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Conical spiral
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<h2 id="parametric-representation">Parametric representation</h2> <p>In the x {\displaystyle x} - y {\displaystyle y} -plane a spiral with parametric representation </p> x = r ( φ ) cos φ , y = r ( φ ) sin φ {\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi } <p>a third coordinate z ( φ ) {\displaystyle z(\varphi )} can be added such that the space curve lies on the <a href="/facts/Cone/2c9GIzYD">cone</a> 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\;} : </p> <ul><li> 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 )}\ .} </li></ul> <p>Such curves are called conical spirals.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> They were known to <a href="/facts/Pappos/UABLMu0m">Pappos</a>. </p><p>Parameter m {\displaystyle m} is the slope of the cone's lines with respect to the x {\displaystyle x} - y {\displaystyle y} -plane. </p><p>A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone. </p> <h3>Examples</h3> 1) Starting with an <i>archimedean spiral</i> 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 <a href="/facts/Helicoid/ekrSN1Pm">helicoid</a>. 2) The second diagram shows a conical spiral with a <i><a href="/facts/Fermat%2527s_spiral/FtRIVkXm">Fermat's spiral</a></i> r ( φ ) = ± a φ {\displaystyle \;r(\varphi )=\pm a{\sqrt {\varphi }}\;} as floor plan. 3) The third example has a <i>logarithmic spiral</i> r ( φ ) = a e k φ {\displaystyle \;r(\varphi )=ae^{k\varphi }\;} as floor plan. Its special feature is its constant <i>slope</i> (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 <i><a href="/facts/Hyperbolic_spiral/Ic4oft2k">hyperbolic spiral</a></i> r ( φ ) = a / φ {\displaystyle \;r(\varphi )=a/\varphi \;} . Such a spiral has an <i>asymptote</i> (black line), which is the floor plan of a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> (purple). The conical spiral approaches the hyperbola for φ → 0 {\displaystyle \varphi \to 0} . <h2 id="properties">Properties</h2> <p>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. </p> <h3>Slope</h3> <p>The <i>slope</i> 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 <i>slope angle</i> (see diagram): </p> 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}}}}\ .} <p>A spiral with r = a φ n {\displaystyle r=a\varphi ^{n}} gives: </p> <ul><li> tan β = m n n 2 + φ 2 . {\displaystyle \tan \beta ={\frac {mn}{\sqrt {n^{2}+\varphi ^{2}}}}\ .} </li></ul> <p>For an <i>archimedean</i> 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}}}}\ .} </p> <ul><li>For a <i>logarithmic</i> 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!}}} ).</li></ul> <p>Because of this property a conchospiral is called an <i>equiangular</i> conical spiral. </p> <h3>Arclength</h3> <p>The <a href="/facts/Length/EwFyAqGH">length</a> of an arc of a conical spiral can be determined by </p> 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 \ .} <p>For an <i>archimedean</i> spiral the integral can be solved with help of a <a href="/facts/Table_of_integrals/FqW2wdwr">table of integrals</a>, analogously to the planar case: </p> 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}}\ .} <p>For a <i>logarithmic</i> spiral the integral can be solved easily: </p> 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 )}\ .} <p>In other cases <a href="/facts/Elliptical_integral/q8EGYzQ3">elliptical integrals</a> occur. </p> <h3>Development</h3> <p>For the <a href="/facts/Development_(differential_geometry)/k8esDhnE">development</a> of a conical spiral<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> 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: </p> 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 \ .} <p>Hence the polar representation of the developed conical spiral is: </p> <ul><li> ρ ( ψ ) = 1 + m 2 r ( 1 + m 2 ψ ) {\displaystyle \rho (\psi )={\sqrt {1+m^{2}}}\;r({\sqrt {1+m^{2}}}\psi )} </li></ul> <p>In case of r = a φ n {\displaystyle r=a\varphi ^{n}} the polar representation of the developed curve is </p> ρ = a 1 + m 2 n + 1 ψ n , {\displaystyle \rho =a{\sqrt {1+m^{2}}}^{\,n+1}\psi ^{n},} <p>which describes a spiral of the same type. </p> <ul><li>If the floor plan of a conical spiral is an <i>archimedean</i> spiral than its development is an archimedean spiral.</li></ul> In case of a <i>hyperbolic</i> spiral ( n = − 1 {\displaystyle n=-1} ) the development is congruent to the floor plan spiral. <p>In case of a <i>logarithmic</i> spiral r = a e k φ {\displaystyle r=ae^{k\varphi }} the development is a logarithmic spiral: </p> 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 }\ .} <h3>Tangent trace</h3> <p>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 <i>tangent trace</i>. </p><p>For the conical spiral </p> ( r cos φ , r sin φ , m r ) {\displaystyle (r\cos \varphi ,r\sin \varphi ,mr)} <p>the tangent vector is </p> ( 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}} <p>and the tangent: </p> 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'\ .} <p>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 </p> <ul><li> ( 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)\ .} </li></ul> <p> 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 <i>circle</i> 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 <a href="/facts/Self-similarity/9Nvo7edY">self-similarity</a> of a logarithmic spiral. </p> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Wenzel_Jamnitzer/7cn5yAs8">Jamnitzer</a>-Galerie: <i><a href="http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html">3D-Spiralen.</a> <a href="https://web.archive.org/web/20210702004420/http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html">Archived</a> 2021-07-02 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></i>.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ConicalSpiral.html">"Conical Spiral"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <ul><li></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Conical helix". MATHCURVE.COM. Retrieved 2022-03-03. <a href="https://mathcurve.com/courbes3d.gb/heliceconic/heliceconic.shtml" target="_blank">https://mathcurve.com/courbes3d.gb/heliceconic/heliceconic.shtml</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Siegmund Günther, Anton Edler von Braunmühl, Heinrich Wieleitner: Geschichte der mathematik. G. J. Göschen, 1921, p. 92. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Theodor Schmid: Darstellende Geometrie. Band 2, Vereinigung wissenschaftlichen Verleger, 1921, p. 229. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>