Radius of an osculating sphere

goteamusa · **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?

outermeasure · **Posted:** Thu, 13 Oct 2011 23:24:25 UTC

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 $\mathbb{E}^3$ , i.e. for any two points on the helix, there exists an isometry of $\mathbb{E}^3$ bring one to the other presering the helix.

goteamusa · **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.

outermeasure · **Posted:** Thu, 13 Oct 2011 23:43:19 UTC

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 $R=\sqrt{\rho^2+\left(\dfrac{\dot\rho}{\tau}\right)^2}$ ?

goteamusa · **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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Radius of an osculating sphere

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