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<title>Conchoïde</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Conchoïde</span></h1>

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<p>Une <b>conchoïde</b> <span title="Alphabet phonétique international" class="API" style="font-family:'DejaVu Sans','Doulos SIL','Lucida Grande','Segoe UI','Arial Unicode MS','Adobe Pi Std','Lucida Sans Unicode','Chrysanthi Unicode',Code2000,Gentium,GentiumAlt,'TITUS Cyberbit Basic','Bitstream Vera Sans','Bitstream Cyberbit','Hiragino Kaku Gothic Pro','Matrix Unicode',sans-serif;"><a href="Alphabet_phon%C3%A9tique_international" title="Alphabet phonétique international"><span class="nowrap">[kɔ̃kɔid]</span></a></span> (du latin <i>concha</i>, coquille) est une courbe obtenue à partir d'un <a href="Point_fixe" title="Point fixe">point fixe</a> O, d'une autre courbe, et d'une distance <i>d</i>. O est alors le pôle de la conchoïde et <i>d</i> son module. Pour chaque droite passant par O qui coupe la courbe donnée en un point P, on trace les points N et Q de la droite situés à une distance <i>d</i> de P. La conchoïde est le <a href="Lieu_g%C3%A9om%C3%A9trique" title="Lieu géométrique">lieu géométrique</a> des points N et Q lorsque P parcourt la courbe donnée.

</p><p>En coordonnées polaires de pôle O, si la courbe donnée a pour équation polaire <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\alpha (\theta )}">

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<annotation encoding="application/x-tex">{\displaystyle r=\alpha (\theta )}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8bb2824cb37566ffb65c7f9589e2da6b042e1437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.534ex; height:2.843ex;" alt="{\displaystyle r=\alpha (\theta )}" loading="lazy"></span> alors la conchoïde aura pour équation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=\alpha (\theta )\pm d}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle r=\alpha (\theta )\pm d}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3e96e02b7b9a8dc90db030f9f4a0960380431fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.591ex; height:2.843ex;" alt="{\displaystyle r=\alpha (\theta )\pm d}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Conchoïde_de_droite_:_la_conchoïde_de_Nicomède"><span id="Concho.C3.AFde_de_droite_:_la_concho.C3.AFde_de_Nicom.C3.A8de"></span>Conchoïde de droite : la conchoïde de Nicomède</h2></div>

<p>La conchoïde la plus simple est la conchoïde de droite, inventée par <a href="Nicom%C3%A8de_(math%C3%A9maticien)" title="Nicomède (mathématicien)">Nicomède</a>, mathématicien grec du <abbr class="abbr" title="2ᵉ siècle"><span class="romain">II</span><sup style="font-size:72%">e</sup></abbr> siècle <abbr class="abbr nowrap" title="avant Jésus-Christ">av. J.-C.</abbr> . Il fut le premier à réaliser une construction mécanique d'une <a href="Courbe_plane" title="Courbe plane">courbe plane</a> (autre que le cercle).

</p><p>C'est la courbe d'<a href="%C3%89quation_polaire" title="Équation polaire">équation polaire</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {a}{\cos \theta }}+d}">

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {a}{\cos \theta }}+d}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78c2a770bc80159e0bc50c777ef1faeb2a2916d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.781ex; height:4.843ex;" alt="{\displaystyle \rho ={\frac {a}{\cos \theta }}+d}" loading="lazy"></span>, où <i>a</i> est la distance du pôle à la directrice (<i>a</i> = OH).

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<div class="mw-heading mw-heading3"><h3 id="Trisection_d'un_angle"><span id="Trisection_d.27un_angle"></span>Trisection d'un angle</h3></div>

<p>Les conchoïdes de Nicomède sont des trisectrices, c'est-à-dire qu'elles permettent de diviser en trois angles égaux un angle. À chaque angle φ à trisecter, correspond une conchoïde différente, avec <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \varphi ={\frac {a}{b}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \sin \varphi ={\frac {a}{b}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88f72334ff25fe78e75bee7df9291296c9cdd25d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.927ex; height:4.843ex;" alt="{\displaystyle \sin \varphi ={\frac {a}{b}}}" loading="lazy"></span>.

</p><p>Afin de réaliser une trisection, construire un <a href="Triangle" title="Triangle">triangle</a> OHI rectangle en H, tel que l'angle φ à trisecter soit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OIH}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {OIH}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaaecde2cefc9ad78e261e6c2d3944cdef124090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.009ex; height:3.176ex;" alt="{\displaystyle {\widehat {OIH}}}" loading="lazy"></span>. Ensuite, construire la conchoïde de la droite (IH) de pôle O et de module OI.

</p><p>On a alors, avec <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {IOH}}=\phi }">

<mover>

<mrow>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {IOH}}=\phi }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4807615599dd0b4c2e7e783c6abc46a8458f144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.493ex; height:3.509ex;" alt="{\displaystyle {\widehat {IOH}}=\phi }" loading="lazy"></span> : <i>a</i> = <i>OH</i> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=OI={\frac {a}{\cos \phi }}}">

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=OI={\frac {a}{\cos \phi }}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25ab70d6010635562e6c3f359169c6f77929ad78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.078ex; height:5.176ex;" alt="{\displaystyle d=OI={\frac {a}{\cos \phi }}}" loading="lazy"></span>. La conchoïde a donc pour équation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {a}{\cos \theta }}+{\frac {a}{\cos \phi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {a}{\cos \theta }}+{\frac {a}{\cos \phi }}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e598697783e3d21501f21ef382062df9e00e2c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.285ex; height:5.176ex;" alt="{\displaystyle \rho ={\frac {a}{\cos \theta }}+{\frac {a}{\cos \phi }}}" loading="lazy"></span>.

</p><p>L'intersection de la courbe avec le cercle de centre I passant par O permet de déterminer deux points M et N, et grâce aux propriétés fondamentales de la conchoïde, on démontre que l'angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {NIP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {NIP}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88c2343688c13e9f521ad262568512e686582fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.981ex; height:3.176ex;" alt="{\displaystyle {\widehat {NIP}}}" loading="lazy"></span> trisecte l'angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OIH}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {OIH}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaaecde2cefc9ad78e261e6c2d3944cdef124090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.009ex; height:3.176ex;" alt="{\displaystyle {\widehat {OIH}}}" loading="lazy"></span> ou encore que l'angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {NIP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {NIP}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88c2343688c13e9f521ad262568512e686582fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.981ex; height:3.176ex;" alt="{\displaystyle {\widehat {NIP}}}" loading="lazy"></span> est le tiers de l'angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OIH}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {OIH}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaaecde2cefc9ad78e261e6c2d3944cdef124090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.009ex; height:3.176ex;" alt="{\displaystyle {\widehat {OIH}}}" loading="lazy"></span>.

</p>

<p><span><strong>Démonstration</strong> — </span>

Dans cette démonstration, on notera α la mesure de l'angle <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {NIP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {NIP}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88c2343688c13e9f521ad262568512e686582fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.981ex; height:3.176ex;" alt="{\displaystyle {\widehat {NIP}}}" loading="lazy"></span>. On sait, d'après les propriétés de la conchoïde, que IN = NP = <i>d</i>. Le triangle INP est donc isocèle avec <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OPH}}={\widehat {NIP}}=\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {OPH}}={\widehat {NIP}}=\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a295acc809f0a69330e363d2ee411733ebb2395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.248ex; height:3.176ex;" alt="{\displaystyle {\widehat {OPH}}={\widehat {NIP}}=\alpha }" loading="lazy"></span> .

</p><p>De plus, en considérant le cercle de centre N passant par P, on utilise le <a href="Th%C3%A9or%C3%A8me_de_l'angle_inscrit_et_de_l'angle_au_centre" title="Théorème de l'angle inscrit et de l'angle au centre">théorème de l'angle inscrit et de l'angle au centre</a> afin de montrer que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {ONI}}=2{\widehat {OPH}}=2\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {ONI}}=2{\widehat {OPH}}=2\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42b7a6e4d0320a07523a331c11b84c692eada844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.6ex; height:3.176ex;" alt="{\displaystyle {\widehat {ONI}}=2{\widehat {OPH}}=2\alpha }" loading="lazy"></span> . Le triangle NOI est isocèle, ce qui permet de déduire que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {ION}}={\widehat {ONI}}=2\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {ION}}={\widehat {ONI}}=2\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/714e2cf739a004c51bb9e0d467bf7f455e3153be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.864ex; height:3.176ex;" alt="{\displaystyle {\widehat {ION}}={\widehat {ONI}}=2\alpha }" loading="lazy"></span>. Les angles <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {yOP}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {yOP}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e21ea885e1beb11f8663ac2535f9d3a00c1422d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.03ex; width:4.726ex; height:3.509ex;" alt="{\displaystyle {\widehat {yOP}}}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OPH}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {OPH}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9630646f854fbf214cff92aaf8e05f9f2a72d5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.582ex; height:3.176ex;" alt="{\displaystyle {\widehat {OPH}}}" loading="lazy"></span> étant alternes-internes, on en déduit que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {yOP}}=\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {yOP}}=\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2d4be7226b6148b58a4e3e1ee5ebee39b5c99a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.282ex; height:3.509ex;" alt="{\displaystyle {\widehat {yOP}}=\alpha }" loading="lazy"></span>.

</p><p>Or, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {yOI}}={\widehat {yOP}}+{\widehat {ION}}=3\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {yOI}}={\widehat {yOP}}+{\widehat {ION}}=3\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07ebe80bb22d10d888d8745903fec1b67bbf5bdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.493ex; height:3.509ex;" alt="{\displaystyle {\widehat {yOI}}={\widehat {yOP}}+{\widehat {ION}}=3\alpha }" loading="lazy"></span>. Par conséquent, les angles <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {yOI}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {yOI}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/778f88f057c8bb42b3900572bf25932e7cbbe518.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.101ex; height:3.343ex;" alt="{\displaystyle {\widehat {yOI}}}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OIH}}}">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {OIH}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eaaecde2cefc9ad78e261e6c2d3944cdef124090.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.009ex; height:3.176ex;" alt="{\displaystyle {\widehat {OIH}}}" loading="lazy"></span> étant alternes-internes, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {OIH}}=3\alpha }">

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {OIH}}=3\alpha }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5d9f8d3c5dc93edb3442aa7c64f6cd06f04fe9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.757ex; height:3.176ex;" alt="{\displaystyle {\widehat {OIH}}=3\alpha }" loading="lazy"></span> .

</p>

</div><div class="clear" style="clear:both;"></div>

</div>

<div class="mw-heading mw-heading3"><h3 id="Duplication_d'un_cube"><span id="Duplication_d.27un_cube"></span>Duplication d'un cube</h3></div>

<p>Les conchoïdes de Nicomède sont également des duplicatrices<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Construction_de_la_tangente_et_de_la_normale">Construction de la tangente et de la normale</h3></div>

<p>Dans son livre <a href="https://fr.wikisource.org/wiki/Page:%C5%92uvres_de_Descartes,_%C3%A9d._Cousin,_tome_V.djvu/363" class="extiw external" title="s:Page:Œuvres de Descartes, éd. Cousin, tome V.djvu/363">La Géométrie</a>, <a href="Th%C3%A9ories_scientifiques_de_Descartes#Géométrie" title="Théories scientifiques de Descartes">René Descartes</a> explique une méthode permettant de tracer la normale, et donc par extension la tangente à la conchoïde de Nicomède.

</p><p>La voici exposée brièvement :

</p><p>On veut tracer la normale d'une conchoïde de Nicomède de pôle A et de module b en un point C. La droite directrice de cette conchoïde sera appelée (BH), où B est de telle sorte que (AB) et (CH) soient perpendiculaires à (BH).

</p>

<ul><li>Tracer le segment [CE] de manière qu'E soit l'intersection entre les droites (BH) et (CA).</li>

<li>Placer le point F tel que F appartienne à [CE] et CF = CH.</li>

<li>Placer le point G sur la droite perpendiculaire à (BH) et passant par F de façon que FG = EA.</li>

<li>La droite (CG) est alors la normale à la courbe en C.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Conchoïde_de_cercle"><span id="Concho.C3.AFde_de_cercle"></span>Conchoïde de cercle</h2></div>

<p>Les conchoïdes de cercle peuvent être utiles pour obtenir la trisection d'un angle. On pourrait également les utiliser pour étudier le mouvement d'une <a href="Bielle_(m%C3%A9canique)" title="Bielle (mécanique)">bielle</a> dans le cas où elle serait astreinte à coulisser en passant par un point fixe et où l'un de ses points parcourrait un cercle.

</p><p>Dans un repère dont l'origine O est le pôle de la conchoïde, l'équation polaire de la conchoïde d'un cercle de centre <span class="texhtml"><i>C</i>(<i>a</i>, 0)</span> et de rayon <span class="texhtml"><i>r</i> = <i>ka</i></span> (<span class="texhtml mvar" style="font-style:italic;">a</span> représente la distance <span class="texhtml mvar" style="font-style:italic;">OC</span>) et de module <span class="texhtml mvar" style="font-style:italic;">d = la</span> est : <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a200af5fe2359ddead886138a3a3436da63018a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.119ex; height:3.509ex;" alt="{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}" loading="lazy"></span> .

</p>

<p><span><strong>Démonstration</strong> — </span>

</p>

<p>On calcule tout d'abord l'équation du cercle de centre C. L'équation cartésienne d'un cercle étant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-c)^{2}+(y-d)^{2}=R^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x-c)^{2}+(y-d)^{2}=R^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9e15e77919a03134ddf62325303a04e06c89cf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.873ex; height:3.176ex;" alt="{\displaystyle (x-c)^{2}+(y-d)^{2}=R^{2}}" loading="lazy"></span>, on a, avec <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\rho \cos \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=\rho \cos \theta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/763223d9d625bb5257ceae8416058c42ddaba7e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.606ex; height:2.676ex;" alt="{\displaystyle x=\rho \cos \theta }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\rho \sin \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\rho \sin \theta }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/263c59eb5544a7595a66214ae976aa3011ba24d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.176ex; height:2.676ex;" alt="{\displaystyle y=\rho \sin \theta }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\cos(0)\times a=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=\cos(0)\times a=a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e4b78566746d48e7e78ace8fed15c826da4b232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.587ex; height:2.843ex;" alt="{\displaystyle c=\cos(0)\times a=a}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=\sin(0)\times a=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=\sin(0)\times a=0}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/efbc7824d4b012bde646300e8f26cb8591d64905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.473ex; height:2.843ex;" alt="{\displaystyle d=\sin(0)\times a=0}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=r=ka}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=r=ka}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/168b14a4d0498141d0681e356b293b40d6eca698.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.451ex; height:2.176ex;" alt="{\displaystyle R=r=ka}" loading="lazy"></span> :

</p>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y^{2}+x^{2}-2xa+a^{2}-r^{2}=0&\Longleftrightarrow &\rho ^{2}(\cos ^{2}\theta +\sin ^{2}\theta )-2a\rho \times \cos \theta +a^{2}-(ka)^{2}=0\\&\Longleftrightarrow &\rho ^{2}-2a\rho \times \cos \theta +a^{2}(1-k^{2})=0.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ae5037859c641d4c5a2e52c0375177d199d22bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:88.019ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}y^{2}+x^{2}-2xa+a^{2}-r^{2}=0&\Longleftrightarrow &\rho ^{2}(\cos ^{2}\theta +\sin ^{2}\theta )-2a\rho \times \cos \theta +a^{2}-(ka)^{2}=0\\&\Longleftrightarrow &\rho ^{2}-2a\rho \times \cos \theta +a^{2}(1-k^{2})=0.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>On résout cette <a href="%C3%89quation_du_second_degr%C3%A9" title="Équation du second degré">équation du second degré</a> :

</p>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {2a\cos \theta \pm {\sqrt {4a^{2}\cos ^{2}\theta -4a^{2}(1-k^{2})}}}{2}}=a\left(\cos \theta \pm {\sqrt {k^{2}-(1-\cos ^{2}\theta )}}\right)=a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01176214ca1bbf4a91b877a607cc04451f7702c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:101.698ex; height:6.176ex;" alt="{\displaystyle \rho ={\frac {2a\cos \theta \pm {\sqrt {4a^{2}\cos ^{2}\theta -4a^{2}(1-k^{2})}}}{2}}=a\left(\cos \theta \pm {\sqrt {k^{2}-(1-\cos ^{2}\theta )}}\right)=a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }})}" loading="lazy"></span></dd></dl>

<p>On peut en conclure que la conchoïde du cercle de centre <i>C</i>(<i>a</i>, 0) a pour équation polaire <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}">

<msqrt>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a200af5fe2359ddead886138a3a3436da63018a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.119ex; height:3.509ex;" alt="{\displaystyle \rho =a(\cos \theta \pm {\sqrt {k^{2}-\sin ^{2}\theta }}\pm l)}" loading="lazy"></span> puisque <i>al = d</i>.

</p>

</div><div class="clear" style="clear:both;"></div>

</div>

<p>On trouve quelques cas particuliers intéressants :

</p>

<ul><li>lorsque le pôle est confondu avec le centre du cercle, la conchoïde correspondante est constituée de deux cercles concentriques dont les rayons sont <span class="nowrap"><i>r</i> = R + <i>d</i></span> et <span class="nowrap"><i>r'</i> = R – <i>d</i></span>.</li>

<li>lorsque le pôle se situe sur le cercle, on obtient alors un <a href="Lima%C3%A7on_de_Pascal" title="Limaçon de Pascal">limaçon de Pascal</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Limaçon_de_Pascal"><span id="Lima.C3.A7on_de_Pascal"></span>Limaçon de Pascal</h3></div>

<p>Les limaçons de Pascal doivent leur nom à Étienne Pascal, père de <a href="Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a>.

</p><p>Un limaçon de Pascal correspond à la conchoïde d'un cercle lorsque le pôle de la conchoïde se situe sur le cercle. Ils ont pour équation en coordonnées polaires :

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =a+b\cos \theta ~}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14ba72a5ff50135e34dd1429227a608861bdde21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.925ex; height:2.676ex;" alt="{\displaystyle \rho =a+b\cos \theta ~}" loading="lazy"></span></dd></dl>

<p>On peut noter que lorsque <i>b</i> = 2<i>a</i>, on obtient le limaçon trisecteur qui possède, à l'instar de la conchoïde de Nicomède, la particularité de permettre d'exécuter la <a href="Trisection_de_l'angle" title="Trisection de l'angle">trisection de l'angle</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Texeira1980"><span class="ouvrage" id="Gomes_Texeira1980">Gomes Texeira, <cite class="italique">Traité des courbes spéciales remarquables plane et gauches</cite>, <abbr class="abbr" title="tome">t.</abbr> I, Gauthier-Villars, <time>1980</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://digitalis-dsp.uc.pt/jspui/bitstream/10316.2/3258/6/Obras%20sobre%20mathematica%20do%20Dr.%20F.%20Gomes%20Teixeira%20Vol.4%20%281908%29.pdf">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 267<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trait%C3%A9+des+courbes+sp%C3%A9ciales+remarquables+plane+et+gauches&rft.pub=Gauthier-Villars&rft.aulast=Texeira&rft.aufirst=Gomes&rft.date=1980&rft.pages=267&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AConcho%C3%AFde"></span></span></span></span>

</li>

</ol></div>

</div>

<div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2></div>

<ul><li><a href="Concho%C3%AFde_de_de_Sluze" title="Conchoïde de de Sluze">Conchoïde de de Sluze</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Liens_externes">Liens externes</h2></div>

<ul><li><span class="ouvrage" id="Weisstein"><span class="ouvrage" id="Eric_W._Weisstein"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, « <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Conchoid.html"><cite style="font-style:normal;" lang="en"><span class="lang-en" lang="en">Conchoid</span></cite></a> », sur <span class="italique"><a href="MathWorld" title="MathWorld">MathWorld</a></span></span></span></li></ul>

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