Clelies curve
in [Math]
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From: I.N. Galidakis on 15 Jul 2010 23:51 Does anyone know if the Clelies curve (http://en.wikipedia.org/wiki/Cl%C3%A9lies) (projected) on the xz and yz planes have any special names? i.e., the curves C1 and C2, with: C1 = {x=a*sin(m*theta)*cos(theta), z=a*cos(m*theta)} and C2 = {y=a*sin(m*theta)*sin(theta), z=a*cos(m*theta)} Information on these curves seems a little sparse on Google. Anyone know of any equivalent curves or refs for these? Many thanks, -- I.
From: David R Tribble on 16 Jul 2010 00:31 I.N. Galidakis wrote: > Does anyone know if the Clelies curve > (http://en.wikipedia.org/wiki/Cl%C3%A9lies) > (projected) on the xz and yz planes have any special names? > > i.e., the curves C1 and C2, with: > C1 = {x=a*sin(m*theta)*cos(theta), z=a*cos(m*theta)} > and > C2 = {y=a*sin(m*theta)*sin(theta), z=a*cos(m*theta)} Just a guess, but could these be related to hypocycloids and epicycloids? -drt
From: I.N. Galidakis on 16 Jul 2010 00:51 David R Tribble wrote: > I.N. Galidakis wrote: >> Does anyone know if the Clelies curve >> (http://en.wikipedia.org/wiki/Cl%C3%A9lies) >> (projected) on the xz and yz planes have any special names? >> >> i.e., the curves C1 and C2, with: >> C1 = {x=a*sin(m*theta)*cos(theta), z=a*cos(m*theta)} >> and >> C2 = {y=a*sin(m*theta)*sin(theta), z=a*cos(m*theta)} > > Just a guess, but could these be related to hypocycloids and > epicycloids? > > -drt I went through every single entry on Wiki: http://en.wikipedia.org/wiki/List_of_curves but I could not match C1 or C2 to anything listed. -- I.
From: David Bernier on 16 Jul 2010 02:29 I.N. Galidakis wrote: > Does anyone know if the Clelies curve > > (http://en.wikipedia.org/wiki/Cl%C3%A9lies) > > (projected) on the xz and yz planes have any special names? > > i.e., the curves C1 and C2, with: > > C1 = {x=a*sin(m*theta)*cos(theta), z=a*cos(m*theta)} > > and > > C2 = {y=a*sin(m*theta)*sin(theta), z=a*cos(m*theta)} > > Information on these curves seems a little sparse on Google. > > Anyone know of any equivalent curves or refs for these? The most helpful definition I found is from a web page in French: "Les cl�lies sont les lieux d�un point M d�un m�ridien d�une sph�re tournant � vitesse constante w autour de l�axe polaire, le point M se d�pla�ant � la vitesse constante nw sur ce m�ridien. " Or: " A clelie is the curve traced on a rotating globe by a point moving on a meridian fixed in space (which passes through the poles of a rotating sphere) , where the sphere or globe moves along its axis of rotation at a fixed rate, and the point moving on the non-rotating meridian moves at a constant rate (degrees of latitude per unit time) on the meridian; once passed a pole, the point on the meridian moves at 1 degree per minute away from North Pole if it was previously moving at one degree per minute towards the North Pole. " The best view is probably the 3-D animated graphic here, with the clelie in red. One could also say that the globe is fixed in space and the meridian circle is moving relative to the fixed globe. The clelie is then the winding curve on the fixed globe. Cf.: < http://www.mathcurve.com/courbes3d/clelie/clelie.shtml > . [ by Robert Ferreol and Jacques Mandonnet]. David Bernier
From: I.N. Galidakis on 16 Jul 2010 02:54 David Bernier wrote: > I.N. Galidakis wrote: >> Does anyone know if the Clelies curve >> >> (http://en.wikipedia.org/wiki/Cl%C3%A9lies) >> >> (projected) on the xz and yz planes have any special names? >> >> i.e., the curves C1 and C2, with: >> >> C1 = {x=a*sin(m*theta)*cos(theta), z=a*cos(m*theta)} >> >> and >> >> C2 = {y=a*sin(m*theta)*sin(theta), z=a*cos(m*theta)} >> >> Information on these curves seems a little sparse on Google. >> >> Anyone know of any equivalent curves or refs for these? > > The most helpful definition I found is from a web page in French: > > "Les cl�lies sont les lieux d�un point M d�un m�ridien d�une sph�re tournant � > vitesse constante w autour de l�axe polaire, le point M se d�pla�ant � la > vitesse constante nw sur ce m�ridien. " > > Or: > " A clelie is the curve traced on a rotating globe by a point moving on a > meridian fixed in > space (which passes through the poles of a rotating sphere) , > where the sphere or globe moves along its axis of rotation at a fixed > rate, and the point moving on the non-rotating meridian moves > at a constant rate (degrees of latitude per unit time) on the > meridian; once passed a pole, the point on the meridian moves > at 1 degree per minute away from North Pole if it was previously > moving at one degree per minute towards the North Pole. " > > The best view is probably the 3-D animated graphic here, with > the clelie in red. > > One could also say that the globe is fixed in space and the meridian > circle is moving relative to the fixed globe. The clelie > is then the winding curve on the fixed globe. > > Cf.: > < http://www.mathcurve.com/courbes3d/clelie/clelie.shtml > . > [ by Robert Ferreol and Jacques Mandonnet]. Thanks David. It appears that C1 and C2 do not have any special names. > David Bernier -- I.
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