Focal surface

<h2 id="special-cases">Special cases</h2>
<ol><li>The focal surface of a <a href="/facts/Sphere/jywYLNM2">sphere</a> consists of a single point, its center.</li>
<li>One part of the focal surface of a <a href="/facts/Surface_of_revolution/yzpQMdzx">surface of revolution</a> consists of the axis of rotation.</li>
<li>The focal surface of a <a href="/facts/Torus/J9SkaV1W">Torus</a> consists of the directrix circle and the axis of rotation.</li>
<li>The focal surface of a <a href="/facts/Dupin_cyclide/2yBu2Ylu">Dupin cyclide</a> consists of a pair of <a href="/facts/Focal_conics/W4p0ttQx">focal conics</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The Dupin cyclides are the only surfaces, whose focal surfaces degenerate into two curves.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li>
<li>One part of the focal surface of a <a href="/facts/Channel_surface/QGo3IH6u">channel surface</a> degenerates to its directrix.</li>
<li>Two <a href="/facts/Confocal_quadrics/sNANGYBF">confocal quadrics</a> (for example an ellipsoid and a hyperboloid of one sheet) can be considered as focal surfaces of a surface.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Focus_(optics)/QuRKfhg6">Focus (optics)</a></li>
<li><a href="/facts/Evolute/TmNQdwFY">Evolute</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Chandru, V.; Dutta, D.; Hoffmann, C.M. (1988), <i>On the Geometry of Dupin Cyclides</i>, Purdue University e-Pubs.</li></ul>

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

<ol>
<li id="fn:1"><p>David Hilbert, Stephan Cohn-Vossen: Anschauliche Geometrie, Springer-Verlag, 2011, ISBN 3642199488, p. 197. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Morris Kline: Mathematical Thought From Ancient to Modern Times, Band 2, Oxford University Press, 1990,ISBN 0199840423 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Porteous, Ian R. (2001), Geometric Differentiation, Cambridge University Press, pp. 198–213, ISBN 0-521-00264-8 <a href="0-521-00264-8" target="_blank">0-521-00264-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Georg Glaeser, Hellmuth Stachel, Boris Odehnal: The Universe of Conics, Springer, 2016, ISBN 3662454505, p. 147. <a href="/wiki/Georg_Glaeser" target="_blank">/wiki/Georg_Glaeser</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>D. Hilbert, S. Cohn-Vossen:Geometry and the Imagination, Chelsea Publishing Company, 1952, p. 218. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Hilbert Cohn-Vossen p. 197. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Focal surface open-in-new

Focal surface