spherical spiral") is the curve on a sphere traced by a ship with constant [[bearing (navigation)|bearing]] (e.g., travelling from one [[Geographical pole|pole]] to the other while keeping a fixed [[angle]] with respect to the [[Meridian (geography)|meridians]]). The loxodrome has an [[Infinity|infinite]] number of [[Orbital revolution|revolution]]s, with the separation between them decreasing as the curve approaches either of the poles, unlike an [[Archimedean spiral]] which maintains uniform line-spacing regardless of radius.
==Verwysings==
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