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 Post subject: Radius of an osculating spherePosted: Thu, 13 Oct 2011 23:17:06 UTC
 S.O.S. Oldtimer

Joined: Fri, 30 Oct 2009 16:33:10 UTC
Posts: 261
Find the radius of osculating sphere of the helix: alpha(t)=(acos(t),asin(t),bt), a,b>0.

Well, the function r(s)=<m(s0)-alpha(s),m(s0)-alpha(s) is the radius of the sphere squared I believe. Then would it just be r^2=(acost,asint,bt), and then r^2=sqrt[(acost)^2+(asint)^2+(bt)^2]=sqrt[a^2(1)+b^2t^2) so r=(a^2+b^2t^2)^(1/4).

Is that correct?

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 Post subject: Re: Radius of an osculating spherePosted: Thu, 13 Oct 2011 23:24:25 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
goteamusa wrote:
Find the radius of osculating sphere of the helix: alpha(t)=(acos(t),asin(t),bt), a,b>0.

Well, the function r(s)=<m(s0)-alpha(s),m(s0)-alpha(s) is the radius of the sphere squared I believe. Then would it just be r^2=(acost,asint,bt), and then r^2=sqrt[(acost)^2+(asint)^2+(bt)^2]=sqrt[a^2(1)+b^2t^2) so r=(a^2+b^2t^2)^(1/4).

Is that correct?

No. For a start, it wouldn't depend on t, since translations on the helix are realised by isometries of , i.e. for any two points on the helix, there exists an isometry of bring one to the other presering the helix.

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 Post subject: Re: Radius of an osculating spherePosted: Thu, 13 Oct 2011 23:32:58 UTC
 S.O.S. Oldtimer

Joined: Fri, 30 Oct 2009 16:33:10 UTC
Posts: 261
Do you know what I should do to try to solve these differential geometry problems. We do not have a book for the class, only my teacher's notes, and they do not seem to be sufficient for me to figure out my homework.

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 Post subject: Re: Radius of an osculating spherePosted: Thu, 13 Oct 2011 23:43:19 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
goteamusa wrote:
Do you know what I should do to try to solve these differential geometry problems. We do not have a book for the class, only my teacher's notes, and they do not seem to be sufficient for me to figure out my homework.

You haven't seen the formula ?

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 Post subject: Re: Radius of an osculating spherePosted: Fri, 14 Oct 2011 00:04:21 UTC
 S.O.S. Oldtimer

Joined: Fri, 30 Oct 2009 16:33:10 UTC
Posts: 261
I do not believe I have. I just looked through all the notes, and the only radius formula I saw was the one I tried to use.

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