Developable surface

<h2 id="particulars">Particulars</h2>
<p>The developable surfaces which can be realized in <a href="/facts/Three-dimensional_space/KAqKWUbS">three-dimensional space</a> include:
</p>
<ul><li><a href="/facts/Cylinder_(geometry)/0wQPPr3k">Cylinders</a> and, more generally, the "generalized" cylinder; its <a href="/facts/Cross_section_(geometry)/g5R1c6v2">cross-section</a> may be any <a href="/facts/Smooth_function/mffL4ch2">smooth</a> <a href="/facts/Curve/xBQ0pTBW">curve</a></li>
<li><a href="/facts/Cone_(geometry)/2c9GIzYD">Cones</a> and, more generally, <a href="/facts/Conical_surface/n6h4xmLu">conical surfaces</a>; away from the <a href="/facts/Apex_(geometry)/SrMxmHJk">apex</a></li>
<li>The <a href="/facts/Oloid/rlvCc8hw">oloid</a> and the <a href="/facts/Sphericon/C2uhkCoL">sphericon</a> are members of a special family of <a href="/facts/Solid_(geometry)/tHH3JFoq">solids</a> that develop their entire surface when <a href="/facts/Rolling/rfmy2YIl">rolling</a> down a flat plane.</li>
<li>Planes (trivially); which may be viewed as a cylinder whose cross-section is a <a href="/facts/Line_(mathematics)/gJqsr4Gj">line</a></li>
<li><a href="/facts/Tangent_developable/e24EwwRp">Tangent developable</a> surfaces; which are constructed by extending the <a href="/facts/Tangent/sF0jH3EI">tangent</a> lines of a spatial curve.</li>
<li>The <a href="/facts/Torus/J9SkaV1W">torus</a> has a metric under which it is developable, which can be embedded into three-dimensional space by the <a href="/facts/Nash_embedding_theorem/a7eLrRLM">Nash embedding theorem</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> and has a simple representation in four dimensions as the Cartesian product of two circles: see <a href="/facts/Clifford_torus/jG2YNVjS">Clifford torus</a>.</li></ul>
<p>Formally, in mathematics, a developable surface is a surface with zero <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a>.  One consequence of this is that all "developable" surfaces embedded in 3D-space are <a href="/facts/Ruled_surface/uAGM9b1X">ruled surfaces</a> (though <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloids</a> are examples of ruled surfaces which are not developable).  Because of this, many developable surfaces can be <a href="/facts/Scientific_visualization/p0yPc1X5">visualised</a> as the surface formed by moving a straight line in space.  For example, a cone is formed by keeping one <a href="/facts/Point_(geometry)/6YPAvX2a">end-point</a> of a line fixed whilst moving the other end-point in a <a href="/facts/Circle/9fy5ggKn">circle</a>.
</p>
<h3>Application</h3>

<p>Developable surfaces have several practical applications.  
</p><p>Many <a href="/facts/Cartography/8YdDAAeX">cartographic</a> <a href="/facts/Map_projections/XYxklzIu">projections</a> involve projecting the <a href="/facts/Earth/HLLmDfj0">Earth</a> to a developable surface and then "unrolling" the surface into a region on the plane.  
</p><p>Since developable surfaces may be constructed by bending a flat sheet, they are also important in <a href="/facts/Manufacturing/3qW7bcdy">manufacturing</a> objects from sheet metal, <a href="/facts/Corrugated_fiberboard/VjDFR6II">cardboard</a>, and <a href="/facts/Plywood/eM5WFvER">plywood</a>. An <a href="/facts/Industry_(economics)/nlB2hvkF">industry</a> which uses developed surfaces extensively is shipbuilding.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p><a href="/facts/Developable_mechanism/qzcowj2C">Developable Mechanisms</a> are mechanisms that conform to a developable surface and can exhibit motion (deploy) off the surface.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 
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<h2 id="non-developable-surface">Non-developable surface</h2>
<p>Most smooth surfaces (and most surfaces in general) are not developable surfaces. Non-developable surfaces are variously referred to as having "double curvature", "doubly curved", "compound curvature", "non-zero Gaussian curvature", etc.
</p><p>Some of the most often-used non-developable surfaces are:
</p>
<ul><li><a href="/facts/Sphere/4Ivb1cTj">Spheres</a> are not developable surfaces under any <a href="/facts/Metric_tensor/tbqek1gJ">metric</a> as they cannot be unrolled onto a plane.</li>
<li>The <a href="/facts/Helicoid/ekrSN1Pm">helicoid</a> is a ruled surface – but unlike the ruled surfaces mentioned above, it is not a developable surface.</li>
<li>The <a href="/facts/Hyperbolic_paraboloid/AUdRV2QH">hyperbolic paraboloid</a> and the <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloid</a> are slightly different doubly ruled surfaces – but unlike the ruled surfaces mentioned above, neither one is a developable surface.</li></ul>
<h3>Applications of non-developable surfaces</h3>
<p>Many <a href="/facts/Gridshell/csk1P7n5">gridshells</a> and <a href="/facts/Tensile_structure/MUlT2xgs">tensile structures</a> and similar constructions gain strength by using (any) doubly curved form.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Development_(differential_geometry)/k8esDhnE">Development (differential geometry)</a></li>
<li><a href="/facts/Developable_roller/TBsJ2pl7">Developable roller</a></li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Developable surfaces.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/DevelopableSurface.html">"Developable Surface"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="http://www.rhino3.de/design/modeling/developable/">Examples of developable surfaces on the Rhino3DE website</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Chalfant, Julie S.; Maekawa, Takashi (September 1998). "Design for Manufacturing Using B-Spline Developable Surfaces". Journal of Ship Research. 42 (3): 207–215. doi:10.5957/jsr.1998.42.3.207. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), New York: Chelsea, pp. 341–342, ISBN 978-0-8284-1087-8 {{citation}}: ISBN / Date incompatibility (help) <a href="978-0-8284-1087-8" target="_blank">978-0-8284-1087-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration", Proceedings of the National Academy of Sciences, 109 (19): 7218–7223, doi:10.1073/pnas.1118478109, PMC 3358891, PMID 22523238. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Nolan, T. J. (1970), Computer-Aided Design of Developable Hull Surfaces, Ann Arbor: University Microfilms International <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Developable Mechanisms | About Developable Mechanisms". compliantmechanisms. Retrieved 2019-02-14. <a href="https://www.compliantmechanisms.byu.edu/about-developable-mechanisms" target="_blank">https://www.compliantmechanisms.byu.edu/about-developable-mechanisms</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Howell, Larry L.; Lang, Robert J.; Magleby, Spencer P.; Zimmerman, Trent K.; Nelson, Todd G. (2019-02-13). "Developable mechanisms on developable surfaces". Science Robotics. 4 (27): eaau5171. doi:10.1126/scirobotics.aau5171. ISSN 2470-9476. PMID 33137737. <a href="https://doi.org/10.1126%2Fscirobotics.aau5171" target="_blank">https://doi.org/10.1126%2Fscirobotics.aau5171</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Developable surface open-in-new

Developable surface